Shared Formulas (Math)

We will calculate the power.

We calculate the ratio based on the base quantity and the comparison quantity.

Calculating Pi (Circumference and Area)

The Collatz conjecture states that for any positive integer n: • If n is even, then n/2; • If n is odd, then 3n+1; When these steps are repeated, the sequence will always reach 1 after a finite number of iterations.

Calculate the power residue

We will use the Pythagorean theorem to find the length of the diagonal of a square.

Find the diameter of a circle from its circumference.

Take any positive integer n. If n is even, divide n by 2. If n is odd, multiply n by 3 and add 1. This illustrates the Collatz conjecture, which states that repeating this process will eventually result in 1. For example, if n = 27, it takes 111 iterations to reach 1. It is said that a reward of 100 million yen is offered for a proof.

The Four Basic Arithmetic Operations

It can generate large prime numbers up to 50 digits long. However, it's slow. You'll have to wait quite a while for a 50-digit number.

Think of any number you like (we recommend a 3- or 4-digit number). Rearrange the digits of that number to create the largest possible number and the smallest possible number. Calculate the difference between them. If you repeat this process, the result may eventually settle on a specific number. (You can use any number up to 10 digits. What happens with a 5-digit number?)

Calculate the check digit from the 11-digit number.

Converts a decimal number to a base-n number (where n is any integer ≤ 34).

The value of S when T lies between the values of T2 and T1

Calculate the x and y coordinates of the vertex of the graph of the quadratic function y = ax² + bx + c (where a is not zero).

I made this because there are opportunities to double numbers as part of a game.

Calculates the reciprocal of the specified number.

Calculate the continued fraction (up to the nth term)

Evaluates the complexity of the 4-digit number entered.

We will perform calculations related to quadratic functions. We will transform the equation y = ax² + bx + c into y = a(x - p)² + q.

Find the circumference of the circle based on its diameter.

This program uses Euclid's algorithm to find the least common multiple of a and b, and displays the intermediate steps of the calculation.

If the integers a and p are coprime and the congruence x² ≡ a (mod p) has a solution, then a is called a quadratic residue modulo p. We will perform this calculation. We will also calculate the quadratic residue symbol (a/p).

"For 'Cubic Functions Passing Through Four Integer Points'"

Two natural numbers are called amicable numbers if the sum of their divisors (excluding themselves) is the same (as featured in *The Equation of Love*). Enter two numbers to see if the sum of their divisors is the same for both.

You can display prime numbers up to 1 million using Eratosthenes' sieve.

Calculate the nth power of a 2x2 square matrix. Although complex numbers can be entered according to the specifications, the correct result will not be obtained.

This tool generates algebraic expression factoring problems. Although it generates only simple expressions with one variable, it can also create problems involving cubic and quartic expressions.

This program factors the entered natural number. It uses the trial division method and the rho method for factorization.

Here is the calculation process for the Collatz conjecture.

Calculate the factorial of 3.

Let’s have the Casio Calculator Site compute the famous formula 6/2(1+2) Last updated February 22, 2022

Generates a table of variations for a function. Although it can only generate simple tables of variations, it should cover the scope of Math II. Please enter the coefficients in the boxes. If there is no term, enter 0. The layout may appear distorted on smartphones. Support for the √ symbol is limited.

There are n beautiful women lined up on the beach. Although they are ranked, you don’t know their rankings, so you meet them one by one. Which one should you ask out? The display shows n-1 people, but the ranking of the last one is obvious, right?

We will calculate the weighted average. Here, we assume there are 10 data points. The weighted average calculated here is the weighted arithmetic mean. Weights must be positive numbers greater than or equal to 0. (No output will be generated for invalid inputs.) If there are fewer than 10 data points, enter the number 0 in the x and w input fields (do not leave any fields blank).

(x^2 + 9xxy - y^2)^3 + (12x^2 - 4xxy + 2y^2)^3 = (9x^2 - 7xxy - y^2)^3 + (10x^2 + 2y^2)^3 This identity, discovered by Ramanujan, allows us to reproduce the moment when, while hospitalized, Ramanujan heard the taxi number 1729 and immediately replied, "1729 = 1³ + 12³ = 9³ + 10³."

Find the arithmetic mean, geometric mean, harmonic mean, logarithmic mean, mean of the sums of the squares, and mean of the sums of the cubes of two numbers, a and b.

Write down six numbers of your choice. Then, write the first digit of the sum of each pair of adjacent numbers one row below. Repeat this process, and the final number that appears is the one you’ll guess the moment you finish writing the six numbers. (From Jun Mitani’s tweet)

In the past, McDonald’s in the UK offered Chicken McNuggets in sets of 6, 9, or 20 pieces. When purchasing multiple sets, the maximum number of non-Chicken McNuggets that cannot be obtained as the total number of nuggets is the “non-Chicken McNugget count.” The Frobenius coin exchange problem is a version of this concept applied to coins of different denominations. We will perform the calculation using three coins.

Calculate the maximum and minimum values of a quadratic function. Assume the domain is a closed interval. Calculate the maximum and minimum values in the corresponding range.

Calculate the slant height of a rectangular prism

We will use the Pythagorean theorem to find the hypotenuse of the triangle.

Handling Division by Zero

If you enter a large number for x, you can calculate an approximate value of pi.

We will examine the number of divisors of a natural number and the sum of those divisors. Here, we will take a straightforward approach and perform a brute-force division to calculate the results. (This also serves as a verification of other algorithms.)

We will calculate the Fibonacci sequence, the Pell sequence, the Padovan sequence, and the Perrin sequence, as well as the ratios of their adjacent terms. As n approaches infinity, the ratios become the golden ratio, the silver ratio, and the other two become plastic numbers.

We will examine the divisors of a natural number n.<\n> List all natural numbers from 1 to n, inclusive, that are divisors of n.<\n> <\n> Output the divisors.

Determines whether a number is prime.

We will calculate the divisor function σx(n), which returns the sum of the xth powers of the divisors d of the natural number n. When σ1(n) = 2n, n is a perfect number.

Determine a circle based on three points. Calculate the radius and the coordinates of the center. For the general form of the equation of a circle, let the equation be x² + y² + lx + my + n = 0.

I don’t think it’ll be very useful, but I’d be happy if you could use it as much as possible.

a^b

Time Zone Calculator (Unnecessarily precise to the second)

Based on the custom formula for calculating the sum from 1² and 1³ to x² and x³, I’ve modified it so that you can change the values of x and n to perform the calculation.

Prime Factorization (Displayed with Prime Factors and Exponents) Last Updated: February 22, 2022

Calculate the Fibonacci numbers.

Calculate the trigonometric functions.

Calculate the rate of change of the proportional constant.

Prime Factorization (Displaying Prime Factors and Exponents Side by Side) (Performance-Enhanced Version) Last Updated: February 22, 2022

Let's do some Egyptian arithmetic.

You can generate simple tables of function values. The table generator is also available under the account "abcdefg85," but it was created by the same person. I've shortened the code and improved readability.

Think of any positive integer you like; if you add up the squares of its digits, the sum will either equal 1 or fall into the cycle of 4, 16, 37, 58, 89, 145, 42, and 20.

"On Cubic Functions Passing Through Four Integer Points" (Including Fractional Representations)

The sum of a through n is…

This is a calculation where you find two numbers using only their sum and difference.

Please express the system of two linear equations after setting the coefficient of y to be the same. Solve for x and y.

Please enter any natural number n.

We will calculate the expansion of the expression. Here, we will consider the case where the expression is a fifth-degree polynomial in x. The form of the expression is: (x+a)(x+b)(x+c)(x+d)(x+e) = k5(x^5) + k4(x^4) + k3(x^3) + k2(x^2) + k1(x^1) + k0 k5, k4, k3, k2, k1, and k0 are the coefficients of the terms in the expanded expression. We will expand the left-hand side into the right-hand side.

Eratosthenes' Sieve Last updated February 22, 2022

Displays the value of pi.

Substituting an even number works, but the answer is not an integer. Possible application to complex numbers (?) (The author did not create this with that in mind.)

This involves some programming concepts. We perform division using a number with twice as many digits as the number being divided to obtain the quotient and remainder. Of course, we’re dealing with integers.

Calculate the factorial of 3.

Calculate the Fibonacci numbers.

Perform polynomial calculations. Assume that each term is of the form x^n multiplied by the coefficient a_n. Here, the highest degree is 10. The n to the right of the coefficient a_n is a subscript ranging from 0 to 10, matching the exponent n of x^n. a₀ is the constant term. Let y = a10 * (x^10) + a9 * (x^9) … + a1 * x + a0, and calculate the value of y and output it. (* denotes multiplication)

Problem 4 in the 2022 National Center Test for University Admissions (Math I and Math A) involved Diophantine equations. The problem asked for the solutions to the equations 5^4*x - 2^4*y = 1, 5^5*x - 2^5*y = 1, and 11^5*x - 2^5*y = 1. In general, the solutions to the equation a*x + b*y = gcd(a, b) are known as Bézout’s identities. We are to find the values of x and y.

Ayaka Riri (a science news commentary account) tweeted about Valeriy G. Narushin et al.'s paper "Egg and Math: Introducing a Universal Formula for Egg Shape" (Ann. N. Y. Acad. Sci., 2021), so I decided to try it out on Casio's high-precision calculation website as well.

Calculate the dynamic range (theoretical value) based on the number of bits.

This formula is a handy shortcut that lets you skip the squaring step when subtracting b² from a². Try working it out in your head first, then use this formula.

We will use the Pythagorean theorem to find the height of an equilateral triangle.

We will calculate the division of polynomials using the synthetic division method. Here, we consider dividing polynomials in x of degree up to 10 by the linear expression (x-α). We treat terms that do not exist as having a coefficient of 0.

A formula for calculating the radius based on the two tangents to two circles and the angle between them

Sum of Permutations

★Construction for Problems Involving Ellipses

Calculate the maximum and minimum values of a cubic function. For a cubic function with a closed domain, calculate the maximum and minimum values of the function within its range.

Calculate the number of times the monkey must hit the typewriter before it types the target string.

Find the value of x for which the limit of the cubic function f(x) = ax³ + bx² + cx + d holds. d cancels out during differentiation.

Formulas for error rates used in voltage and current calibration

A formula that calculates the roots of a quartic equation just by pasting it into an Excel cell

Let's solve the quadratic equation. If x and y are equal, it is a repeated root.

We will determine the number of natural numbers that are "mutually prime." For a given natural number, we will determine the number of consecutive natural numbers that are mutually prime.

Rationalize the denominator of the cube root.

We will expand the expression. We will consider the case where the expression is a sixth-degree polynomial in x. The form of the expression is p(x+k1)(x+k2)(x+k3)(x+k4)(x+k5)(x+k6) = p{a6(x^6) + a5(x^5) + a4(x^4) + a3(x^3) + a2(x^2) + a1(x^1) + a0(x^0)}.We expand the left-hand side into the right-hand side. The subscripts (numbers) to the right of k, a, and A correspond to the powers of x. A represents the numerical value of the coefficient after p-distribution.

Eratosthenes' Sieve Last updated February 22, 2022

This program performs arithmetic operations on fractions. It performs addition, subtraction, multiplication, and division on two fractions, a and b. It also displays the result of each operation in decimal form for reference. This program does not handle mixed fractions. The output fractions are "fractions with integer denominators."

I went ahead and calculated the speed in Å/s, even though it was unnecessary.

As a method for quickly calculating products, a technique called prosthaphaeresis—which involves converting the problem into a sum using the formula cos(a)cos(b) = (cos(a+b) + cos(a-b))/2—was reportedly used before the 16th century. Later, the process became even easier by utilizing the fact that log(ab) = log(a) + log(b) from logarithm tables. Let’s take a look at that.

Find x to the power of y

We will perform calculations involving fractions. Here, we will consider fractions raised to the power of n. We assume that the numerator of the fraction is an integer. We assume that the denominator of the fraction is a non-zero integer. We assume that "n" in the power of n is an integer greater than or equal to 1. We will calculate the fraction raised to the power of n.

Graphs of (sin x)÷x and (tan x)÷x Last updated February 22, 2022

Displays the greatest common divisor and least common multiple of 11 integers　　　　　 Last updated February 27, 2022

It tends to be a prime number relatively easily. There is a limit to 50 digits.

Fractions → Decimals

For testing purposes

We will find a common denominator for the fractions.

Enter the number of questions and the number of correct answers to calculate your score.

Calculation for the forum question "√(2012 - n²...)"

This is a bit of a programming-related topic. When dealing with integers, it’s a problem that numbers get rounded to a certain number of significant digits, isn’t it? Ideally, you need twice as many significant digits to calculate the exact result of a multiplication.

We will calculate the partial fraction expansion. Here, we will consider the case where a fraction with a numerator that is a linear expression and a denominator that is the product of two linear expressions is expanded into two terms: one with a constant numerator and the other with a linear denominator. Let the form of the expression be: (Lx+M)/(x+P)(x+Q) = a/(x+P) + b/(x+Q). We will perform partial fraction decomposition of the left-hand side into the right-hand side.

Let's expand the expression.

Calculate the values of the trigonometric functions. In older trigonometric functions, terms like "positive slope" are also used to refer to "slope height."

Find the remainder of x divided by y.

★ Problems involving ellipses (Arc length C and angle n° between two points on an ellipse)

The first item is numbered 1.

Check the value inside the radical. Here, the radical is used to calculate the nth root. Enter the value x you want to check. Enter the value of n for the nth root. Output the value k that can be taken out of the radical. Output the value y that remains inside the radical. As a verification step, raise k to the power of n, place it inside the radical, and calculate its product with y. Confirm whether x has been successfully reconstructed.

Solve the system of equations ax + by = m and cx + dy = n.

Find the area of the circle

We will perform an Egyptian fraction calculation. This involves converting a fraction into a sum of unit fractions. A unit fraction is a fraction whose numerator equals 1. Both the numerator and denominator of the fraction are natural numbers. Here, we will consider the case of decomposing a fraction into two unit fractions. The algorithm we use is a loop-based inverse simplification method. This type of algorithm increases the flexibility of decomposition by repeatedly multiplying the numerator and denominator of the fraction being examined by the same natural number. Although this is the reverse of prime factorization—where we construct composite numbers to search for solutions—using natural numbers 1, 2, 3, … for inverse simplification is a good method for mimicking manual calculations. The results may differ from those of a sequential subtraction algorithm. This is because the decomposition of unit fractions does not always yield a unique result. Since the search range generally has no upper limit, and the numbers involved may become too large or cause a timeout, we set an upper limit on the number of loop iterations.

Find the Nth term of an arithmetic sequence

Create a geometric sequence modulo N.

Generally, one of them will be a prime number. Depending on the value of a, the number of answers increases exponentially.

Calculate the area of quadrilaterals (parallelograms, squares, rectangles, and rhombuses).

We will perform an Egyptian fraction calculation. This involves converting a fraction into a sum of unit fractions. A unit fraction is a fraction whose numerator is 1. Both the numerator and denominator of the fraction are natural numbers. Here, we will consider the case of decomposing a fraction into three unit fractions. The algorithm we will use is a loop-based inverse simplification method. This type of algorithm increases the flexibility of decomposition by repeatedly multiplying the numerator and denominator of the fraction being examined by the same natural number. Although this is the reverse of prime factorization—where we construct composite numbers to search for solutions—using natural numbers 1, 2, 3, … for inverse simplification is a good method for mimicking manual calculations. The results may differ from those of a sequential subtraction algorithm. This is because the decomposition of unit fractions does not always yield a unique result. (Note that the search range generally has no upper limit, so if the numbers become too large, a timeout is also expected.)

Ramanujan's Identities (with source code) Last updated February 28, 2022

Find the y-intercept of the line. Assume that the slope of the line is known. Calculate the y-intercept when this line passes through a point on the xy-plane.

This function performs calculations involving fractions. It converts the entered number into a fraction expressed as a ratio of integers.

This program searches for specific solutions to an indeterminate equation. Let the equation be of the form ax + by = c. It handles integers. It sequentially substitutes integers for x, and if y is an integer that satisfies the equation, it outputs the result. If no solution is found or if the input is outside the specified range, nothing is displayed.

Archimedes' Method of Calculation

We will perform calculations using the Factor Theorem. We will consider polynomials up to the 10th degree. We assume that all coefficients are integers. Let the form of the expression be f(x) = a₁x¹⁰ + a₉x⁹ + … + a₂x² + a₁x + a₀.　

The Tribonacci sequence is 0, 0, 1, 1, 2, 4, 7, 13, 24..., but please treat the first 0 as the 0th term.

For a 57-sided polygon, it is approximately 3.14.

Let's do the Crane-Turtle Problem.

We will calculate the sequence a_n = (n^(n^n) / number of digits), which is inspired by James Joyce's *Ulysses*.

We will calculate the expansion of the expression. In this case, we are dealing with a fourth-degree polynomial. Let the form of the expression before expansion be k{(x+a)(x+b)(x+c)(x+d)}. Let the form of the expanded expression without distributing k be: k{p4(x^4)+p3(x^3)+p2(x^2)+p1(x^1)+p0}. Let the form of the expanded expression with k distributed be: t4(x^4)+t3(x^3)+t2(x^2)+t1(x^1)+t0.

Perform cross-factoring. The domain consists of integers. The expression is of the form Px^2 + Qx + R = (ax + b)(cx + d). Factor the left-hand side into the right-hand side. If a suitable set of values is found during the search, output it.

The discriminant D of the quadratic formula. We will examine which numbers can be factored out of the radical. We assume D is a natural number, and the numbers we consider are also natural numbers.

We will use regular polygons to calculate the area of a circle.

We will perform calculations involving fractions. Here, we will consider outputting the results of complex number division as fractions. When calculating the output fraction, ensure that the real and imaginary parts of the numerator are integers. When calculating the output fraction, ensure that the denominator is a natural number. For reference, also output the value obtained by dividing the numerator of the output fraction by the denominator of the output fraction. Enter a non-zero value for the denominator of the input fraction. (Ensure that either the real part or the imaginary part of the denominator of the input fraction is not zero.)

Find the area of the sector.

Number of diagonals in a regular polygon

We will calculate the weighted average. Here, we assume the dataset consists of 30 pairs of data. The weighted average calculated here is the weighted arithmetic mean. Weights must be positive numbers greater than or equal to 0. (No output will be generated for invalid input.) If the dataset contains fewer than 30 entries, please leave the default value of 0 in the x and w input fields (do not leave any fields blank). The sum of the entered weights does not need to equal 1.

Determine whether a permutation is even or odd. Here, we consider permutations of the sequence 12345. This determines whether a 5-character permutation σ is even or odd. If even, sgn(σ) = 1; If odd, sgn(σ) = -1; and outputs 1 or -1 as the result. A permutation that can be expressed as the product of an even number of transpositions is an even permutation, while a permutation that can be expressed as the product of an odd number of transpositions is an odd permutation. If the permutation is even, the parity check returns "even" and outputs 1. If it is an odd permutation, the parity check returns "odd" and outputs -1. The parity check of a permutation is sometimes used in the definition of determinants. Enter the result of permuting 12345 (the sequence for which you want to check the parity) in the input field. In permutation notation, this is often written in the bottom row. For example, if you want to check the parity of 12354—which is the result of permuting 12345— enter 1=1, 2=2, 3=3, 4=5, and 5=4 from left to right. The program will output the parity result.

We will perform calculations related to cubic curves.

Calculate the Napierian constant using continued fractions.

Calculates the coefficients of the tangent line.<\n Let f(x) be a polynomial of degree 2 or higher, up to degree 10.<\n Given a specific x-coordinate, the program outputs the coefficients of the equation of the tangent line at the point (x, f(x)) on the curve.<\n

This program calculates the addition and subtraction of multiple fractions. Here, we'll limit it to 10 fractions. For subtraction, enter a negative number in the numerator. In this case, enter a positive number (not zero) in the denominator.

You can express any number as a power.

We will calculate the expansion of the expression. Here, we will expand the product of two polynomials. Let the two polynomials be f(x) and g(x). There is only one variable, x, and both polynomials are of degree 5 or less.

Ramanujan's Formula Last updated May 1, 2025

We will perform an Egyptian fraction calculation. This involves converting a fraction into a sum of unit fractions. A unit fraction is a fraction whose numerator is 1. Both the numerator and denominator of the fraction are natural numbers. Here, we will consider the case of decomposing a fraction into four unit fractions. The algorithm we will use is a loop-based inverse simplification method. This type of algorithm increases the flexibility of decomposition by repeatedly multiplying the numerator and denominator of the fraction being examined by the same natural number. While this is the reverse of prime factorization—where we construct composite numbers to explore possibilities—using natural numbers 1, 2, 3, … for inverse simplification is a good method for mimicking manual calculations.

Determine whether a permutation is even or odd. Here, we will consider permutations of the sequence 1234. This determines whether a 4-character permutation σ is even or odd. If even, sgn(σ) = 1; if odd, sgn(σ) = -1; and outputs 1 or -1 as the result. A permutation that can be expressed as the product of an even number of transpositions is an even permutation, while a permutation that can be expressed as the product of an odd number of transpositions is an odd permutation. If it is an even permutation, the parity is determined to be even, and 1 is output. If it is an odd permutation, the parity is determined to be odd, and -1 is output. Determining the parity of a permutation is sometimes used in the definition of determinants.

We will expand the expression. The result will be a ninth-degree polynomial in x.

Calculate the number of multiples of a and b contained in the set of natural numbers from 1 to n. This is based on Gauss's symbol. n, a, and b are natural numbers. c is a natural number such that c = a * b. (* denotes multiplication)

Divide a quartic polynomial by a quadratic polynomial. Divide a quartic polynomial by a quadratic polynomial. Let the form of the quartic polynomial being divided be ax^4 + bx^3 + cx^2 + dx + e. Let the dividing quadratic polynomial be lx^2 + mx + n. Let the form of the quotient be Q2x^2 + Q1x + Q0. Let the form of the remainder be R1x + R0. The numbers to the right of Q and R are subscripts, and they correspond to the degree of x.

Determine a curve of degree three or less from four points. Find the coefficients of a cubic equation from four distinct x-y coordinates. However, since we allow for the possibility that a is zero, we will also output equations of degree three or less. Let the equation be of the form y = ax³ + bx² + cx + d. (We do not exclude the case where a = 0) For reference, calculations are performed in the complex number domain.

Be careful—if there aren't enough of them, they'll scatter.

This formula solves the equation "z^N = a + b*i".

Let’s factor this expression. Here, we will only consider the set of integers. The form of the expression is: x^2 + (a+b)x + a*b = (x+a)(x+b) (* denotes multiplication, and + denotes addition. x^2 means “x squared.”) We will find the sum (a+b) and the product (a*b).

Let's calculate pi!

This involves some programming concepts. When the result of an addition exceeds the number of significant digits, it is converted to exponential notation, and the least significant digit is rounded off. However, by performing a special operation during the addition, we can avoid rounding and achieve a carry.

For a given natural number N, list the natural numbers (in the range from 2 to N-1) that are relatively prime.

Lower bound of the 95% confidence interval for a binomial distribution approximated by a normal distribution

Determine whether a permutation is even or odd. Here, we will consider the permutation of the sequence 123. This determines whether the three-letter permutation σ is even or odd. If it is even, sgn(σ) = 1; if it is odd, sgn(σ) = -1; and outputs 1 or -1 as the result.

We will perform partial fraction decomposition. Assume that both the numerator and denominator are numbers, not algebraic expressions. This is not the partial fraction expansion performed in preparation for integration, but rather the decomposition of a fraction with specific numerical values for the numerator and denominator into two fractions. The two resulting fractions are unit fractions. The numerator of the original fraction to be decomposed is the difference between the denominators of the two resulting unit fractions. The denominator of the original fraction to be decomposed is the product of the denominators of the two unit fractions resulting from the decomposition. The form of the expression is n/m = (1/a) - (1/b). We expand the left side into the right side. n = b - a m = a * b We calculate a and b within the range of integers. (b - a is subtraction, a * b is multiplication)

Calculating infinitely nested square roots

We will calculate pi using a regular polygon.

It is not necessarily a prime number.

It converges really fast.

We will perform partial fraction decomposition. Here, we will decompose a fraction with a quadratic numerator and a cubic denominator into a sum of three terms, where the numerator is a constant and the denominator is a linear expression. Let the form be: (Lx^3+Mx^2+N)/(x+P)(x+Q)(x+R) = (a/(x+P)) + (b/(x+Q)) + (c/(x+R)) We will perform the calculation to factor the left-hand side into the right-hand side.

Calculate the value of the discriminant of a cubic equation. Let the cubic equation be of the form ax³ + bx² + cx + d = 0. We consider the case where all coefficients are real numbers. We will calculate the value of the discriminant in this case. When the value of the discriminant is 0, the equation has repeated roots; when it is positive, it has distinct real roots; and when it is negative, it has one real root and two complex roots that are conjugate to it.

Please enter any natural number for n.

Given a range of consecutive natural numbers, determine whether each number is divisible by two distinct natural numbers, and summarize the results.

Same as above; upper limit.

We will perform factorization calculations. Let the expression be of the form (x^2 - m) = (x + a√b)(x - a√b), where m ≠ 0. We will search for a and b within the range of integers and display the results.

We will perform a unit conversion calculation.<\n Each digit of a natural number n is a number between 0 and 9. Calculate the sum of these digits; if the result is 10 or greater,<\n treat that number as the new n and add its digits together.<\n Repeat this process until the result is between 1 and 9, inclusive.<\n The unit conversion for 0 is 0.

Determine whether a permutation is even or odd.<\n Here, we will consider the permutation of the sequence 123456.<\n We will determine whether the 6-character permutation σ is even or odd. If it is even, sgn(σ) = 1 (one); if it is odd, sgn(σ) = -1 (minus one); and, the result is output as either 1 or -1. A permutation that can be expressed as the product of an even number of transpositions is an even permutation, while a permutation that can be expressed as the product of an odd number of transpositions is an odd permutation. If it is an even permutation, the parity is determined to be even, and 1 is outputted. If it is an odd permutation, the parity is determined to be odd, and -1 is outputted. Determining whether a permutation is even or odd is sometimes used in the definition of a determinant. In the input field, enter the sequence of numbers (a string of numbers resulting from permuting 123456) for which you want to check the parity. In permutation notation, this is often written in the bottom row. For example, if you want to check the parity of 123564—which is the result of permuting 123456— set Input 1 to 1, Input 2 to 2, Input 3 to 3, Input 4 to 5, Input 5 to 6, and Input 6 to 4. The program outputs the result of the parity check. For Inputs 1, 2, 3, 4, 5, and 6, enter any of the digits 1, 2, 3, 4, 5, or 6. For Inputs 1, 2, 3, 4, 5, and 6, enter distinct digits with no duplicates.

Test

Calculate the number of prime numbers less than or equal to x.

We will calculate the number of positive divisors and their sum. For a composite number whose prime factors are three primes P, Q, and R, we will consider the number of positive divisors and their sum. P, Q, and R are all limited to primes less than 1000.

A segment of the circumference of a unit circle

We will calculate the force applied to the fulcrum to balance the lever at the point of action. Please use this information for your lever experiments.

We will show how large the nth prime number becomes when n ≥ 6.

Compute d(N) = max{τ(x) | 1 ≤ x ≤ N} N ≤ 10³⁶

We will perform partial fraction decomposition. Here, we will convert a rational expression where the numerator is a cubic polynomial and the denominator is a quartic polynomial into a sum of four terms, where the numerator is a constant and the denominator is a linear polynomial. Let the expression be of the form: (Kx^3+Lx^2+Mx^1+N)/(x+P)(x+Q)(x+R)(x+S) = a/(x+P) + b/(x+Q) + c/(x+R) + d/(x+S). We will perform partial fraction decomposition of the left-hand side into the right-hand side.

When you're doing calculations in Excel or something, that "e" keeps popping up—it's so annoying, isn't it!? Well, this will take care of it in a snap!!

Find the number of digits in the natural number n (in the decimal system). Count each digit as one and calculate the sum.

Calculate the volume of the sphere accurately.

We will perform calculations related to the sum of unit fractions. In electromagnetism, the formulas for the equivalent capacitance in a series circuit and the equivalent resistance in a parallel circuit take the form: 1/R = (1/R1) + (1/R2) + … + (1/Rn), 1/C = (1/C1) + (1/C2) + … + (1/Cn) where both the left and right sides are expressed as equations involving unit fractions. (A unit fraction is a fraction with a numerator of 1.) Here, we will consider up to 20 terms on the right side. We will find the denominator on the left side.

We will transform a rational function. Let the original rational function be of the form y = (ax + b) / (cx + d). Let the transformed form be y = {k / (x - p)} + q.

One of the equations is a system of two linear equations containing quadratic terms: ax² + by² + cx + dy = m, fx + gy = n. Can you solve for y in just one of the equations?

We will calculate pi using the gamma function. Although the result approaches pi as n increases, convergence is slow, and errors occur when n is large, so this method is not practical.

Find the area within a radius of 1 using the trapezoidal rule.

We will expand the expression. We will calculate the expansion into a cubic expression. Let the form of the expression before expansion be k(x+a)(x+b)(x+c). Let the form of the cubic expression in x after expansion be A(x^3) + B(x^2) + C(x^1) + D. Please enter k, a, b, and c. Calculate and output the expanded values of A, B, C, and D.

We will perform calculations related to the rules for square roots. Here, we check the number inside the radical and determine whether there is a number that can be factored out of the radical. Let the number inside the radical be n. We consider n to be a non-negative real number. Let the form of the expression before checking be √n. Let the form of the expression after checking be k√m. The checking process involves treating n as m*(k^2) and bringing k out of the radical. As a verification, we recalculate m*(k^2) and display it alongside n. Finally, we display the values of √n and √m*(k^2) side by side.

Determine whether a permutation is even or odd. Here, we will consider permutations of the sequence 123456789. This determines whether a 9-character permutation σ is even or odd. If even, sgn(σ) = 1 (one); If odd, sgn(σ) = -1 (minus one); and output 1 or -1 as the result. A permutation that can be expressed as the product of an even number of transpositions is an even permutation, while a permutation that can be expressed as the product of an odd number of transpositions is an odd permutation. If the permutation is even, the parity check returns "even" and outputs 1. If it is an odd permutation, the parity check returns "odd" and outputs -1. The parity check of a permutation is sometimes used in the definition of determinants. Enter the sequence of numbers whose parity you want to check (the string of numbers resulting from permuting 123456789) into the input field. In permutation notation, this is often written in the bottom row. For example, if you want to check the parity of 876512349—which is the result of permuting 123456789— enter the following values: Input 1=8, Input 2=7, Input 3=6, Input 4=5, Input 5=1, Input 6=2, Input 7=3, Input 8=4, Input 9=9.

Let’s expand the expression. Here, we will perform the inverse of cross-factoring. Let the expression be of the form (ax+b)(cx+d) = L(x^2) + Mx + N. We will expand the left-hand side into the right-hand side.

Calculate the expansion of the expression. This applies to quadratic expressions. Let the form of the expression before expansion be (ax+b)(cx+d). Let the form of the expression after expansion be P*(x^2)+Q*(x^1)+R. (* denotes multiplication) Enter a, b, c, and d as fractions. Please enter integers in the numerator and denominator fields. Enter non-zero values for the numerators of a and c. Enter non-zero values for the denominators of a, b, c, and d. Neither input nor output supports mixed fractions. Complex numbers (non-real numbers with a non-zero imaginary part) are not supported.

We will solve a system of linear equations. Here, as an example of a two-variable system of equations involving x and y, we will consider the product and sum cases. Let the equations be of the form: u = kxy + c m = ax + by We assume that the coefficients k, a, and b are not zero.

We will calculate the partial fraction expansion. Here, we will consider the case of expanding a fraction where the numerator is a constant and the denominator is a linear expression into two terms. Let the form of the expression be (Lx + M)/(x + P)(x + Q) = a/(x + P) + b/(x + Q). We will perform a partial fraction expansion of the left-hand side into the right-hand side.

We will examine the discriminant D of a quadratic equation. Consider the quadratic equation ax² + bx + c = 0. a and b are coefficients. c is the constant term. We assume that a is not zero. Here, we will consider cases where the value of the discriminant D is positive and represents a fraction with integer ratios. We will check whether there are numerators and denominators that can be factored out of the radical √D in the quadratic formula. This refers to cases where both the numerator and denominator are natural numbers. (The numerator may be 0.) Let the form before checking be √D. Let the form after checking be k√E. k is the fraction that was factored out of the radical. E is the fraction that remains inside the radical.

Let’s expand the expression. It is the product of polynomials in x. Here, we will consider cases where the product of two quadratic polynomials results in a quartic polynomial. Let the form of the expression be: (ax² + bx + c)(px² + qx + r) = k4x⁴ + k3x³ + k2x² + k1x + k0 We will expand the expression on the left side into the expression on the right side. The numbers to the right of the coefficient k are subscripts, corresponding to the degree of x.

Calculate the expansion of the expression. After expansion, it is a seventh-degree polynomial. The form of the expression is: On the left-hand side: k(x+a1)(x+a2)(x+a3)(x+a4)(x+a5)(x+a6)(x+a7) On the right-hand side, the expression is in unsimplified form: k(p7x^7 + p6x^6 + ... + p1x^1 + p0x^0) After factoring out k: kp7x^7 + kp6x^6 + ... + kp1x^1 + kp0x^0 We will expand the left-hand side into the right-hand side.

List the common divisors of two natural numbers x and y. Let t be the number of common divisors.

Divide the cubic polynomial by the quadratic polynomial.

We will search for specific solutions to the system of linear equations. The range is limited to integers. The equation is of the form ax + by = c.

Determine whether a permutation is even or odd. Here, we consider permutations of the sequence 12345678. This determines whether an 8-character permutation σ is even or odd. If even, sgn(σ) = 1 (one); if odd, sgn(σ) = -1 (minus one); and outputs 1 or -1 as the result. A permutation that can be expressed as the product of an even number of transpositions is an even permutation, while a permutation that can be expressed as the product of an odd number of transpositions is an odd permutation. If the permutation is even, the parity check returns "even" and outputs 1. If it is an odd permutation, the parity check returns "odd" and outputs -1. The parity check of a permutation is sometimes used in the definition of determinants. Enter the sequence of numbers whose parity you want to check (the string of numbers resulting from permuting 12345678) into the input field. In permutation notation, this is often written in the lower row. For example, if you want to check the parity of 87651234 (the permutation of 12345678), enter the following values from top to bottom: Input 1=8, Input 2=7, Input 3=6, Input 4=5, Input 5=1, Input 6=2, Input 7=3, Input 8=4. The program will output the parity result.

Formulas used for calibrating resistance meters

In the January 2025 issue of Nikkei Science, we examine the accuracy of the approximation formula for the area of a circular arc presented in the feature article “The True Portrait of Seki Kōwa, the Sage of Mathematics” by Ken’ichi Sato, from the “Rediscovering Wasan” series, by comparing it with the exact value. Note that the diameter of the circle is fixed at 1.

Approximate the number of prime numbers less than or equal to any integer x. The first result is rounded, while the second result displays the calculation exactly as it is.

We will perform calculations related to systems of two linear equations. The equations are of the form: ax + by = e cx + dy = f We will transform both equations into the form "y = ~" and, when visualizing them as graphs in the xy-plane, we will examine whether the two lines are parallel or whether they are the same line and overlap. The equations take the form: y = m1x + n1 y = m2x + n2 where n1 and n2 are the y-intercepts. m1 and m2 are the slopes of the lines. If the slopes are the same, the lines are parallel. Furthermore, if the y-intercepts are the same, the two lines are identical and coincide. Here, we do not use inverse matrices, nor do we use the back-substitution method with the scale matrix. First, we eliminate y, find x, and then substitute it into y = m1x + n1. If the lines are not parallel, we output the solution x, y. We do not handle cases where there is no solution, an indeterminate solution, or an arbitrary constant. We do not handle complex numbers (non-real numbers with a non-zero imaginary part).

cubicEquation function

Given a geometric sequence with first term a0 and common ratio r, where an = a0*(r)^n, output the values of an for n = 1..N (N <= 40).

Calculate the weighted geometric mean. The dataset is limited to 10 entries. When entering data, the sum of the weights does not have to equal 1. Calculate and display the redistribution of weights so that their total equals 1. Examples of the weighted geometric mean include the effective exchange rate. Enter each data point in the dataset as a fraction. Output the weighted geometric mean as a decimal.

For two natural numbers A and B, this program outputs the results of the procedure for finding the least common multiple and the greatest common divisor.

I am using continued fractions to calculate an approximation of pi.

For each natural number from n0 to n1, determine whether it is divisible by each of the three natural numbers a, b, and c, and count the number of such numbers.

Find the denominator of a unit fraction that is the sum of two other unit fractions. This relates to Egyptian arithmetic. This is the inverse of unit fraction decomposition. Let the form of the expression be (1/A) + (1/B) = 1/C. (/ denotes division.) The value of each numerator is 1. A, B, and C are denominators. A and B are known, and C is the unknown variable to be found. A, B, and C are all natural numbers greater than or equal to 2. A and B are distinct natural numbers.

Quadratic Equation Function

Calculate the number of prime numbers less than x. The accuracy is not very good.

cubic_equation_formula

Determine whether a permutation is even or odd. Here, we will consider permutations of the sequence 1234567. This determines whether a 7-character permutation σ is even or odd. If even, sgn(σ) = 1 (one); if odd, sgn(σ) = -1 (minus one); and output 1 or -1 as the result. A permutation that can be expressed as the product of an even number of transpositions is an even permutation, while a permutation that can be expressed as the product of an odd number of transpositions is an odd permutation. If the permutation is even, the parity check returns "even" and outputs 1. If it is an odd permutation, the parity check returns "odd" and outputs -1. The parity check of a permutation is sometimes used in the definition of determinants. Enter the sequence of numbers whose parity you want to check (the string of numbers resulting from permuting 1234567) into the input field. In permutation notation, this is often written in the lower row. For example, if you want to check the parity of 7651234—which is the result of permuting 1234567— enter the following values: Input 1=7, Input 2=6, Input 3=5, Input 4=1, Input 5=2, Input 6=3, Input 7=4.

We will calculate the common denominator for a fractional expression. Assume that both the numerator and denominator are linear expressions in x. Let the original form be (mx+n)/(ax+b)+(px+q)/(ax+b). Let the form of the expression after finding a common denominator be: (k0(x^2) + k1x + k2) / (t0(x^2) + t1x + t2) We will display the expression as is, without simplifying or rearranging it.

When a natural number n is greater than two distinct natural numbers a and b, we investigate the largest and smallest natural numbers such that, when divided by a, the remainder is Ra, and when divided by b, the remainder is Rb.

Calculate the coefficients of the derivatives. Consider differentiating polynomials up to the fifth degree. (The exponent is an integer between 0 and 5, inclusive.) Find the coefficients of the first and second derivatives. (The coefficients are real numbers.)

Find the solutions to the quadratic equation ax² + bx + c = 0. When the square root is removed and the expression becomes simplified, the result will appear in the fraction in the bottom half.

We will expand the expression. We will consider the product of four linear expressions. Let the expression be of the form: (ax+b)(cx+d)(ex+f)(gx+h) = k4x^4 + k3x^3 + k2x^2 + k1x + k0 We will expand the left-hand side into the right-hand side.

We will calculate the partial fraction decomposition. We will consider the case where the numerator of the fractional expression is a quadratic expression in x, and the denominator is the product of two linear expressions. Let the expression be: (x^2 + Lx + M) / (x + P)(x + Q) = a / (x + P) + (x + b) / (x + Q). The coefficient of x^2 is 1. We will perform partial fraction decomposition of the left-hand side into the right-hand side.

We will calculate the inverse factorial. For natural numbers greater than or equal to 2, find the values for which x = n!. Let M be the value for which x = n!. Output the values of L, M, and H, as well as L!, M!, and H!, such that L < M < H and L! < M! < H!.

We approximate the number of prime numbers less than x.

Output a table of values for trigonometric functions with coefficients. Let the form of the expressions be: y0 = a0 * cos(n0 * x) y1 = a1 * sin(n1 * x) y2 = a2 * tan(n2 * x)

We will calculate the point of intersection between a direct and an inverse proportional equation. Let the equations be of the form: y = ax y = b/x where a and b are non-zero real numbers. Additionally, assume that b/a is greater than or equal to 0. We will find the coordinates of the point of intersection.

We will expand the expression. Here, we will consider the nth power of the binomial x + a. Let us write the expression in the form k{(x + a)^n} = k{polynomial in x} = a polynomial with k distributed to each term.

We will calculate the tangent line to an irrational function. Let the form of the irrational function be y = k + a√x. We will find the coefficients and the constant term of the equation of the tangent line.

We will calculate the weighted average. We will assume there are two data points.

We will calculate the expansion of the expression. Here, we consider the case where the expression expands into a quadratic form. The form of the expression is: P(x+a)(x+b) = P(j2x^2 + j1x + j0) By factoring out P, we get: = k2x^2 + k1x + k0 We will expand the left-hand side into the right-hand side. The numbers to the right of the letters j and k are subscripts corresponding to the degree of x. (j0 and k0 are constant terms)

Rearranging quadratic functions using the completing the square method. Part 2.

Calculate the length of the line segment cut off from the x-axis by the graph of a quadratic function. This is the distance between the points where the parabola (perpendicular to the axis) intersects the x-axis.

We will perform algebraic manipulation on a logarithmic expression. Let the original expression be of the form log_a x. Let the resulting expression be of the form r*log_a M. In the equation log_a x = r*log_a M, we will factor out the coefficient r and rearrange the left-hand side to match the right-hand side. We will then determine the value of M.

Rewrite the logarithmic expression. Convert the coefficient into the exponent of the argument and cancel it out.

Let's expand this expression. It is the product of three linear expressions. Let the form of the expression be (ax+b)(cx+d)(ex+f) = k3x^3 + k2x^2 + k1x + k0 . Expand the left side into the right side.

The accuracy isn't bad.

List the common divisors of the three natural numbers x, y, and z.<\n Let t be the number of common divisors.

There are commercial programs available for calculating suspension alignment changes, but since they are expensive, I decided to create my own.

We will perform the inverse operation of simplifying fractions. Suppose we have two fractions connected by an equals sign, and the numerator of one of them is unknown. In this case, the program will output the value of the unknown numerator.

This function approximates the number of prime numbers less than or equal to a given number. The larger the value of n, the higher the accuracy.

We will calculate the expansion of the expression. Here, we consider the case where the expression expands into a cubic polynomial. The form of the expression is: P(x+a)(x+b)(x+c) = p(j3x^3 + j2x^2 + j1x + j0) Factoring out P, we get: = k3x^3 + k2x^2 + k1x + k0 Let’s expand the left-hand side into the right-hand side. The numbers to the right of the letters j and k are subscripts corresponding to the powers of x. (j0 and k0 are constant terms)

Output a table of cosine values with coefficients. Let the expressions be: y0 = cos(x); y1 = a * cos(x); y2 = cos(n * x); y3 = a * cos(n * x).

We will calculate the weighted average. Let's assume there are 5 data points in the dataset.

We will plot the graph of a rational function (linear/linear). Let the form of the expression be y = g(x)/f(x), g(x) = ax + b, f(x) = cx + b (where a and c ≠ 0).

We will perform the inverse operation of simplifying fractions. Suppose we have two fractions connected by an equals sign, and one of them has an unknown denominator. Assume that all known values are non-zero. In this case, the program will output the value of the unknown denominator.

Calculate the transformation of a rational function. This applies to cases where both the numerator and denominator are linear expressions.

We will calculate the translation of a conic section.

Calculate the Fourier series. Enter the function f(x) (with domain -L ≤ x ≤ L and period 2L), and use integration to find the coefficients.

For a positive integer n, the number of natural numbers from 1 to n that are coprime to n is denoted by φ(n) and is called Euler's φ-function or the Torsion function.

Solve a fifth-degree equation using the DKA method. Complex numbers can be used for the coefficients.

Solve a sixth-degree equation using the DKA method. Complex numbers can be used for the coefficients.

Search the Stern-Brocot tree to find the approximate fraction of a given positive real number.

The Lambert W-function is the inverse of z = w * exp(w), i.e., w = W(z). In addition to the principal solution, you can calculate up to the kth solution. For the principal solution, set k = 0.

This calculates the Ackermann function Ack(m,n). It is known that as m increases, the result becomes an incredibly large number. Unfortunately, we can only calculate up to m=4 and n=2... (the limit of this site's capabilities)

Find the nth-degree polynomial that passes through n+1 points, all of which have distinct x-coordinates, using Lagrange interpolation. Even if n is less than 11 and some points are not needed, please enter 0 for those points.

We will compare Ramanujan’s formula and the Chudnovsky brothers’ formula, both of which are known as formulas for calculating pi. Both are known for converging extremely quickly.

Two prime numbers whose difference is 2 are called twin primes. Let’s calculate these twin primes.

Find the smallest integer solution (X, Y) to Pell's equation, X^2 - NY^2 = ±1, and the next largest solution. Here, N is a positive integer that is not a perfect square.

Solve the nth-degree equation a(n)*x^n + a(n-1)*x^(n-1) + ... + a1*x + a0 = 0 using the DKA method. n is at most 20. When solving equations of degree less than 20, set any unnecessary higher-order coefficients to 0. Complex numbers may be used for the coefficients.

If you flip 56 to make 65 and add them together, the result is 121, which is a palindromic number. By repeating this process—for example, 57 → 57 + 75 = 132 → 132 + 231 = 363—it becomes a palindromic number in just two steps; 89 takes 24 steps, and 10911 takes 55 steps. Lychrel numbers are those that never become palindromic, no matter how many times you repeat the process.196 is the smallest candidate.

Quaternions

We will examine the AGM method (Gauss-Legendre algorithm, arithmetic-geometric mean method), the algorithm used by Mr. Takahashi and his colleagues at the University of Tsukuba to calculate pi to 2,576,980,370,000 digits. Using the initial values a0 = 1, b0 = 1/√2, t0 = 1/4, and p0 = 1, an+1 = (an + bn) / 2 bn+1 = √(anbn) tn+1 = tn - pn * (an+1 - an)^2 pn+1 = 2*pn Thus, π ≒ (an + bn)^2 / (4*tn) is used for the calculation.

Generates random numbers following the Poisson distribution: P(X=n) = exp(-λ)λ^n / n!

Using the two-variable Newton-Raphson method, we will calculate the values of x and y that satisfy f(x,y) = 0 and g(x,y) = 0. We also need ∂f/∂x, ∂f/∂y, ∂g/∂x, and ∂g/∂y.

The sum of the numbers is the area of a right triangle whose side lengths are all rational numbers; we will determine whether this is true. Assume that the number in question has no square factors (i.e., it cannot be written as n = qk²). We use Tunnell's theorem to determine this.

The partition number p(n) represents the number of ways a natural number n can be expressed as the sum of natural numbers (including n itself), in any order. Since the film *The Formula*, about Ramanujan and Hardy, featured this concept prominently, I’m doing this to commemorate its release. I will calculate the exact partition number p(n) and the Ramanujan–Hardy asymptotic formula. I’ll cover values of n up to 250.

Arnab Priya Saha and Aninda Sinha, researchers in string theory from India, derived a new formula for pi in their paper “Field Theory Expansions of String Theory Amplitudes.” Let’s take a look at it.

Prime Number Test

This is a calculation of the Pell's sequence (numbers that return to their original value when squared) as described in *Numbers Reaching to the Sky* (by Fumimoto Kato and Yasuyuki Nakai).

How many zeros are there at the end of n!?

Jesús Guillera has discovered a formula for the mathematical constant π that converges incredibly quickly. Let’s give it a try. It might be interesting to compare it with Ramanujan’s formula and the Chudnovsky brothers’ formula.

dx/dt = f1(x, y, z, w, t) dy/dt = f2(x, y, z, w, t) \frac{dz}{dt} = f3(x, y, z, w, t) \frac{dw}{dt} = f4(x, y, z, w, t) We will solve these four-variable ordinary differential equations using the Runge-Kutta method (four-stage, fourth-order).

In his paper titled “A formula for the nth decimal digit or binary of π and powers of π,” Simon Plouffe presents a formula for calculating the value of any digit of pi (π). We will verify this. However, this site is limited to 50 digits.

This is Dirichlet's theta function.

We will perform numerical diagonalization on real symmetric matrices. Here, we will diagonalize a 3×3 real symmetric matrix.

Finds an approximate fraction for a given open interval of real numbers.

Using high-precision arithmetic, I'll try to brute-force a solution to: Google Code Jam 2008 - Round 1A - Problem C. Numbers http://code.google.com/codejam/contest/dashboard?c=32016#s=p2 Since integer precision is guaranteed up to n = 69, this approach might be sufficient to pass the Small dataset (2 ≤ n ≤ 30).

You can calculate the area of a spherical triangle.

Calculate the product of three 3x3 matrices. Calculate the product of the three 3x3 matrices A, B, and C as follows: AB = A * B ABC = (A * B) * C and output the results. (Multiply the left matrix by the right matrix.)

Rational Approximations of Real Numbers

Tables and Graphs of the Trigonometric Functions sin(x), cos(x), and tan(x)

Normalize a three-dimensional vector. Let the components of the vector v to be normalized be v1, v2, and v3. Let the magnitude of v be r. Let the components of the normalized vector u be u1, u2, and u3.

We will calculate the square root of a 2×2 matrix. Given a 2×2 matrix M, we will find a 2×2 matrix X such that M = X². Here, we will refer to X as the square root of M. This involves considering an operation analogous to the square root of a real number (represented by the radical symbol √).

Perform calculations related to 3x3 square matrices. 1) Output the coefficients of the characteristic equation. The form of the characteristic equation is assumed to be k0λ³ + k1λ² + k3λ + k4 = 0. 2) Calculate the value of the minor. 3) Calculate the value of the cofactor. 4) Calculate the cofactor matrix. 5) Calculate the value of the determinant using Sarrus's formula. 6) If the value of the determinant calculated using Sarrus's formula is not zero, calculate the inverse.

This program performs calculations related to 4x4 matrices. Here, we calculate the minor, the cofactor, and the cofactor matrix. Additionally, for reference, it outputs the determinant of the input 4x4 matrix and its inverse.

We will calculate the scalar triple product. We consider the xyz coordinate system to be right-handed. Let bc be the cross product of vectors b and c. We calculate the dot product of a and bc to find the value of the scalar triple product.

Calculate the direction cosine of a 3D vector. Let the vector be a. The origin is the starting point.

Find the value of c using the values of a, b, and cos.

Let’s review the following from Fermat’s Library: Liouville proved the following property regarding the divisors of the divisors of a number: 1) Start with a number (e.g., 10) 2) List its divisors (1, 2, 5, 10) 3) List the number of divisors of each divisor (1, 2, 2, 4) 4) (1 + 2 + 2 + 4)² = 81 = 1³ + 2³ + 2³ + 4³

We will derive the coefficients of a fourth-degree polynomial. Let the polynomial be of the form k⁴*λ⁴ + k³*λ³ + k²*λ² + k*λ + k₀.

Normalize a two-dimensional vector to a unit vector. Let the components of the vector v before normalization be v1 and v2. Let the magnitude of the vector be r. Let the components of the normalized vector u be u1 and u2.

Specify Input 1 and Input 2 to generate any Pythagorean triple. However, m must be greater than n.

Calculate the characteristic equation of a 2x2 matrix. Display the coefficients of the characteristic equation. Although the characteristic equation is a quadratic equation, calculate the value of the discriminant D. Also, display the square root of the discriminant. Output the solutions to the characteristic equation. If there are repeated roots, display two solutions. If the value of the discriminant is 0, there are repeated roots.

Find the vertex of the quadratic function.

We will perform quaternion calculations. We will calculate the magnitude, inverse, and product (Q1Q2, Q2Q1) of two quaternions, Q1 and Q2.

For y = 1/(1+x²), the integral values of the equation of a circle over the interval [0, 1] are both π/4 (1/4 of a circle), and thus equal.

Display Pythagorean triples in a table

Please enter any natural number n.

a

We will calculate the result of multiplying a 2D vector by a rotation matrix. Let A and B be 2D vectors. R is a 2x2 matrix and a rotation matrix. We will calculate B = R * A.

Calculate the coefficients of the cubic characteristic equation. Since we are considering the eigenvalue problem for a 3×3 square matrix, the characteristic equation is derived from the determinant of the 3×3 matrix. The form of the cubic characteristic equation is -(λ³) + m(λ²) + n(λ¹) + a constant term. We assume the coefficient of the highest-order term (λ³) remains -1. We calculate and output the coefficients of the (λ²) term, the (λ¹) term, and the constant term. The form of the characteristic equation is displayed in its expanded form.

We will perform calculations related to 3x3 square matrices. We will list the coefficients of the characteristic equation of a 3x3 square matrix and calculate its solutions. Let the form of the characteristic equation be a(λ³) + b(λ²) + cλ + d = 0. (1) Find the coefficients of the characteristic equation of the 3x3 square matrix. (2) Divide each coefficient by the coefficient (-1) of the highest power of λ (3), and display the results. (3) Display the approximate solution to the characteristic equation. (4) As a verification, display the results of substituting the approximate solution into the equation. (5) When the coefficients b, c, and d are integers, apply the factor theorem; if a solution is found, display the result.

Please enter any natural number n. We do not support numbers 16 or higher.

Calculate the value of the cosine cosθ of the dot product of two vectors a and b.

Calculates the value of the dilogarithm (double logarithm). Calculations are possible for x ∈ (-∞, 1]. For x > 1, Li(2, x) is not a real number; since this is not implemented, 0 is returned as an error.

We will perform the calculation of multiplying a 2×2 square matrix by a rotation matrix. We multiply the 2×2 square matrix A by the inverse of the rotation matrix P, Q, from the right to compute the matrix QA, and then multiply the rotation matrix P by QA from the right to compute QAP. The elements of P are (the numbers to the right of Q represent row and column indices): P11 = cosθ P12 = -sinθ P21 = sinθ P22 = cosθ (where θ is in radians).

Normalize a four-dimensional vector. Let the components of the vector v to be normalized be v1, v2, v3, and v4. Let the magnitude of v be r. Let the components of the normalized vector u be u1, u2, u3, and u4.

Calculate the value of the cosine cosθ of the dot product of vectors a and b.

Calculate the product of three 4x4 matrices. Consider three 4x4 matrices A, B, and C. First, calculate AB = A * B, and then calculate ABC = AB * C. AB and ABC are also 4x4 matrices. Output the result, ABC. (* denotes matrix multiplication)

We will calculate the determinant of a 2x2 matrix. Here, we will consider the case where the elements are complex fractions. Let a be the element in row 1, column 1. Let b be the element in row 1, column 2. Let c be the element in row 2, column 1. Let d be the element in row 2, column 2. In the input, a, b, c, and d are entered as fractions. Enter non-zero numbers for the denominators of a, b, c, and d. Enter complex numbers for both the numerators and denominators of a, b, c, and d.

We will calculate quadratic expressions involving the 2×2 matrix x. Let’s consider polynomials involving a 2×2 matrix. This is a quadratic polynomial in the 2×2 matrix x. Let the form of the expression be y = aL*x*x*aR + bL*x*bR + c. (* denotes the product of matrices, and + denotes the sum of matrices. The product is obtained by multiplying the left matrix by the right matrix in order.) aL, aR, bL, bR, c, x, and y are all 2x2 matrices.

Calculate the product of three 2x2 matrices. Calculate the product of the three 2x2 matrices A, B, and C as follows: AB = A * B ABC = (A * B) * C and output the results. (Multiply the left matrix by the right matrix.)

We will calculate the cubic expression of a 2x2 matrix. Let y, x, aL, aR, bL, bR, cL, cR, and d be a 2x2 matrix. We define the expression as: y = aL * x * x * x * aR + bL * x * x * bR + cL * x * cR + d This mimics the cubic equation y = ax³ + bx² + cx + d for "ordinary numbers" with respect to x. L and R denote the left and right positions relative to the power of x. * denotes the matrix product, where the right matrix is multiplied by the left matrix. Multiple * operations are performed sequentially, multiplying the right matrix by the left matrix step by step. Finally, the terms are added together.

We will calculate the orthogonal projection of a vector.

We will derive the coefficients of the characteristic equation. Here, we will consider the case of a 5×5 square matrix. (Characteristic equation of a 5×5 matrix)

This is a numerical diagonalization method for real symmetric 2×2 matrices. Since the matrix is both 2×2 and symmetric, the process is simplified because we can skip the step of finding the "diagonal element with the largest absolute value." Furthermore, converting the problem into linear expressions involving `cos2θ` and `sin2θ` reduces the number of iterations, improving the efficiency of interval selection and reduction via the bisection method. We perform Newton's approximation within the reduced interval. The precision settings are as follows: the number of interval divisions is set to 100 each time, the interval width is set to 0.1^15, and for Newton's approximation, it is reduced to 0.1^20.

We will calculate the nth power of a quaternion. Let the expression be of the form q = a + bi + cj + dk, and y = q^n. i, j, and k are the imaginary units. a, b, c, and d are the real and imaginary parts, respectively.

This is a generalization of the volume of an n-dimensional sphere.

This outputs the signs and indices during the expansion of a determinant. Here is an example for a 5x5 determinant. It lists the signs and the indices of the elements, n and σ(n). n = 1, 2, 3, 4, 5.

Display the values of the Levi-Civita symbol. Here, let ε be the symbol. Let {1, 2, 3, 4} be a set of numbers. When this set is replaced by the set {t1, t2, t3, t4} using the indices t1, t2, t3, and t4, the value changes depending on whether the permutation is even, odd, or otherwise. List the values of ε alongside t1, t2, t3, and t4. However, only output the table for cases where ε = +1 or -1; omit combinations where ε = 0.

Displays the value of Levi-Civita's symbol. Here, we set it to ε. Let {1, 2, 3, 4, 5} be a set of numbers. When this set is permuted by the indices t1, t2, t3, and t4 to form the set {t1, t2, t3, t4, t5}, the value changes depending on whether the permutation is even, odd, or neither. List the values of ε alongside t1, t2, t3, t4, and t5. However, only output the table for cases where ε = +1 or -1; omit combinations where ε = 0.

Display the values of Levi-Civita symbols. Here, we use ε. Let {1, 2, 3} be a set of numbers. When this set is replaced by the set {i, j, k} using the indices i, j, and k, the value changes depending on whether the permutation is even, odd, or otherwise. List the values of ε along with i, j, and k.

This program outputs a table of values for the gamma function. Imagine the complex plane. The input value $s$ for the gamma function is taken from a line segment on the complex plane. Assume that the real part of the complex number at the starting point $P$ is a nonzero positive number. Assume that the real part of the complex number representing the endpoint Q is also a nonzero positive value. Divide the line segment into n equal parts, with the starting point as the 0th input and the endpoint as the nth input, and calculate the values of the gamma function. The output is in table format.

1. Definition and Role The sinc function is defined as \(\frac{\sin (x)}{x}\). It is an essential mathematical formula used in communication theory and signal processing (sampling theorem) to smoothly interpolate waveforms. 2. Key Points of the Program Avoiding Division by Zero: When \(x=0\), the expression becomes \(\frac{0}{0}\), which is an invalid calculation (error).Using Limit Values: Since mathematically \(\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1\), the code uses an `if` statement to force a return value of \(1\) when \(x=0\). 3. Behavior: The function reaches its maximum value of \(1\) at \(x=0\). As \(x\) moves away from \(0\), the function oscillates and converges to \(0\).

This outputs the signs and indices during the expansion of a determinant. Here is an example for a 4x4 determinant. It lists the signs and the indices of the elements, n and σ(n). n = 1, 2, 3, 4.

This outputs the signs and indices during the expansion of a determinant. This is the case for a 3x3 determinant. The simplified form is known as "Sarrus' formula." It lists the signs and the indices of the elements, n and σ(n). n = 1, 2, 3.

Calculate the product of two octonions. This calculates the product of two octonions, a and b.

Calculate the value of a cubic polynomial using quaternions. Ln is the coefficient of the left-hand side of x^n. Rn is the coefficient of the right-hand side of x^n. M is the constant term. The product is calculated by multiplying the right-hand side by the left-hand side. For each term of degree n, multiply Ln by x n times from the right, and then multiply by Rn from the right. Finally, sum all terms, including the constant term. Output this as the value of the polynomial.

We will calculate the value of a quadratic expression using quaternions. Let the expression be of the form y = (aL * x * x * aR) + (bL * x * bR) + c, where * denotes multiplication. y, x, aL, aR, and b are quaternions. Multiplication is performed by multiplying the right-hand side by the left-hand side.

We will calculate the value of a linear expression involving quaternions. Let the expression be of the form y = aL * x * aR + b (where * denotes multiplication). y, x, aL, aR, and b are quaternions. Multiplication is performed by multiplying the right-hand side by the left-hand side.

Average score → Standardized score

We will simplify fractions. We will also display the greatest common divisor. (Mixed fractions are allowed.)

Grades for the First Semester of the Third Year

The number of times a multiple of 3 or a number containing the digit 3 appears in an n-digit natural number

Display recurring decimals in fractional form.

Find a common denominator for the three fractions.

Calculate distances on a map and actual distances based on the scale. (Includes unit conversion)

This is for the "!!" on the bulletin board.

Question 6 on the 2014 National Center Test for University Admissions (Mathematics II and Mathematics B) involved factoring the factorial (N!). I have ported the BASIC program to my own code.

We will find a common denominator for the two mixed fractions.

f(x) = (ax + bP) / (bx + a) … Basic form where a, b, and P are positive integers. a/b > √P Properties of the basic form 1. √P = f(√P) … Has a fixed point at √P. 2. f(∞) · f(0) = P ... When x is a positive integer, the rational function f(x) yields an approximate rational value of the fixed point within the range f(∞) > f(x) > f(0). 3. Let F1(x) = f(x) and Fn(x) = Fn-1(f(x)). Then, Fn(x) = (Ax + BP) / (Bx + A). Fn(x) possesses the same properties as f(x). 4. Fn(∞) − Fn(0) < Fn-1(∞) − Fn-1(0) ... As the number of substitutions from x to f(x) increases, the range of approximation to the fixed point narrows. 5. When b = 1, the basic formula can be transformed into f(x) = a − (a² − P)/(a + x), and F_n(x) yields an approximate continued fraction. Using the properties described above, we derive more complex expressions by repeatedly applying substitutions starting from a simple basic form. Starting with a = (the integer part of √P) + 1 and b = 1, we calculate Fn(x) = (Ax + BP) / (Bx + A).

We will find a common denominator for the two fractions.

This is a problem from the mathematical tablet at Gokō-gū Shrine in Fushimi, Kyoto. There are three squares, A, B, and C. The sum of their areas equals a certain value; the cube of side A, the fifth power of side B, and the seventh power of side C also equal a certain value; and the difference between side A and side B is equal to the difference between the side length of B and the side length of C are equal. What are the side lengths of A, B, and C? Note that A > B > C.

We will calculate the first and second derivatives of the function f(x) and plot them.

When you want to look up cos, sin, and tan.

Let M and N be natural numbers such that M > N and their greatest common divisor is 1. For the fraction M/N, consider the following steps (i) through (iii): (i) Let Q be the quotient and R the remainder when M is divided by N. (ii) If R > 0, set M equal to N, then set N equal to R, and return to step (i). (iii) If R = 0, terminate. Using this sequence of quotients, we can write the following:

Calculate the rate of change and the range of y for y = ax². Enter the range of x (called the domain) and the constant of proportionality. It displays the rate of change, as well as the "increments of x and y."

We will calculate quadratic functions.<\n\ Let’s consider quadratic functions in the xy-plane. The x-axis is the horizontal axis, and the y-axis is the vertical axis.<\n\ Let’s use the following form for the equation:<\n\ y = f(x)<\n\ f(x) = ax² + bx + c<\n\ .

We will calculate the "relationship between the roots and coefficients" of a quadratic equation. Let the equation be of the form ax² + bx + c = 0 (a ≠ 0). Let the two roots be α and β. We will express the sum and product of the roots. These are the fundamental forms.

We will perform calculations related to Cardano's formula. This is the formula for solving cubic equations. We will display the intermediate steps, including the Chirlhouse variable and variable transformations.

It can literally perform addition, subtraction, multiplication, and division

We will calculate the “relationship between the roots and coefficients” of a cubic equation. Let the equation be of the form ax³ + bx² + cx + d = 0 (a ≠ 0). Let the three roots be α, β, and γ.

We will calculate the "relationship between the roots and coefficients" of a quadratic equation. Let the equation be of the form ax² + bx + c = 0. Let the two roots be α and β. We will express the sum and product of the roots. These are the fundamental forms.

Let’s perform division. Suppose that a ≥ b, a ≥ 0, and b > 0. Let the expression be of the form a = (b × Q) + R. Q is a natural number. Let R be the remainder when the quotient Q is calculated to the units digit.

We will calculate the "relationship between the roots and coefficients" of a cubic equation. Let the equation be of the form ax³ + bx² + cx + d = 0 (a ≠ 0). Let the three roots be α, β, and γ.

If you travel the same distance back and forth at different speeds, why isn’t the average speed equal to the sum of the two speeds divided by two?

We will examine changes in the argument of a complex number using the four basic arithmetic operations on complex numbers, with two complex numbers a and b. The unit of the argument is degrees. Let c be the complex number resulting from the arithmetic operations. Here, the change in the argument refers to the increase or decrease when the argument of a changes to the argument of c as a result of the arithmetic operations. (Change = argument of c – argument of a)

Determine the magnitude of the argument θ of the complex number z in degrees. Output the result within the range 0 ≤ θ < 360.

Here's how to read large numbers using the manjin system: one, ten, hundred, thousand, ten thousand, million, billion, trillion, quadrillion, quintillion, sextillion... an immeasurable number?

Given the equation of a linear function, find its y-intercept and x-intercept.

We will examine how the phase angle changes when complex numbers are multiplied. Let the form of the expression be c = a * (b^n). Here, the magnitude of the change in phase angle is defined as the phase angle of c minus the phase angle of a. We use degrees as the unit of phase angle. The range of phase angles θ is 0 ≤ θ < 360.

We will perform a test of the difference in proportions between two groups. This can be used for analyzing A/B tests and similar scenarios.

This tool estimates a confidence interval for the population proportion—such as whether a candidate is certain to win based on the vote share from an exit poll. Enter the sample size (number of respondents in the exit poll), the number of cases meeting the condition (number of people who voted for a specific candidate), the confidence interval percentage (e.g., 95%), and the estimation method (choose between Wald and Agresti-Coull), and the confidence interval will be calculated. *Note: On June 24, 2020, we fixed an issue where the results differed from those obtained using R.

Perform a t-test for the difference between two groups under the assumption of equal variances. Enter the sample sizes, means, and standard deviations for each of the two samples, as well as the significance level and whether equal variances can be assumed (if not, Welch’s t-test will be used).

When conducting a test for the difference between two means, given the effect size (Δ = (μ1 - μ2) / σ; i.e., how many standard deviations is the difference between the means?) and the significance level, calculate the required sample size n (where n = n1 = n2, representing the sample sizes of each group).　Null hypothesis: μ1 = μ2; for the alternative hypothesis, you can choose from three options: μ1 ≠ μ2, μ1 > μ2, or μ1 < μ2.

We will perform a test of the correlation coefficient. The data set must contain up to 21 values. Please enter 0 (not leave blank) for any missing data. The null hypothesis is r = r0. If r0 = 0, we will perform a t-test; otherwise, we will perform a test based on a normal distribution using Fisher’s Z-transformation.

Find the probability of a random outcome in a series of trials.

This function calculates Shannon entropy (average information content) when there are two possible outcomes from a source. Perfect for when you need to know the return value of the binary entropy function! Essentially, it’s a function that calculates the value of x*log₂(1/x) + (1-x)*log₂(1/(1-x)).

You can calculate the probability of rolling the same number consecutively x times when rolling a die.

Determine the "sample size" that satisfies the specified conditions in a non-replacement sampling from a finite population.

The relationship between the number of types available as random bonuses and the number of items purchased

We will perform Pearson's chi-square test (test of independence). There are two categories, and you can enter up to 8 conditions for each category and up to 3 conditions for the other. Please enter 0 for any unnecessary data.

We will perform a one-sample t-test. (Null hypothesis: μ = μ₀)　

Find the probability of a random outcome in a series of trials.

This is for calculating the standard "t-test for the difference between the means of two independent groups (two-sample t-test)." It displays results for both the assumption of equal variances and the case where this assumption is not made (Welch's t-test) simultaneously. Additionally, it uses standard deviation as the input variable. --- The formula was based on "Statistics WEB" (see Related Links). A similar formula created by another user (see Related Links) required the input of the unbiased variance, but since the (sample) standard deviation is often reported in research papers, I created this tool so that you can immediately verify the results of an unpaired t-test as seen in such papers. (Created July 27, 2022; Updated July 29)

This graph illustrates the probability of matching birthdays. It is well known that the probability is high when two out of N people share a birthday, but as the number increases to three, four, and so on, the probability drops sharply. The calculation uses the approximation formula employed by the R functions `pbirthday` and `qbirthday`, as described in Diaconis, P. and Mosteller, F., "Methods for studying coincidences," Journal of the American Statistical Association.

Overall Calculation of Rock-Paper-Scissors

When conducting a one-sample t-test, calculate the required sample size given the effect size (Δ = (μ - μ₀)/σ; i.e., how many standard deviations is the difference in means?) and the significance level. The null hypothesis is μ = μ₀, and there are three possible alternative hypotheses: μ ≠ μ₀, μ > μ₀, and μ < μ₀.

Calculates the power curve (1-β for a difference in means μ1 - μ2). You can choose from three alternative hypotheses: that μ1 is greater than μ2, that μ1 is less than μ2, or that μ1 is equal to μ2, given the null hypothesis that μ1 and μ2 are equal.Enter the sample sizes n1 and n2, the significance level α, and the effect size Δ (defined as (μ1 - μ2) / σ, i.e., how many standard deviations the difference in means is), along with the maximum and minimum values and the number of partitions, to generate a power curve graph.

This calculates the probability of rolling the same number when b six-sided dice are rolled c times in a row.

The number of attempts required to drop n. x represents the percentage of players who dropped n on their first attempt, or the percentage of attempts where n was obtained after multiple tries. The larger the sample size, the higher the reliability of x. Please refer to players who keep records of their drops.

We calculate the probability that the birthdays of at least k people out of N will coincide. It is well known that the probability of two people out of N having the same birthday is high, but it drops sharply when the number rises to three, four, and so on. The formula used is an approximation also employed by the R functions `pbirthday` and `qbirthday`: Diaconis, P. and Mosteller, F. "Methods for studying coincidences." J. American Statistical Association.

The odds of winning the first prize in the lottery are 1 in 10 million or 1 in 5 million. I had them calculate the odds of winning the first prize based on the last five digits of the ticket number.

I calculated the probability of winning based on the last six digits of a single New Year's postcard. Since the postcard numbers are displayed in hexadecimal, the possible selections are naturally listed in hexadecimal as well.

This is a program for beginners in statistics that calculates the t-value and p-value for a paired two-sample t-test.

We will perform Pearson's chi-square test (goodness-of-fit test). You can enter observed frequencies for up to 8 categories. You can also enter probabilities as fractions. Please enter 0 for any missing data.

Calculates the power curve (1-β for a single mean). You can choose from three alternative hypotheses: that μ is different from μ0, that μ is greater than μ0, or that μ is less than μ0, given the null hypothesis that μ equals μ0.Enter the sample size, significance level α, and effect size Δ (defined as (μ - μ₀)/σ, i.e., how many standard deviations the difference in means is), along with the maximum and minimum values and the number of groups, to generate a power curve graph.

This tool calculates the t-test and confidence intervals for the regression coefficients a and b (slope and intercept) in simple linear regression (y = ax + b). Up to 21 data points are supported. Please enter 0 (not a blank) for any data points you wish to exclude. The calculation assumes the null hypothesis that the slope and intercept are equal to the values you entered.

This program calculates the t-value, degrees of freedom, and significance probability for an independent two-sample t-test, designed for beginners in statistics.

Here is the formula I devised to calculate the probability that your food scraps will stain the kotatsu blanket: Area of the trapezoid * probability Use this formula to calculate the probability.

We calculate the distribution of the ratio (quotient) of two normal distributions, X1 and X2, namely X1/X2. If the means of X1 and X2 are both 0, the distribution follows a Cauchy distribution; otherwise, it becomes quite complex. (Note) It appears there was a typographical error in the equations in the original paper, so I have corrected them to match the equations in another paper (see the last link) (June 2, 2023).

This is a program for beginners in statistics that calculates the t-value and significance level for a one-sample t-test.

Determine the "margin of error" for a non-replacement sample drawn from a finite population that satisfies specified conditions.

We will simulate the Monty Hall problem.

The probability of winning after performing n% of the process m times

This calculator uses a random number function to generate random integers within any specified range and in any quantity.

Let’s calculate a artillery duel between one gun with 100% accuracy and 100 guns with 10% accuracy. Let’s designate the single gun with 100% accuracy as Team A, and the 100 guns with 10% accuracy as Team B. In the initial state, Team A has one gun, and Team B has 100 guns, and we assume they fire simultaneously each turn. For both Team A and Team B, if even a single shot hits, the number of guns is reduced by one. (We will not consider partial destruction, repairs, or reinforcements.) Therefore, Team B is guaranteed to lose one gun per turn (after firing). In the next turn, Team B will fire the number of shots corresponding to its remaining guns, each with a 1% hit rate. Team A will be reduced to zero if it takes a hit after firing.

This is also known as Condorcet’s Jury Theorem, and we will calculate the probability of correctness—a concept that also appears in Yuto Ichikawa’s novel *The Closed Room of God and the Ripples*. If p > 0.5, the group’s probability of getting the answer correct exceeds that of an individual (approaching God’s answer); if p < 0.5, it falls short. Of course, each individual must answer completely independently.

In a non-replacement sample from a finite population, calculate the "population proportion" that satisfies the specified conditions.

Find the probability that two out of three people will join the same club.

We will simulate rolling a 10-sided die. We will use random numbers. The die will land on one of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Use this whenever you find yourself thinking, "I've tried the gacha so many times, but I never win anything... How many items are actually in there?"

Find the probability that three people will join the same club.

The probability that three people will be assigned to three different groups

Calculate the expected score for the test. Based on the point values (how many points each question is worth) and the total number of questions, calculate the point value for each question, as well as the expected score and the maximum and minimum scores.

Determine the "confidence level" for a non-replacement sample drawn from a finite population that satisfies specified conditions.

Calculate the overall probability when an event with a certain probability is repeated

Determine the "population size" that satisfies the specified conditions in a non-replacement sample from a finite population.

Roll a die with between 2 and 100 sides. Let n be the number of sides on the die. Here, we assign numbers to each face of the die using an arithmetic sequence. The first term corresponds to the first face, and we define a general term up to the nth face. Perform 10 rolls of the die and output the results. The default is a six-sided die.

Roll a six-sided loaded die. This means that the probability variables for each face can be set to be unequal. Here, we assign a "likelihood" to each face of the die, and normalize the sum of the probability variables to 1 by dividing each "likelihood" by their total sum. If you assign the same "likelihood" value to every face of the die, it becomes a standard, non-biased setup; if you assign them unevenly, you create a biased die.

Place white and red balls into a bag, take one out, record its color, and return it to the bag. This process is repeated. A random number generator is used. The number of balls placed in the bag at each step is at least one, and the total number of balls must be ten or fewer.

I would like to request a new rating.

We will conduct an experiment involving rolling four dice. We will use random numbers. Since the minimum and maximum values for each die can be changed, the number of sides will change accordingly.

We will conduct a simulation of rolling a six-sided die. Here, you can freely assign values to each face of the die. We will use random numbers to simulate the act of rolling the die. The number of trials will range from 1 to 12.

Roll a n-sided die 12 times. Use random numbers. The results are integers. Determine the minimum and maximum possible outcomes for the die, then enter them and click the "Calculate" button to run the simulation. The results for all 12 rolls will be displayed.

We will conduct an experiment involving rolling three dice. We will use random numbers. Since the minimum and maximum values for each die can be changed, the number of sides will change accordingly.

This is a program for beginners in statistics that calculates the F-value and p-value for a one-way ANOVA with between-subjects factors.

Generates random complex numbers. Here, we assume that both the real and imaginary parts are integers. Generates 10 numbers and outputs them.

We will perform an experiment involving rolling two dice. We will use random numbers. Since the minimum and maximum values for each die can be changed, the number of sides will change accordingly.

95% confidence interval

We will calculate the number of possibilities. Consider the experiment of rolling three distinct dice. For the sum and product of the dice rolls, we will determine the number of cases where the result falls within the range of integers from n₀ to n₁, and the number of cases where the result is divisible by a natural number d. The rest is for reference.

We will examine the number of possible outcomes. Here, when rolling two distinct dice, we will determine the number of cases where the sum and product of the numbers on the dice fall within a certain range. The range is defined by two integers. Let the lower bound be n0. Let the upper bound be n1. We assume that n1 is an integer greater than or equal to n0.

We will examine the number of possible outcomes. Here, we consider the experiment of rolling two distinguishable dice. For the sum and product of the dice rolls, we count the number of values that are divisible by a given natural number n.

This is a program for calculating the F-value and p-value for a one-way ANOVA (within-subjects) designed for beginners in statistics.

This is a program for beginners in statistics that calculates the F-value and p-value for a two-way ANOVA with between-subjects factors.

Here, we will discuss the expansion of (A+B+C)^n. This is a trinomial. We will calculate the coefficients using the binomial theorem (for trinomials) and list them. We will find the coefficients of the (A^a)(B^b)(C^c) terms in the expansion. The lowercase letters a, b, and c represent the exponents of the uppercase letters A, B, and C.

We will now calculate the binomial theorem. Here, we will consider the expansion of (A+B+C)^n. This is a trinomial. We will find the coefficients of the terms (A^a)(B^b)(C^c) in the expanded form. The lowercase letters a, b, and c represent the exponents of the uppercase letters A, B, and C.

Shuffle 13 playing cards. Suppose you have 13 playing cards. The suits do not matter. The deck is stacked in numerical order from top to bottom: 1, 2, …, 13. Take this deck, cut off the top p cards, and place them under the remaining stack of 13–p cards. Let q be the number of shuffles. Display this operation. [Number of cuts] = 0 represents the initial state of the deck. (1), (2), …, (13) represent the positions of the cards: the first, second, …, thirteenth cards from the top.

Generates complex random numbers. Here, the real and imaginary parts are not restricted to integers. Generates and outputs 10 complex random numbers. (Note that the imaginary unit i appears at the end of the output numbers.)

We will conduct an experiment involving the roll of two distinguishable dice. Here, we will determine the number of cases where the result is an integer n when we subtract the smaller number from the larger one.

95% Confidence Interval (Add-on)

We will calculate conditional probabilities. There are two boxes, A and B, and each box contains a certain number of winning tickets mixed in with losing tickets. We randomly select a box and draw one ticket.

We will discuss the binomial theorem. We will calculate and list the coefficients of the terms in the expression expanded using the binomial theorem. We will find the coefficient of the term (A^a)(B^b) in the expanded expression. Here, a and b are the exponents of A and B, respectively.

Outputs a list of nCr values.

Outputs a list of nCr, nPr, and r!.

Enter values for any two of the following: arc length (L), chord length (d), arc height (h), or radius (r). If you enter 0 for the remaining two (do not leave them blank), the program will calculate those two values and the central angle.The calculation uses the formulas L=r*θ, d=2*r*sin(θ/2), and h=r*(1-cos(θ/2)), and is performed using the Newton-Raphson method. *Support has been added for central angles greater than 180° and cases where there are two solutions. *January 2026* We have created a new version with significantly improved design and usability. Please feel free to use this one as well. https://sci.tea-nifty.com/blog/files/circlecalc.html

The formula is: Distance = sqrt((x1 - x2)² + (y1 - y2)²)

This is a program that calculates the circumference of an ellipse. The basic idea is to divide the radius into 1,000 equally spaced segments, identify the points where the horizontal lines at those segments touch the ellipse, and calculate the total length of the lines connecting those points. In short, it uses a polygon to obtain an approximate value for the circumference of the ellipse. For more details, please refer to the link below.

This calculates the required dimensions when shaping thin materials, such as paper, into a funnel shape.

Find the circumference of the circle.

Find the values of l, m, and n in the equation x² + y² + lx + my + n = 0, which passes through the points (a, b), (c, d), and (e, f), as well as the coordinates of the center of the circle and its radius.

Calculate the vertices of a regular polygon. To rotate the entire shape, enter an appropriate angle (in degrees) in the "Angle Offset" field.

Calculate the height, length, and width of an arc based on its radius and opening angle

The calculation results are based on the corrected route.

Calculate the other radius of an ellipse given its circumference and one of its radii (either the major or minor radius). *The circumference of an ellipse is given by the semi-major axis a, the eccentricity k = sqrt(1-b^2/a^2), and the second kind of complete elliptic integral E(k): L = 4a E(k) We use the Newton-Raphson method to solve this inverse problem.

Enter the major axis radius, minor axis radius, and central angle of the ellipse to calculate the arc length L, chord length d, and height h.

Calculate the area and perimeter of a rounded rectangle based on its width, height, and the radius of the rounded corners.

This is for the forum post titled "Formulas for the distance between a line and a point..."

Find the point of intersection of y = ax + b and y = cx + d.

Enter the coordinates of the vertices of triangle ABC, and the program will calculate the coordinates of the five centers of the triangle (centroid, incenter, circumcenter, orthocenter, and excenter).

If the sum of a pair of diagonals is known, Bretschneider's formula can be used, but here we are using only "length information."

X and Y coordinates and distance of an ellipse

Hyperellipse (Lame curve) |x/a|^n + |y/b|^n = 1 (a standard ellipse when n=2) We will calculate the circumference and area of this hyperellipse.

The function y = (√(cos(x)) * cos(400x) + √(|x|) - 0.4) * (4 - x²)^0.1 is a heart-shaped function (familiar from Google's graphing tool). Extending this, we plot the graph of y = (sqrt(cos(a*x)) * cos(b*x) + sqrt(abs(x)) - c) * (d - x*x)^e.

Find the arc opening angle, arc height, and arc width from the arc length and radius.

We know the sum of the interior angles of a regular polygon.

This is a tool for quickly finding the angles in the “Langley Problem” and “Integer-Angle Quadrilateral.” For more information on the “Langley Problem” and “Integer-Angle Quadrilateral,” please search Google for “biglobe Langley Problem” and refer to the page titled “Langley Problem, Integer-Angle Quadrilateral .”(Since I changed internet providers in 2026,) please note that the homepage is “Math and Gauss.”

We calculate the area using the central angle and the arc length.

Determine the parabola from three points. Assume that the axis of the parabola is vertical (parallel to the y-axis). The equation of the parabola is given in the form y = ax² + bx + c. Here, results where a = 0 are allowed. Output results that are straight lines as well.

This is for the "Circle Circumscribed Around a Rounded Rectangle" forum.

This is for the forum thread titled "On the Distance Between Two Lines."

Given the lengths of the six edges of a tetrahedron OABC, find its volume.

This is the formula for finding the height at which a frustum of a cone is divided into two equal parts—similar to the problem of determining where to cut a carrot to reduce its volume by half. If we consider the carrot to be a cone (R1 = 0), we simply need to cut it at the point where the ratio of the two parts is 1:4, starting from the thicker end.

Find the volume V of the shaded portion on the right side of the cross-section when a right circular cone with radius R and base angle δ is cut vertically (parallel to the central axis).

We'll calculate the area of a triangle using two sides and one angle.

Calculate the points of intersection between a circle and a line. Determine the number of points of intersection and their coordinates. The number of points of intersection is determined in the real xy-plane. This is based on the discriminant. The coordinates of the points of intersection are the solutions to a system of equations. These are output as complex numbers. Please check carefully to ensure that the imaginary unit \(i\) is not appended to the end of the output. Let the equation of the circle be \((x-a)^2 + (y-b)^2 = r^2\) and the equation of the line be \(lx + my + n = 0\)

Calculate the coordinates of a parabola

This is for the "Tangents and Curves" forum.

We will calculate the area of the triangle using the lengths of its two sides and the radius of the circumcircle.

Find the area of the circle

We will calculate the points of intersection between a parabola and a line. We will consider the real xy-plane (where the y-axis is vertical and the x-axis is horizontal). Assume that the axis of the parabola is vertical. Let the equation of the parabola be y = ax² + bx + c. Let the equation of the line be y = mx + n. The number of common points is determined by the discriminant of the quadratic equation after eliminating the y-term.

For the elementary geometry problem titled “Triangle with Integer Angles,” please search Google for “Biglobe Langley Problem” and refer to the “Math and Gauss” section on the top page. (I changed my internet service provider in 2026.)

When a right circular cone with radius R and base angle δ is cut by a plane that is tangent to the edge of the base and intersects the base at an angle γ, calculate the volume V of the portion to the right of the cut (the shaded area) and the volume Vu of the remaining portion(the portion to the left of the cross-section, not shaded) are to be calculated.

Calculate the volume V of the portion on the right side of the cross-section (the shaded area) when a right circular cone with radius R and base angle δ is cut by a plane that intersects the base at an angle γ.

We will calculate the tangents to an ellipse. Given that the slope of the tangent is m, we will find its equation. We will calculate the constant terms in the equations of the two tangents. We assume that the equation of the ellipse is in standard form.

This is a program to calculate the area of a parallelogram.

Calculate the volume V of the section to the right of the cross-section (the shaded area) when a right circular cone with radius R and base angle δ is cut by a plane intersecting the base at an angle γ. Since this calculation only considers cases where the cutting angle γ exactly matches the base angle δ, the input value for γ should generally be the same as δ.

Calculate the volume V of the portion to the right of the cross-section (the shaded area) when a right circular cone with radius R and base angle δ is cut by a plane intersecting the base at an angle γ.

This is a program to calculate the area of a parallelogram.

Determine the parabola. Given the position of the axis at p and two distinct points, determine the parabola that passes through those two points and has the axis at x = p. Let the equation of the parabola be of the form y = ax² + bx + c. After completing the square, let the equation be of the form y = a(x - p)² + q.

This is a program to calculate the area of a square.

Let the coordinates of point A be (x1, y1). Let the coordinates of point B be (x2, y2). Calculate the x and y coordinates of the point that divides line segment AB in the ratio m:n. Calculate the x and y coordinates of the point that divides line segment AB in the ratio m:n from the outside.

These are the angles between two cells with extended dihedral angles.

Find the area of the rectangle.

"The Langley Type B Problem." For more information on this, please search Google for "Biglobe Langley problem" and check out the "Math and Gauss" section on the first page. (I switched internet service providers in 2026.)

Find the equation of the tangent line drawn from a given point to a parabola. Let the equation of the parabola be y = ax² + bx + c. Let p₁ be the x-coordinate of point of tangency ①. Let q₁ be the y-coordinate of point of tangency ①. Let the equation of tangent line ① be y = m₁x + n₁. Let the x-coordinate of point of contact ② be p₂. Let the y-coordinate of point of contact ② be q₂. Let the equation of tangent line ② be y = m₂x + n₂.

Calculate the equation of the tangent line to the ellipse. The equation of the ellipse is in standard form.

We will calculate the intersection point of an ellipse and a line. The ellipse is in standard form. Let the equation of the ellipse be in the form ((x^2)/(a^2)) + ((y^2)/(b^2)) = 1. Let the equation of the line be in the form lx + my + n = 0.

Calculate the number of common points between two parabolas. Consider the xy-plane (the real plane with the y-axis as the vertical axis and the x-axis as the horizontal axis). Let the equation of Parabola 1 be y = ax² + bx + c. Let the equation of Parabola 2 be y = lx² + mx + n. Both parabolas are vertical. The coefficients and constant terms are real numbers. If there are common points, calculate their x-coordinates and y-coordinates. If there is one common point, output it to the x1 and y1 fields. If there are two common points, also output them to the x2 and y2 fields. In this exercise, we will not determine whether the common points are points of intersection or points of tangency. Additionally, we will omit the display of results regarding whether the two parabolas are identical (completely overlapping) or whether they have the same axis positions but different vertex heights (parallel above and below).

Find the x and y coordinates of the intersection point of two lines expressed in Hesse's standard form. Let θ be the slope of Line 1. Let φ be the slope of Line 2. Let h1 be the distance from the origin to the starting point of the unit normal vector of Line 1. Let h2 be the distance from the origin to the starting point of the unit normal vector of Line 2. Positive values correspond to the direction of the unit normal vector, and negative values correspond to the opposite direction.

Calculate the length of a line segment along a parabola. Consider a certain interval along the x-axis and divide it into n equal parts. For the starting point, ending point, and intermediate x-coordinates within that interval, calculate the y-coordinates of the parabola, and then calculate the length of the line segment connecting the points (x, y) with a line.

A parabola is determined by its vertex and another point through which the curve passes. Let the equation of the parabola be of the form y = ax² + bx + c.

We will calculate the length of a line segment that follows a cubic curve. Let the equation be of the form y = f(x), where f(x) = ax³ + bx² + cx + d. We divide the interval from the starting point’s x-coordinate to the ending point’s x-coordinate along the x-axis, calculate the value of f(x) at each point, and connect them with a line. We will find the length of this line.

Given the equation x² + y² = a, find the maximum and minimum values of k for the equation x + y² = k. Assume that a is a positive number not equal to 0.　

Calculate the radius from the volume of a sphere.

When an ellipse is expressed in polar coordinates, we perform the relevant calculations.

We will calculate the equation of a parabola with a slanted axis.