Big Number Calculator

About the Big Number Calculator

Ordinary calculators and spreadsheets keep only about 15 significant digits — Excel’s limit is 15 — so 2^100 or 12345678901234567890 × 98765432109876543210 comes back rounded. This calculator keeps every digit. Type whole numbers or decimals of up to 100,000 digits each and add, subtract or multiply them exactly; divide to get the quotient with its remainder and the decimal value to as many as 10,000 places; raise to whole-number powers (2^1000 has all 302 digits); take square, cube and higher roots; and find A mod B, the greatest common divisor with Bézout’s coefficients, and the least common multiple. Answers can have up to a million digits.

Factorial n! gives the exact value of n! for n up to 100,000 — 100,000! has 456,574 digits — with its length, the number of trailing zeros by Legendre’s formula, the double factorial n!! and Stirling’s approximation with its error. Copy a result, download it as a text file, or use it as the next first number. The calculations run in your browser, in the background, so even very long ones never freeze the page.

Common uses

• Checking the exact value of a huge power, product or factorial for homework, a puzzle or a programming exercise.
• Testing big-integer code (RSA-style numbers, hash values, modular arithmetic) against exact answers.
• Counting arrangements: 52! ≈ 8.07 × 10⁶⁷ ways to shuffle a deck of cards.
• Getting thousands of decimal places of a quotient or a square root.

How the arithmetic stays exact

Each number is held as a whole number of any length (JavaScript’s BigInt) together with the position of its decimal point, so 12.5 is 125 with one decimal place. Adding and subtracting line the decimal points up; multiplying multiplies the digits and adds the decimal places; dividing works on whole numbers too. These are the multiple-precision methods described by Donald Knuth in The Art of Computer Programming, Volume 2, §4.3. Nothing is converted to floating point, so no digit is lost or invented. The only rounding is where you ask for a decimal that does not end, and that last digit is rounded half away from zero.

Trailing zeros of n! — Legendre’s formula

Legendre’s formula gives the exponent of a prime p in n! as ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + …, stopping when the power of p is bigger than n. A trailing zero needs a factor 10 = 2 × 5, and n! always has more factors 2 than 5, so the number of trailing zeros is the exponent of 5. For 1,000! that is 200 + 40 + 8 + 1 = 249. The calculator works it out this way, checks it against the exact digits, and lists the exponent of every prime in n! for n up to 2,000.

Stirling’s approximation

n! ≈ √(2πn) · (n/e)ⁿ. The estimate is always slightly low, by a relative error close to 1/(12n): about 0.83% for n = 10 and 0.083% for n = 100. Multiplying by (1 + 1/(12n)), the next term of Stirling’s series, removes most of the remaining error. Because the calculator also has the exact value, it shows how far off the estimate is.

Remainders of negative numbers

For negative numbers “A mod B” has three common definitions. The mathematical (Euclidean) remainder is never negative: −7 mod 3 = 2. The % operator of JavaScript, C, C++, Java and C# keeps the sign of A: −7 % 3 = −1. Python’s % operator and Excel’s MOD function keep the sign of B: 7 mod −3 = −2. The calculator gives the Euclidean remainder and lists the others whenever they differ.

Double factorials

n!! multiplies every other number down to 1 or 2: 9!! = 9 × 7 × 5 × 3 × 1 = 945 and 10!! = 10 × 8 × 6 × 4 × 2 = 3,840, with 0!! = 1. It is not (n!)!. Double factorials turn up in the volumes of spheres in many dimensions and in integrals of powers of sine and cosine.

Limitations

• Numbers you type can have up to 100,000 digits and results up to 1,000,000 digits. GCD and LCM accept up to 20,000 digits (Bézout’s coefficients up to 5,000), factorials go up to n = 100,000 and decimal answers up to 10,000 places.
• Exponents and root indices must be whole numbers. For fractional or decimal exponents use the exponent calculator; for fractions such as 1/3 use the fraction calculator.
• Very long results can take a few seconds in some browsers. The page stays usable and Stop cancels the calculation.
• The digits box shows the start and the end of results longer than 30,000 characters until you ask for every digit; Copy and Download always give the whole number.

Privacy

Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.