Big Number Calculator
Performs exact arithmetic on very large numbers: add, subtract, multiply, divide, powers, factorials, square roots, GCD and LCM with up to 1,000 digits per number and no floating-point rounding. Choose an operation, enter two numbers, and get every digit.
Exact Arithmetic on Very Large Numbers
Big Number Formula
Where:
- \(V\) — every digit of the number written as one exact BigInt integer (up to 1,000 digits)
- \(S\) — the decimal scale: \(x = V/10^S\), so 2.5 is stored as 25/10 and 0.125 as 125/1000
- \(s\) — the common scale that aligns the two operands for addition, subtraction, remainder, GCD and LCM
Because each input is rewritten as a scaled integer, all ten operations run entirely in exact integer arithmetic and the decimal point is restored only for display. Multiplication, addition, powers, factorials, GCD and LCM are always exact; division and square root round half-up at the requested decimal place, and the result box says so.
Big Number – Calculation Example
Example: the default inputs, with Operation set to a × b (multiply): a = 123456789012345678901234567890 and b = 987654321098765432109876543210.
\[ 123456789012345678901234567890 \times 987654321098765432109876543210 = 121\,932\,631\,137\,021\,795\,226\,185\,032\,733\,622\,923\,332\,237\,463\,801\,111\,263\,526\,900 \]
Result: 121932631137021795226185032733622923332237463801111263526900 — an exact 60-digit product, reported as 60 significant digits and equal to 1.2193263114 × 1059 in scientific notation. A floating-point calculator would keep only the first 15–17 of those digits.
How to Use the Big Number Calculator
- Operation — pick one of the ten operations from the dropdown: add, subtract, multiply, divide, remainder (mod), power, square root, factorial, GCD or LCM.
- First number (a) — enter up to 1,000 digits: an integer, a decimal with a dot or a comma, or scientific notation such as
3.2e50or-1.7E-30. For √x this is x; for n! this is n. - Second number (b) — the second operand in the same format. For x ^ y it must be a whole number from 0 to 10,000; for √x and n! the field is hidden because it is not used.
- Decimal places (÷ and √) — set how many decimal places the quotient or square root should show (0 to 10,000). This field appears only for those two operations.
- Read the results: the full Result with a copy button, Significant digits in the result, Scientific notation with 11 significant digits, the Exact or rounded? verdict, and a note on How the result was computed.
Operations at a Glance
| Operation | Second number (b) | Decimal places | Result |
|---|---|---|---|
| a + b, a − b, a × b | used | not used | always exact |
| a ÷ b (divide) | used, cannot be 0 | 0–10,000 | exact if possible, otherwise rounded half-up and flagged as rounded |
| a mod b (remainder) | used, cannot be 0 | not used | exact; the sign follows the first number |
| x ^ y (power) | whole number 0–10,000 | not used | exact (x0 = 1, and 00 = 1 by convention) |
| √x (square root) | hidden | 0–10,000 | exact for perfect squares, otherwise rounded half-up |
| n! (factorial) | hidden | not used | exact, for n = 0–5,000 |
| GCD(a, b), LCM(a, b) | used | not used | exact; decimals are allowed |
Why Exact Big-Number Arithmetic Matters
Ordinary calculators and spreadsheets store numbers as 64-bit IEEE 754 floating-point values, which hold only about 15–17 significant decimal digits. Everything beyond that is silently rounded: on most systems 1e20 + 1 evaluates to exactly 1e20, and a 60-digit product keeps at most 17 correct digits. The rounding is invisible on screen but real, and it compounds through every subsequent calculation — a serious problem when you need to verify a large product, count combinations, or check an algorithm digit by digit.
This calculator avoids floating point entirely. Each input is rewritten as one BigInt integer \(V\) with a decimal scale \(S\), and every operation — addition, multiplication, exponentiation by squaring, Newton-iteration square roots, the Euclidean algorithm for GCD, term-by-term factorials — runs on those integers with no limit of 15–17 digits. The only place rounding can occur is division and square root at the requested decimal place, and the Exact or rounded? box tells you plainly which case you are in, so an “Exact” verdict means every single digit shown is correct.
The remaining output boxes help you handle long answers: Significant digits in the result gives the exact length, and Scientific notation shows the magnitude with 11 significant digits — handy for comparing sizes, and easy to convert further with the scientific notation converter. Typical uses include checking big multiplications by hand, combinatorics, verifying homework on huge numbers, and any task where a double-precision tool would quietly round your answer.
Limits, Conventions and Practical Tips
- Input size: each number may have up to 1,000 digits before the exponent, with an exponent up to ±10,000. Results longer than 1,000,000 digits are refused with a warning rather than freezing the page.
- Powers and factorials: for x ^ y the exponent must be a whole number from 0 to 10,000 (negative powers give non-terminating fractions and are rejected); n! accepts whole numbers from 0 to 5,000. For smaller n, the dedicated factorial calculator shows the step-by-step product.
- Division and roots: a ÷ b is rounded half-up to the requested decimal places, and the calculator reports whether the division came out exact. √x first checks whether x is the square of a terminating decimal; if not, the root is rounded half-up and flagged.
- Remainder sign: a mod b takes the sign of the first number, like the % operator in JavaScript and C. If you need the always-non-negative (Euclidean) remainder, adjust the result by adding |b| when it is negative.
- GCD and LCM on decimals: both operands are scaled to a common power of ten first, so GCD(2.5, 1.5) = 0.5 works naturally. For everyday whole numbers you can also use the GCF calculator.
- Modular arithmetic: if you only need x ^ y mod m, the modular exponentiation calculator avoids building the huge power in the first place.
- Formatting: use either a dot or a comma as the decimal separator, but never both in the same number; minus signs and spaces are cleaned up automatically. Results use an ASCII minus so they paste cleanly into other tools, and every answer has a Copy result button.
Frequently Asked Questions
How do I multiply big numbers by hand?
Split each number into chunks of 3–4 digits from the right, multiply every chunk of the first number by every chunk of the second, track the place value of each pair (chunk \(i\) of one times chunk \(j\) of the other contributes at position \(i+j\)), and add the partial products column by column with carries. The calculator is the fastest way to verify your result: enter both numbers, choose a × b, and every digit of the exact product appears with a copy button.
Why does a normal calculator give a wrong answer for large numbers?
Ordinary calculators and spreadsheets use 64-bit IEEE 754 floating point, which stores only about 15–17 significant decimal digits — everything past that is silently rounded. For example, on most systems 1e20 + 1 evaluates to exactly 1e20. This happens invisibly on screen but compounds through later steps. This calculator uses exact BigInt integers instead, so a 60-digit product, a huge factorial or a long power comes out digit-for-digit correct.
What is the largest number this calculator can handle?
Each input may have up to 1,000 digits before the exponent, and the exponent may range from −10,000 to +10,000. So the largest accepted value is close to 1e11000. The result itself is refused once it would exceed 1,000,000 digits — for example a power that grows too large — and the calculator shows a warning instead of freezing, so shrink the numbers or lower the exponent.
Can it handle decimal numbers like 2.5 and 0.125, not just integers?
Yes. Every input is rewritten internally as an exact scaled integer, for example 2.5 = 25/10. Addition, subtraction, multiplication, powers, GCD, LCM and remainder all accept decimals and scientific notation such as 3.2e50. Even the GCD and LCM work on decimals: GCD(2.5, 1.5) = 0.5. Factorial and the exponent need whole numbers, since non-integer cases give non-terminating fractions that cannot be written exactly.
How many decimal places can division and square root show?
Both a ÷ b and √x let you request anywhere from 0 to 10,000 decimal places, and the calculator first checks whether the exact answer terminates. If 1 ÷ 8 gives 0.125, the result is flagged exact; if the quotient or root continues forever, such as 1 ÷ 3 or √2, the answer is rounded half-up at your requested place and the Exact or rounded? box says so, so you always know whether the last digit was rounded.
What is the sign of the remainder a mod b?
The remainder follows the sign of the first number, exactly like the % operator in JavaScript and C: −7 mod 3 = −1, while 7 mod −3 = 1. If you need the always-non-negative (Euclidean) remainder used in number theory and modular arithmetic, add |b| whenever the result comes out negative — for −7 mod 3 that gives −1 + 3 = 2.
Can I enter a comma as the decimal separator?
Yes — you can use either a dot or a comma, so 3,5 is accepted just like 3.5. Just never mix both in the same number: 3.5,5 triggers an error. Spaces and typographic minus signs are cleaned up automatically, and results use a plain ASCII minus sign, so they paste cleanly into spreadsheets, code editors and other calculators without confusing them.