🔢Big Number Calculator
Perform addition, subtraction, multiplication, division, powers, square roots, factorials, modulo, GCD, and LCM on integers and decimals of any size. Supports E-notation (2.5e50, 3E100). Uses BigInt arbitrary-precision arithmetic — no floating-point rounding errors.
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Result
111111111011111111100
Digit Count — Input vs. Result
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Big Number Calculator: Exact Arithmetic Beyond Standard Limits
Standard floating-point arithmetic (IEEE 754 double precision) has only 53 bits of significand — roughly 15–16 significant decimal digits. Any integer above 9,007,199,254,740,992 (2⁵³) is rounded. A Big Number Calculator bypasses this with arbitrary-precision integer arithmetic, computing exact results with hundreds or millions of digits limited only by memory.
Formula (division): result = floor(X_coeff × 10^(precision + Y_scale − X_scale) / Y_coeff)
| Operation | Example | Result (exact) |
|---|---|---|
| 20! | 20 × 19 × … × 1 | 2,432,902,008,176,640,000 (19 digits) |
| 2^64 | 2 × 2 × … (64 times) | 18,446,744,073,709,551,616 (20 digits) |
| √2 (20 dp) | Newton–Raphson BigInt | 1.41421356237309504880… |
Our big number calculator uses JavaScript's native BigInt type — which supports integers of arbitrary size — combined with scaled-integer BigDecimal arithmetic for decimal operations. Every integer result is exact. For division and square roots, precision is configurable up to 200 decimal places.
Why Floating-Point Fails for Large Numbers
IEEE 754 double-precision floats store numbers as ±mantissa × 2^exponent, where the mantissa is 52 bits plus an implicit leading 1 — giving 53 bits of precision ≈ 15.95 decimal digits. Numbers outside this resolution are silently rounded. The classic symptom: in standard JavaScript, 9007199254740993 === 9007199254740992 returns true — one is added, but the result is unchanged because the precision ran out. BigInt (introduced in ES2020) solves this for integers: 9007199254740993n === 9007199254740992n returns false as expected.
For decimal operations, scaling converts the problem back to integer arithmetic. To compute 1÷3 with 20 decimal places: multiply 1 by 10²⁰, then perform integer division by 3, yielding 33333333333333333333, then insert the decimal point at position 20 to get 0.33333333333333333333. No floating-point is involved — the result is as accurate as the number of digits requested.
E-Notation for Extremely Large or Small Inputs
Scientific notation allows concise entry of very large or very small numbers. The input 3.5e50 means 3.5 × 10⁵⁰ = 350...0 (50 zeros minus 1). The calculator parses E-notation to exact integers or scaled integers before any arithmetic, so operations like 2.5e50 + 1.5e50 produce the exact integer 4 × 10⁵⁰. Negative exponents like 1e-10 are stored as scaled integers (coeff = 1, scale = 10), enabling exact decimal arithmetic on very small quantities.
Factorials Grow Catastrophically Fast
While 10! = 3,628,800 fits in a 32-bit integer, 100! has 158 digits, 1,000! has 2,568 digits, and 10,000! has 35,659 digits — none of which can be represented with standard floating-point. The exact value of 52! ≈ 8.07 × 10⁶⁷ (the number of ways to shuffle a deck of cards) requires 68 digits for exact representation. Stirling's approximation ln(n!) ≈ n·ln(n) − n gives the order of magnitude, but only BigInt factorial gives the exact value down to the last digit.
Frequently Asked Questions
What is the largest number this calculator can handle?
JavaScript BigInt has no fixed upper limit — it grows as large as available memory allows. Practically, integers with millions of digits are computable (factorials up to 10,000 and powers up to exponent 10,000 are supported). Division and square root results are capped at 200 decimal places. 10,000! has 35,659 digits; 2^10000 has 3,011 digits — all computed exactly.
What is E-notation and how does this calculator handle it?
E-notation (scientific notation) writes numbers as mantissa × 10^exponent. Examples: 2.5e19 = 2.5 × 10¹⁹ = 25,000,000,000,000,000,000; 1e-5 = 0.00001. The calculator converts E-notation to exact BigInt scaled integers before computation — so operations on E-notation inputs are exact. Both uppercase (2.5E19) and lowercase (2.5e19) are accepted. Negative exponents become scaled integers (1e-5 → coeff=1, scale=5).
Why does 0.1 + 0.2 not equal 0.3 in standard programming?
Because 0.1 and 0.2 cannot be represented exactly in binary floating-point. In base 2, 1/10 is a repeating fraction (like 1/3 in base 10), so the nearest representable double is 0.1000000000000000055511151231257827021181583404541015625. When 0.1 and 0.2 (both slightly off) are added, the result is 0.30000000000000004, not 0.3. This calculator solves this for integer and scaled-integer arithmetic: 0.1 is stored as { coeff: 1, scale: 1 }, 0.2 as { coeff: 2, scale: 1 }, and their sum is exactly { coeff: 3, scale: 1 } = 0.3.
Can I compute very large powers like 2^1000?
2^1000 = 10715086071862673209484250490600018105614048117055336074437503883703510511249361224931983788156958581275946729175531468251871452856923140435984577574698574803934567774824230985421074605062371141877954182153046474983581941267398767559165543946077062914571196477686542167660429831652624386837205668069376. This 302-digit number is computed exactly. Exponents up to 10,000 are supported, though 2^10000 has 3,011 digits and may take a second to compute.
How does the square root calculation work for large numbers?
Square roots use the Newton–Raphson integer method: starting from an initial estimate, repeatedly compute x_{n+1} = (x_n + M/x_n) / 2 where M is the input scaled to give the desired decimal precision. The iteration converges quadratically (each step roughly doubles the correct digits) and terminates when x_{n+1} ≥ x_n, which happens when x_n = floor(√M). For √2 with 20 decimal places: M = 200000000000000000000 (2 × 10²⁰), integer square root = 14142135623730950488, insert decimal to get 1.41421356237309504880.