ABC Tools

Big Number Calculator, Explained

After reading this you will know how to compute exact integer arithmetic with thousands of digits, read the results, and avoid the rounding traps that ordinary calculators fall into.

Type 2^512 - 1 and press Calculate. You get a 155-digit integer, every digit correct, no rounding and no scientific notation:

13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084095

A typical floating-point calculator would report this as 1.34078079299426e+154 and throw away the last 138 digits. This tool keeps all of them.

What arbitrary precision means

Ordinary hardware arithmetic stores a number in a fixed 64 bits. That gives you exact integers only up to 2^{63} - 1 = 9223372036854775807, about 19 digits. Beyond that, either the value overflows or it is stored as a floating-point approximation with roughly 15 to 16 significant digits.

Arbitrary precision (also called bignum) drops the fixed width. An integer is stored as a list of digit-chunks and grows to whatever size it needs. Adding two 4000-digit numbers produces a 4000- or 4001-digit answer, with no loss. The cost is speed: bignum arithmetic runs in software, so a multiplication of two 10000-digit numbers takes measurable time where a 64-bit multiply is a single instruction. For the sizes you will meet here, the wait is a fraction of a second.

Every operator here returns an exact result. There is no hidden rounding step anywhere. If an answer looks surprising, it is arithmetically true, not a display artifact.

The operators and functions

The expression grammar covers the operators you expect plus a few number-theory helpers. Precedence follows normal math: ^ binds tightest, then × and division, then + and −, and parentheses override everything.

+ − ×
Addition, subtraction, multiplication of integers. Results are always exact.
÷
Exact division. It returns a fraction in lowest terms, not a decimal. So 7 ÷ 2 gives 7/2, and 6 ÷ 3 gives 2.
//
Floor division. 7 // 2 = 3. The result is the largest integer not greater than the true quotient.
%%
Modulo, the remainder that pairs with floor division. 7 %% 2 = 1.
^ or **
Exponentiation. 2^10 = 1024. Exponents can be large; the answer grows fast.
fact(n)
Factorial, the product 1 \cdot 2 \cdot 3 \cdots n.
isqrt(n)
Integer square root: the largest integer m with m^2 \le n. So isqrt(1000) = 31 because 31^2 = 961 and 32^2 = 1024.
gcd(a, b), lcm(a, b)
Greatest common divisor and least common multiple.
abs(n)
Absolute value.

Why factorials and powers explode

The reason you need thousands of digits is that a few operations grow far faster than intuition suggests. The number of digits in a value n is about \log_{10} n + 1.

\text{digits of } n! \approx \sum_{k=1}^{n} \log_{10} k

Each term \log_{10} k is how many digits the factor k contributes. Add them up and the total climbs steeply. At n = 100 the sum is about 158, so fact(100) has 158 digits. At n = 1000 it is about 2568 digits. Powers behave similarly: the digit count of 2^n is n \log_{10} 2 + 1 \approx 0.301 n + 1, so 2^512 has 0.301 \times 512 + 1 \approx 155 digits, matching the hook example.

The number of digits in n! rises faster than linearly. By n = 1000 you already need 2568 digits, which no fixed-width type can hold.

Worked example: fact(100)

Computing 100 factorial exactly

Enter fact(100) and press Calculate. The result is:

93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

  1. Count the digits: there are 158, which agrees with the estimate \sum_{k=1}^{100} \log_{10} k \approx 158 from the previous section.
  2. Look at the tail: it ends in 24 zeros. Those trailing zeros count how many times 10 divides the number, which equals the number of factors of 5 among 1, 2, \dots, 100.
  3. Check the factor-of-5 count directly: \lfloor 100/5 \rfloor + \lfloor 100/25 \rfloor = 20 + 4 = 24. That is exactly the 24 zeros you see, a claim you can verify by eye.

This is the value of an exact tool. The trailing-zero structure is real information about the factorization, and it survives only because nothing was rounded.

For reference, the digit count of n! for a few values: n=10 gives 7 digits, n=50 gives 65, n=100 gives 158, n=500 gives 1135, n=1000 gives 2568. The count rises faster than a straight line because each new factor adds more digits than the last.

Reading fraction and floor results

Division is the one place where the output type changes. Plain ÷ gives an exact fraction, so 1 ÷ 3 returns 1/3, not 0.3333. That is the honest answer, since one third has no finite decimal form.

When you want an integer quotient and remainder instead, use floor division and modulo. They satisfy the identity below for every pair of integers with b \ne 0:

a = (a \mathbin{//} b) \cdot b + (a \mathbin{\%\%} b)

Here a \mathbin{//} b is the floor of the true quotient and a \mathbin{\%\%} b is the leftover. Check it with a = 17, b = 5: 17 // 5 = 3 and 17 %% 5 = 2, and indeed 3 \times 5 + 2 = 17. The remainder always has the same sign as the divisor, so -17 %% 5 = 3 and -17 // 5 = -4, since -4 \times 5 + 3 = -17.

When to use this and when not to

Reach for exact integer arithmetic when correctness of every digit matters: cryptographic exponents, factorial and binomial counts, checking a modular identity, or confirming a hand computation. Use it whenever a normal calculator would show e+ notation and you actually need the digits.

This is an integer and fraction engine, not a decimal one. It will not give you \pi to 1000 places or evaluate sin(2). If you type a decimal point where the grammar expects an integer, expect an error rather than a rounded float.

For irrational values, transcendental functions, or floating-point work, use a scientific calculator instead. For turning a huge exact result into readable scientific notation or a fixed number of significant figures, see the tool linked in the next section.

Common mistakes

Three errors account for most confusing outputs.

  • Expecting a decimal from ÷. If you write 22 ÷ 7 and want 3.142857..., you will instead get 22/7. That is correct; the tool refuses to round. Convert the fraction elsewhere if you need digits.
  • Confusing %% with a percent sign. 50 %% 7 is the remainder 1, not "50 percent of 7". There is no percent operator here.
  • Underestimating growth. An expression like fact(fact(10)) asks for the factorial of 3628800, a number with over 22 million digits. That is legitimate but slow and hard to display. Sanity-check the digit count with the formulas above before you run it.

Related tools

Once you have an exact integer, these calculators pick up where this one stops.

Frequently asked questions

How many digits can it handle?

There is no fixed cap on correctness. Results with several thousand digits compute in well under a second. Beyond a few hundred thousand digits the computation and the display both slow down, so treat that as a practical ceiling rather than a hard limit.

Why does 1 ÷ 3 give a fraction instead of 0.3333?

Because one third has no finite decimal expansion. Rounding it would introduce error, and this tool never rounds. It reports 1/3, the exact value. Use the fraction converter to get repeating-decimal digits.

What is the difference between // and ÷?

Exact division ÷ keeps the full value as a fraction, so 7 ÷ 2 is 7/2. Floor division // throws away the fractional part and rounds down, so 7 // 2 is 3. Pair // with %% to recover the remainder.

Can I compute binomial coefficients here?

Yes, as a ratio of factorials. For example \binom{52}{5} is fact(52) ÷ (fact(5) × fact(47)), which returns the exact integer 2598960. The dedicated combinatorics tool does this in one step and handles larger inputs more cleanly.

Does it handle negative numbers and signs correctly?

Yes. Modulo follows the divisor's sign, so -17 %% 5 = 3, and the identity a = (a \mathbin{//} b) \cdot b + (a \mathbin{\%\%} b) holds for negatives too. Use abs when you want the magnitude only.

Try it yourself in the tool

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