How do you find 1000 factorial?

Answer 1

This goes well beyond the range of a scientific calculator (almost 10100), or most computer software (about 10300). Some special software can handle an arbitrary number of digits; for example, in the Java programming language, you can use the Bigint class.

With a calculator, the following will be extremely slow; try to do it in Excel: calculate the logarithm of each factor, add them, then take the antilog. In Excel:

In Cell A1, put the number 1. In A2, put the formula =A1+1. This should give you the number 2. Copy the formula to cells A3...A1000. This should give you all the numbers you have to multiply: 1, 2, 3, 4, 5, ... 1000.

In Cell B1, calculate the base-10 logarithm of A1: =log(A1). Copy to cells B2...B1000.

In Cell C1, put the following formula: =B1. In Cell C2, the formula should be =C1+B2. Copy this down, to cells C3...C1000.

This last column gives you the antilog (base 10) of the factorial. You can easily verify this by putting, in cell D1, the formula =10^C1, and copying down a few cells. This will give you factorials you can easily verify, for small numbers - for example, the factorial of 5 should be 120, if you did all the steps correctly.

However, for the larger numbers, you can no longer take the antilog, because you will get numeric overflow.

Let's take the factorial of 5 as an example of how to handle this. The sum of the logs is 2.07918124604762. Separate this into two parts, 2 and 0.07918.... Calculate 100.07918, but don't calculate (in Excel) 102, since this is the part that will give you numeric overflow for larger factorials. The first part is 1.2, so the expression is 1.2 x 102.