What is the factorial of 140?

1 Answer
Dec 5, 2015

140! =
1346201247571752460587607385894161555835585114819396719
0051391468057460367090535696797920946629681836680869
097041958983702264048370902871114013579941370766400374
327741701139895604871545254810788060989321379840000000
000000000000000000000000000

But as a rule, you'd just write 140!.

Explanation:

In general, the factorial of an integer n is given by
n! = n(n-1)(n-2)...(2)(1)
that is, the product of all integers from 1 to n.

While it may occur in a problem, the number 140! is in general too large to be worth actually calculating. To get some idea of why, let's look at how quickly the factorial function grows.

1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10! = 3628800
and so on.

By the time we calculate 10! we are already at a 7 digit number, and each subsequent multiplication is going to raise the number of digits by at least 1 (more as we get further along). It turns out, if you work it out, that the number 140! has 242 digits.

By comparison, the dollar value of all of the money on earth has fewer than 16 digits, the number of possible board positions in chess has fewer than 48 digits, and the number of atoms in the observable universe is estimated to be roughly 81 digits. If everyone on earth was to add 1,000,000,000,000 (one trillion) to a collective pot every second, the universe would die long before they reached even 1% of 140!

While we can certainly calculate the value of 140!, either through use of a calculator or computer, or by long, tedious multiplication by hand, there is little practical value in doing so in most cases. Instead, if it occurs in a problem, it is preferable to leave it in the form 140!.

Given all that, though, here is what 140! looks like in decimal digits:

140! =
1346201247571752460587607385894161555835585114819396719
0051391468057460367090535696797920946629681836680869
097041958983702264048370902871114013579941370766400374
327741701139895604871545254810788060989321379840000000
000000000000000000000000000