## Comparing huge numbers

Comparing huge numbers is often times not easy since you practically cannot write them out to compare their digits. (By ‘compare’ here we mean telling which number is greater (or smaller).) So it can sometimes be a challenge to determine.

Notation: recall that N! stands for “N factorial,” which is defined to be the product of all positive whole numbers (starting with 1) up to and including N. (E.g., 5! = 120.) And as usual, M^{n} stands for M raised to the power of n (sometimes also written as M^n).

Here are a couple examples of huge numbers (which we won’t bother writing out!) that aren’t so easy to compare but one of which is larger, just not clear which. I don’t have a technique except maybe in an *ad hoc* manner.

In each case, which of the following pairs of numbers is larger?

(1) (58!)^{2} and 100!

(2) (281!)^{2} and 500!

(3) (555!)^{2} and 1000!

(4) 500! and 10^{1134}

(5) 399! + 400! and 401!

(6) 8^{200} and 9^{189}

(The last two of these are probably easiest.)

Have fun!