The beautiful math trick to compare HUGE numbers

Ajouté : 2026-05-23

[00:00:00]Which is larger?

[00:00:01]99 factorial or 50 to the 99th power?

[00:00:05]Let's expand both.

[00:00:07]99 factorial is the product from 1 to 99, while the other is just 99 50s.

[00:00:14]Let's pull out the outer numbers to compare them. Look at this.

[00:00:18]1 * 99 is exactly 50 squared minus 49 squared.

[00:00:24]Geometrically, if we take a perfect 50 by 50 square and cut out a 49 squared corner, the remaining area transforms into our rectangle.

[00:00:35]It is clearly less than a pure 50 squared. This rule beautifully applies to all other pairs. 2 * 98, 3 * 97, all the way to 49 * 51.

[00:00:48]Every single pair is strictly less than 50 squared.

[00:00:52]In total, we have 49 such pairs. Now, let's assemble everything.

[00:00:59]99 factorial is the product of these 49 pairs multiplied by the lonely 50 in the center.

[00:01:06]Replacing the pairs with 50 squared gives us less than 50 to the 98th power times 50, which equals exactly 50 to the 99th power. The proof is absolute. 50 to the 99th power wins. Math is truly powerful.