Those are very rough approximations: 24! is about 60% of 10^24 and 25! is about 155% of 10^25.

It's simpler to remember that 24.5! ≈ 10^24.5 (accurate to within +/- 3%; see https://www.wolframalpha.com/input?i=solve+x%21+%3D+10%5Ex)

The old saying:

    2 + 2 = 5 for large values of "2"

It's an approximation. The sentence is perhaps confusing since it might imply that n! ≈ 10^n but that's not the case, the article uses 24 and 25 as the cut point where the approximation is true.

The linked post is more informative: https://www.johndcook.com/blog/2023/06/22/mentally-approxima...

A simpler way to think of the approximation is that 24! multiples together all fifteen numbers from 10 to 24, all of which add at least another power of 10, and you get approximately another nine powers of ten from multiplying in 1 through 9. You end up around 6.2e23 which rounds up to the approximation.

Going up to the next factorial bumps it up (approximately) another power of ten since you are multiplying by 2.5 and 10. In reality it’s around 1.5e25 which is also a reasonable approximation.

Over simplification: a number times 25 is closer to the number times 10 than it is to the number times 100.

Because 25! = 25*(24!), and 25 itself is 100/4. So 25! = 10*(10/4)*24! Now since we are given 24! ~10^24, it follows that 25! ~ (10/4)*10^25. But 10/4 is just a bit over 2, not a strong enough multiplicative factor to break the "approximately equals relationship". Therefore 25! ~ 10^25. QED

Took me a while to get it. The point is more that if you want x! ≈ 10^x then these are as good as you'll get. The ≈ is very very approximate though!

They are not equal, nor approximate either. In this text they kinda use the approximation sign (≈) more like "same order of magnitude". Because if you do a comparison you find out that it's a 60% error between those numbers.