Here’s the trick I would use if I needed to calculate a factorial and didn’t have a computer on hand. It has a powers of 2 bias, so lg x refers to the base-2 logarithm.

7! = 5040 ~ 2^12

lg 8 + lg 9 + ... lg 15 has 8 terms of “about 3.5”, so is about 2^28.
lg 16 + ... lg 31 has 16 terms of “about 4.5”, so it’s about 2^72.
lg 32 + ... lg 52 has 21 terms of “about 5.33”, so it’s about 2^112.

So you get a total of around 2^224. The actual answer is 2^225.56..., so it’s not bad for an approximation can be done on a bus without a calculator.

Someone else has taken the title literally and computed the error when you divide by 3000 instead of (54x53). However, that’s not the point of the article.

You can get a handle on the error in the article by identifying all the approximations and estimating from there. I’ll be doing that in a follow-up, but the main error comes from using e ~ 2.7. It’s closer to 2.72. That error is about 0.7%. When we raise to the power of 54 that gives an error of about 40% to 50%.

But that is offset by the approximation that 2^10 ~ 10^3, and raising that to the power of 5. It’s an interesting and useful exercise to refine the original by chasing down estimate of the errors. I might do that in a follow-up post.

By the definition of factorial as 1×2×…×n, you can get 52! exactly if you divide 54! by 53×54=2862. If you divide by the approximation 3000 instead of 2862, the result is exactly 95.4% of 52!.

Here’s the trick I would use if I needed to calculate a factorial and didn’t have a computer on hand. It has a powers of 2 bias, so

`lg x`

refers to the base-2 logarithm.`7! = 5040 ~ 2^12`

`lg 8 + lg 9 + ... lg 15`

has 8 terms of “about 3.5”, so is about`2^28`

.`lg 16 + ... lg 31`

has 16 terms of “about 4.5”, so it’s about`2^72`

.`lg 32 + ... lg 52`

has 21 terms of “about 5.33”, so it’s about`2^112`

.So you get a total of around

`2^224`

. The actual answer is`2^225.56...`

, so it’s not bad for an approximation can be done on a bus without a calculator.Actual error bound?

Someone else has taken the title literally and computed the error when you divide by 3000 instead of (54x53). However, that’s not the point of the article.

You can get a handle on the error in the article by identifying all the approximations and estimating from there. I’ll be doing that in a follow-up, but the main error comes from using e ~ 2.7. It’s closer to 2.72. That error is about 0.7%. When we raise to the power of 54 that gives an error of about 40% to 50%.

But that is offset by the approximation that 2^10 ~ 10^3, and raising that to the power of 5. It’s an interesting and useful exercise to refine the original by chasing down estimate of the errors. I might do that in a follow-up post.

By the definition of factorial as 1×2×…×

n, you can get 52! exactly if you divide 54! by 53×54=2862. If you divide by the approximation 3000 instead of 2862, the result is exactly 95.4% of 52!.There’s a great deal more approximation being done than just 3000 ~ 2862. The calculation of 54! is itself very approximate.

Anyway, now that I’m back at my laptop: (10^68 - 52!) / 52! ~ 0.24.

I

reallyhope you understand that that’s not the point of the article.