6 ms·

Am I misreading the article, or is it actually saying that every factorial is approximately a power? The title makes it sound like it's exactly a power.

gautamcgoel 3y ago
Yeah, you are correct.
qbit42 3y ago
It’s just a power of a non integer base, which he gives a formula for
- dietrichepp 3y ago
  This is either incorrect or it does not really mean anything, depending on how you look at it. It’s Stirling’s approximation, as the article mentions. The formula uses the ≈ character. The Wikipedia article gives bounds on the error— https://en.wikipedia.org/wiki/Stirling's_approximation https://en.wikipedia.org/wiki/Stirling's_approximation
- mqus 3y ago
  but then isn't every integer a power of a (non integer) base? In this interpretation the statement is pretty useless.
- ted_dunning 3y ago
  And he really doesn't give a very nice formula for it. Numerically, the point of the article is the fact that n! \approx (0.826523 + n*0.373522)^n For n in the range from 10 to 50, this is within 5% of the right answer. But I really don't see that this is all that much simpler than Stirling's approximation and it is considerably less accurate and not asymptotically very good. Stirling's formula is within about 1% for all n>10 and within 0.1% for n>90. Actually computing gamma(n) or lgamma(n) to high precision requires a bit more effort than this, but as an approximation, Stirling's formula is really pretty good.
sapiogram 3y ago
I think it really is exactly a power, as long as `b` is allowed to be any real number.
- FartyMcFarter 3y ago
  Every positive number is exactly a power if you allow that.
  - c7b 3y ago
    Every positive number is in fact (uncountably) infinitely many powers in this case.
  - kdmccormick 3y ago
    Yup. I got hung up on that too. I think the article could have better stated it as: For every natural n, there exists a natural b, such that: n! ≈ b^n I don't quite know the precise criteria for "≈" here, that's a bit beyond my understanding of the article...
    - throwawaymaths 3y ago
      He also mis-typesets it as = on the same line! Very confusing.
    - gowld 3y ago
      ≈ is a loose approximation. The fact is that `e * (n!)^(1/n)` is quite close to `n`. But John didn't factor out the `e`, causing the confusion in this HN post's threads.
  - ftxbro 3y ago
    so you're saying the title is accurate
    - FartyMcFarter 3y ago
      It's either inaccurate or trivial depending on how you interpret it: - If you only allow the base to be an integer, then it's inaccurate. - If you allow the base to be a real number, then you're really just saying "every factorial is a positive number", which is a trivial fact if you know what a factorial is. Either way it's a bad title IMO.
Aardwolf 3y ago
Every number is a power if you allow non-integers. But it seems indeed like it means _near_ an integer power in this case. The factorial of 3 is 6 but 6 is not a non-trivial integer power of anything. I don't like the title either. The article also says, you can find a base b such that n! = b^n for any factorial. Guess what, you can do this for any number x and any n to find b such that x = b^n, whether x is a factorial or not: b is x ^ (1/n). Not sure what Stirling's approximation should have to do with this, and why = and ≈ are being used inconsistently in this post. Ignoring my complaints of this particular post, this is a pretty great blog though, and I remember being helped by the linked older blog post article about log1p and expm1 before!
- mistercow 3y ago
  The Stirling's approximation part of it is about why exp(log(n!)/n) is nearly linear. I think what he's saying is basically: if you take the log version of Stirling's approximation, ln(n!) = n*ln(n) - n + O(ln(n)), ignore the big O term, then divide both sides by n and take e to the power of each side, you get the formula for the `b` from the article on the left, and n/e on the right, which is linear. I've never actually worked with Stirling's approximation, but I guess if you have a certain set of context, this fact would be obvious?
  - Aardwolf 3y ago
    The Stirling's approximation is a good way to implement an accurate gamma function approximation on PC/calculators/..., but it's just a formula to implement, it doesn't necessarily mean you think about fun facts like this while implementing it...
- hinkley 3y ago
  Rational numbers yes. Irrational Number: I'm a non-integer, Greg. Can you square root me?
ftxbro 3y ago
well it got your attention enough that you appeared to have read at least some of the article so i guess that's a win. everyone got to hustle.
- glitchc 3y ago
  > everyone got to hustle. Why?
  - i-use-nixos-btw 3y ago
    Scroll down below the article: > My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, math, statistics, and computing. > Let’s talk. We look forward to exploring the opportunity to help your company too. Getting more attention then clarifying the intent in the body is sometimes mischievous (read: every tabloid newspaper), but it nonetheless brings more good attention that it would have without the click bait. If that leads to more consulting work for the author, good for him. (it kind of feels weird calling a high-number approximation “clickbait” but, in some circles…)
    - gowld 3y ago
      "Did you like my article wherein I misunderstand a mathematical concept? Hire me to do math for you!" doesn't seem to be a strong pitch...
      - i-use-nixos-btw 3y ago
        I think it’s clear to the author and every reader that it wasn’t a misunderstanding.
  - whyhustle 3y ago
    [flagged]
bannedbybros 3y ago
[dead]
shultays 3y ago
Well, worst case they are exactly something power to 1
aimor 3y ago
The integer examples really threw me off too. 6.20 ≈ 10.00; 1.55 ≈ 1.00; 2.37 ≈ 2.65
- hinkley 3y ago
  > 6.20 ≈ 10.00 That's like when my friend tortured her kid (and me) by asking how old I was, which was answered with "75". That is a very wide margin of error there.
gcanyon 3y ago
I spent approximately as much time trying to understand this aspect as I did what the author was actually saying. The language is annoyingly misleading, and the seemingly random use of = and ≈ is frustrating. Either my math or my english is failing me.
dang 3y ago
Ok we've made the title approximate now.

gautamcgoel 3y ago

Yeah, you are correct.

qbit42 3y ago

It’s just a power of a non integer base, which he gives a formula for

dietrichepp 3y ago
This is either incorrect or it does not really mean anything, depending on how you look at it. It’s Stirling’s approximation, as the article mentions. The formula uses the ≈ character. The Wikipedia article gives bounds on the error— https://en.wikipedia.org/wiki/Stirling's_approximation https://en.wikipedia.org/wiki/Stirling's_approximation
mqus 3y ago
but then isn't every integer a power of a (non integer) base? In this interpretation the statement is pretty useless.
ted_dunning 3y ago
And he really doesn't give a very nice formula for it. Numerically, the point of the article is the fact that n! \approx (0.826523 + n*0.373522)^n For n in the range from 10 to 50, this is within 5% of the right answer. But I really don't see that this is all that much simpler than Stirling's approximation and it is considerably less accurate and not asymptotically very good. Stirling's formula is within about 1% for all n>10 and within 0.1% for n>90. Actually computing gamma(n) or lgamma(n) to high precision requires a bit more effort than this, but as an approximation, Stirling's formula is really pretty good.

sapiogram 3y ago

I think it really is exactly a power, as long as `b` is allowed to be any real number.

FartyMcFarter 3y ago
Every positive number is exactly a power if you allow that.
- c7b 3y ago
  Every positive number is in fact (uncountably) infinitely many powers in this case.
- kdmccormick 3y ago
  Yup. I got hung up on that too. I think the article could have better stated it as: For every natural n, there exists a natural b, such that: n! ≈ b^n I don't quite know the precise criteria for "≈" here, that's a bit beyond my understanding of the article...
  - throwawaymaths 3y ago
    He also mis-typesets it as = on the same line! Very confusing.
  - gowld 3y ago
    ≈ is a loose approximation. The fact is that `e * (n!)^(1/n)` is quite close to `n`. But John didn't factor out the `e`, causing the confusion in this HN post's threads.
- ftxbro 3y ago
  so you're saying the title is accurate
  - FartyMcFarter 3y ago
    It's either inaccurate or trivial depending on how you interpret it: - If you only allow the base to be an integer, then it's inaccurate. - If you allow the base to be a real number, then you're really just saying "every factorial is a positive number", which is a trivial fact if you know what a factorial is. Either way it's a bad title IMO.

Aardwolf 3y ago

Every number is a power if you allow non-integers. But it seems indeed like it means _near_ an integer power in this case. The factorial of 3 is 6 but 6 is not a non-trivial integer power of anything. I don't like the title either. The article also says, you can find a base b such that n! = b^n for any factorial. Guess what, you can do this for any number x and any n to find b such that x = b^n, whether x is a factorial or not: b is x ^ (1/n). Not sure what Stirling's approximation should have to do with this, and why = and ≈ are being used inconsistently in this post. Ignoring my complaints of this particular post, this is a pretty great blog though, and I remember being helped by the linked older blog post article about log1p and expm1 before!

mistercow 3y ago
The Stirling's approximation part of it is about why exp(log(n!)/n) is nearly linear. I think what he's saying is basically: if you take the log version of Stirling's approximation, ln(n!) = n*ln(n) - n + O(ln(n)), ignore the big O term, then divide both sides by n and take e to the power of each side, you get the formula for the `b` from the article on the left, and n/e on the right, which is linear. I've never actually worked with Stirling's approximation, but I guess if you have a certain set of context, this fact would be obvious?
- Aardwolf 3y ago
  The Stirling's approximation is a good way to implement an accurate gamma function approximation on PC/calculators/..., but it's just a formula to implement, it doesn't necessarily mean you think about fun facts like this while implementing it...
hinkley 3y ago
Rational numbers yes. Irrational Number: I'm a non-integer, Greg. Can you square root me?

ftxbro 3y ago

well it got your attention enough that you appeared to have read at least some of the article so i guess that's a win. everyone got to hustle.

glitchc 3y ago
> everyone got to hustle. Why?
- i-use-nixos-btw 3y ago
  Scroll down below the article: > My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, math, statistics, and computing. > Let’s talk. We look forward to exploring the opportunity to help your company too. Getting more attention then clarifying the intent in the body is sometimes mischievous (read: every tabloid newspaper), but it nonetheless brings more good attention that it would have without the click bait. If that leads to more consulting work for the author, good for him. (it kind of feels weird calling a high-number approximation “clickbait” but, in some circles…)
  - gowld 3y ago
    "Did you like my article wherein I misunderstand a mathematical concept? Hire me to do math for you!" doesn't seem to be a strong pitch...
    - i-use-nixos-btw 3y ago
      I think it’s clear to the author and every reader that it wasn’t a misunderstanding.
- whyhustle 3y ago
  [flagged]

bannedbybros 3y ago

[dead]

shultays 3y ago

Well, worst case they are exactly something power to 1

aimor 3y ago

The integer examples really threw me off too. 6.20 ≈ 10.00; 1.55 ≈ 1.00; 2.37 ≈ 2.65

hinkley 3y ago
> 6.20 ≈ 10.00 That's like when my friend tortured her kid (and me) by asking how old I was, which was answered with "75". That is a very wide margin of error there.

gcanyon 3y ago

I spent approximately as much time trying to understand this aspect as I did what the author was actually saying. The language is annoyingly misleading, and the seemingly random use of = and ≈ is frustrating. Either my math or my english is failing me.

dang 3y ago

Ok we've made the title approximate now.