Hacker News Reader

Comment by est | Hacker News Reader

If pi == 6.28 [1] then we could have a more elegant version

e^(i*pi) == 1

[1]: http://www.math.utah.edu/~palais/pi.pdf, http://www.math.utah.edu/~palais/pi.html

Part of the beauty in the original equation is that it contains every object thought to be elemental in arithmetic and moreover, it contains every one of them exactly once.

It contains 0, 1, +, , ^, which are sufficient to express all of elementary natural arithmetics (unless I'm mistaken) -- all natural numbers can be expressed by adding enough ones to a zero (0, 0+1, 0+1+1, ...), and if I recall correctly, + and aren't enough to state all the truths in natural arithmetics so ^ is needed.

There is a very interesting paper on this called "The saga of the High School Identities" by S. Burris (http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PREPRI...)

Moreover, it contains three very attractive (if not the most attractive) mathematical constants we know. There is so much we don't know about their relationship. We have no idea if any of these numbers is transcendental or not: pi + e, pi − e, pi*e, pi/e, pi^pi, e^e, pi^e. We don't even know if half of them are even irrational or not (therefore, we don't know if pi is just another facet of e, or maybe an entirely different object!)

So to see all this (and more; one could say so much more) in one equation so terse is so surreal a feeling.

> 0, 1, +, , ^, which are sufficient to express all of elementary natural arithmetics (unless I'm mistaken)

Well, personally I don't favor zeros in equation. What's the point of 1+0=1 ?

Leaving 0 on the right side looks so unbalanced.

The reason I subconsciously did it was that natural numbers have an inherent structure to them. You can describe all natural numbers using only the objects 0 and succ, where succ = (+ 1).

An interesting fact is that this is sufficient to express + and :

a + 0 = a a + (succ b) = succ (a + b)

a 1 = a a * (succ b) = a + a * b

One way to look at structures (like, say, natural numbers) is too look at what really defines them, as in, what is absolutely necessary and sufficient to define them? In this case, 0 and succ is.

Note that I'm not saying you're wrong, everyone is entitled to their own definition of beauty. Isn't that the beauty of it?

> I subconsciously did it was that natural numbers have an inherent structure to them.

I understand and concur you :)

But I don't think Euler's identity is meant to serve this purpose. Maybe some other formula.

It's hard to un-like something once you already liked it very much. First impression matters. How about e^(i*pi)-1==0 instead?

I think your aesthetic sense in this case may have been affected by years of pro-pi propaganda. The 'beautiful' equation requires a rearrangement (i.e., moving the -1 to the other side), which is a bit kludgy, to say the least. Moreover, the superior (IMHO) version in the parent comment can be put into words in a rather compelling way: the exponential of the imaginary unit times the circle constant is unity. (If you're still hung up on the appearance of 0, just add it to the RHS. ;-) The same can't be said for the standard version.

P.S. Please email me if you're interested in pursuing this further; I have a project (launching June 28) that I think might convince you of the error of your ways. :-)

I have some secret plans in this direction. Please email me if you'd like to be part of it (my address is in my profile). Hint: It's launching on June 28. :-)

Your version contains less information, since it's a result of taking a square of both sides of e^(i3.14...) = -1 (which is equivalent to the original), and taking a square is not injective. In other words, your version would be true also if e^(i3.14...) was equal to 1. So it's arguably less interesting then the original.

Debatable, I suppose.

I think the inclusion of 1 is pretty significant, in terms of the contrast between 0 and 1. Maybe that sounded strange -- oh well, I'm not a mathematician by any means. :)

Reply on news.ycombinator.com