What Archimedes definitely did do first was show that pi was also the proportional constant between the square of the radius and the area of the circle. It had long been known that the area of a circle was proportional to the square of the radius, but he showed that the constant was the same as that which connected the diameter to the circumference.
Ed
Here’s one I like:
The infinite product (2/1)(2/3)(4/3)(4/5)(6/5)(6/7)(8/7)(8/9)… converges to pi/2.
I second js_africanus’s recommendation of A History of Pi.
Would it be possible to assume that π =1 and make all other numbers fractions of π? Well possible, of course, since I just assumed it, but could it serve any pratical or theoretical purposes?
Possible, but why?
Strike that. pi = 1 can’t be done because pi is not a multiplicative identity.
But you could make a number system of base-pi, right? That way, pi would be 10.
Well, you could write numbers that way—just write x/pi in place of x—but they’d be confusing-as-all-get-out to work with because, as you note, the multiplicative identity would not be the number written “1” but the number written “1/pi.”
It’s already common in certain contexts to write angles as fractions of pi (e.g. pi/6 = 30 degrees, pi/4 = 45 degrees, 2pi = 360 degrees).
I don’t think so. But I’m not sure. Pi is irrational and that is independent of what base you put it in. I think.
Well, not really. (pi/6) is a certain number of radians, so you have degrees being converted into radians because they’re better to work with mathematically. (Hence the Fox Trot where the kid said that golf would be easier if distances and angles were in metric & radians instead of English & degrees.) One radian is the angle described by an arc equal in length to the radius of the circle, and since 2pi*r is the circumference (sp?), the number of radians in a circle equals 2pi.
Since pi is irrational, I think that if you somehow set it to be you basic unit, other stuff wouldn’t work out right and you’d have a whole mess. Borrowing a description of the hypoteneus of a right triangle that I once read, there is no measuring unit that you can use to measure the radius of a circle in whole units that will also measure the circumference in whole units. So if you set a unit equal to one circle circumference, there’s no way you could get a sensible unit for the radius. I think.
I better shut up and let the more knowledgeable actually drop some phat mathematics before stupidify the place too much.
My favorite? gamma(1/2) = sqrt(pi)
If you ever need to remember pi to 740 places, just start reciting…
Sir, I send a rhyme excelling
In sacred truth and rigid spelling.
Numerical sprites elucidate
For me the lexicon’s dull weight.
~or~
See, I send a rhyme assisting
My feeble brain, its tasks oft resisting.
Not as many digits as FlippyFly’s or Sublight’s, but it’s a helpful mnemonic.
Sublight, that work has been extended by the same author to about 4000 digits!
Hey, the BBP formula! I think one of those B’s is named for my old mailing-list pal, David H. Bailey, who I thought was just some random guy until I found out he was a famous mathematician. … and looking at the link, I see that it is indeed named for him, because that’s his site.
Somebody’s got to mention, and apparently it must be me, “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”
Blatant hijack: I know one for e (hope I get this right): “He repeats: I shouldn’t be tippling, I shouldn’t be toppling here…”. Any other irratinal constant mnemonics out there?
Now I will a rhyme construct
By chosen words the young instruct
Cunningly devised endeavour
Con it and remember ever
Widths of circle here you see
Sketched out in strange obscurity