I could probably look this up on my own but that might involve some actual work. So I’d rather ask here what’s the deal with Pi? It never ends? How is it different from random numbers? Does it explain infinity?
Pi is an irrational number which means that no matter what two numbers you tried to use to make a fraction you would never find two that, when one is divided by the other, equal Pi exactly. It also means that its decimal expansion never ends and never repeats.
I’m not exactly sure what you mean by “how is it different from random numbers?”
It could be used by a computer to generate random numbers but so could any other number.
I also don’t know what you mean by “does it explain infinity?” What about infinity do you want explained?
Alternatively, Pie is yummy and I think I will go have some right now. Mmmm, infinite pie…
Pi is (probably) a normal number, which means that its digits have the same probability distribution as if they were random. (The digits are not random, of course, because there are formulas that can calculate any number of them.)
But if you’re looking for infinity, think of this. If pi is a normal number, then it contains every possible finite string of digits. This means that if you converted pi to binary, it would contain every digital file that was ever created and will ever be created, and an infinite number of other files that will never be created, but could be.
Somewhere in the digits of pi, there is a DVD containing all the episodes of the second season of Firefly — even though they were never filmed. And there is another DVD containing exactly the same episodes, only with Martin Luther King Jr. playing River.
When I was a kid, I realized that if images on the computer were just a grid of a finite number of pixels each with one of a finite number of colors, then if you set up a grid and generate every possible combination, you’d have a collection of any possible image you could want to see. Then I thought about all the porn that would generate.
[quote=“friedo, post:4, topic:556330”]
Ahhhh… the good ol’ days.
Archimedes approximated π by approximating circles with many-sided polygons. When he got up to 96 sides, he determined that π was close to (but not exactly) 223/71. (The Greek’s pursuit of π almost led them to discovering calculus.)
It’s always seemed somewhat intuitive to me that π must be irrational because if it could be represented by a ratio of integers, that would seem to describe some sort of polygon rather than a circle. This obviously isn’t a rigorous proof of π’s irrationality, but rather the way I’ve always looked at it.
I think what the OP really needs to know is that pi is the ratio of a circle’s circumference to it’s diameter. It pops up lots of other places in mathematics too. That’s why it’s different from any random number.
Why is it that every time I come up with a great, clear expalanation to a GQ, someone gets in riiiiight before me?
OP, think of a slice of pie that’s perfectly circular. Do you agree that the crust is a curve? Well if it’s a curve, then we can’t perfectly measure it without breaking it down into smaller bits. So let’s do that.
Cut your pie slice in half. Do you see that the edge is still curved, although slightly less? So cut it again. Still curved. Cut again. Dammit, it’s STILL curved!
It goes on like that forever, meaning that no matter how many times you cut the pie, it’ll always be immeasurable. As you go smaller and smaller, the curve gets closer and closer to a straight line. That’s why as you get more and more precise on pi, the digits just add onto the end instead of changing earlier-placed digits.
If pi ever were to terminate, then that would mean that further dividing your slice of pie wouldn’t yield a slightly less-curved edge, and the only way to do that would be if the edge were a straight line at some point. After all, the slope of a line is the same as the slope of half the line, right? Well, we know that your pie will never show a straight line, and thus pi can’t ever terminate.
Got it now?
What’s the deal with Pi?
I don’t know but it always seems to get called against my favorite football team at the most inopportune time…
You can read about Pi on Wikipedia, or math websites, or a number of books. But I still recommend Petr Beckman’s a History of Pi
It’s well-written and easy to understand, and shows you the importance of Pi and how it has been measured and calculated with increasing accuracy through the ages. On top of which, Beckman was undoubtedly a crank – a well-Educated and well-written infdividual who knew about his topic, but it’s colored by his intense and odd hatred of Aristotle, the Soviet Union, and sundry other things he considers enemies of clear and free thinking.
If you put two people in seperate rooms and had them generate random numbers, they would come up with two different sets of numbers.
But pi is not random. If you had two people in seperate rooms each figuring out the value of pi independently, you could compare their answers afterwards and they would be the same.
You found a refusal of the theory that says that “you can’t have your pi and eat it too!”
Someone should call the Nobel committee…
Good analogy, but not quite true. True of Pi, but not of pie.
If you cut a slice of pie in half about 90 times, you could cut no further unless you started splitting atoms.
And there’s nothing particularly special about that. There are lots of irrational numbers (e.g. the square root of anything that isn’t a perfect square), which can’t be written exactly as a fraction or as a terminating (or repeating) decimal. The ancient Greeks were the first to discover (and prove) this, and it blew their minds.
By contrast, it wasn’t proved that pi is irrational until 1761 (many years after the ancient Greeks became ancient), though it had long been suspected/believed that it was.
And in case you’re wondering, you don’t, in practice, figure out the value of pi by drawing a circle and measuring its circumference and diameter. There are various mathematical formulas which have been proven to give pi as an answer. Usually, these take the form of an infinite series, which is to say a sum of an infinite number of terms. Obviously, you can’t add up all the terms, but the more of them you add up, the closer you’ll get. The simplest of these formulas (though not, in practice, the most efficient or practical one to use) is that pi = 4/1 - 4/3 + 4/5 - 4/7 + …
Wha? Why is this true? Are you saying all curves are irrational? That can’t be right. A circle with a radius of 1/pi has a circumference of 2, which is a rational number.
No, I’m saying that the ratio of circumference to radius (or diameter) is always irrational. 1/pi : 2, aka half-pi, is still irrational.
It’s true because the only way to measure (rather, “prove the length of”) a curve is to use calculus. You can always approximate, like we do with “3.14”, but we’re talking about irrational numbers here, so approximation is cheating. My point is that in order to measure a curve, you have to add up its parts. Turns out, though, that the parts of a circle are just curves. And the parts of those curves are straighter curves. There’s no point where a fraction of a curve is just a measurable straight line that you can then sum up with all the other straight lines- it’s curves all the way down! So no matter how precise you get, you’ll never get a terminating decimal because you’ll never get something that’s precisely measurable.
Even the “4/1-4/3+4/5…” thing is just using (measurable) lines to estimate the circumference, but realizing that the circle is inscribed in the resulting polygon, so there has to be a negative term to adjust for it…but wait, then the polygon is inside the circle, so you have to add some more…but wait, then the circle is inside the polygon again…and on and on. As was said up above, Archimedes stopped at 96. But the point remains- at some point, you’ve got to just use lines.
Just because there are curves involved, doesn’t mean it has to be irrational. Consider, for instance, the Lune of Hippocrates, a two-dimensional figure composed entirely of circular arcs, but whose area is nonetheless rational. And there are curves with rational length, too, though I don’t remember any of those off the top of my head.
You can also use yarn.
A “slice of pie” isn’t circular - a pie is. Nitpicky, but it’s a confusing term to start your (excellent) example off on.
You can’t measure a radius of 1/pi, but you can measure something that’s 1/“something really really close to pi”.