All through school, my math teachers told me that if the circumference of a circle is 22, the diameter would be 7… therefore pi can be expressed as 22/7. If I whip out my trusty Casio and divide 22 by 7 I come up with 3.1428571, but pi is traditionally expressed as 3.1415926… WHAT GIVES? Could the public school system have failed me again? Given their track record I’m assuming that they have fibbed once more, and 22/7 is not actually an accurate depiction of pi. So is there a more accurate fraction? I’ve asked several teachers about this and all I get is something to the effect of “they found out long ago that pi doesn’t matter past 3.14” and they all refuse to admit that 22/7 is not actually pi. so what’s the deal?
It’s an approximation, one that’s only off by one one-thousandth. Big deal. If you want an accurate way to calculate pi, you’ll need to find one of the many infinite series out there, and solve it. Until then, I think 3.14159 will suffice for my needs.
Pi is the quotient of the circumference of a circle divided by its diameter. It has been carried out by computers to an almost infinite number of numbers to the right of the decimal point. I don’t remember many of the numbers, but I know you just can’t say it’s 22/7 if you have to be precise. That’s just a fractional approximation.
If you’re at all interested in pi, allow me to reccommend the book and website, http://www.joyofpi.com
Welcome, DEUS1, to the SDMB.
The short answer is that, yes, your teacher fibbed if you were told that 22/7 was exactly equal to pi. It ain’t. 22/7 is just an approximation that’s close enough for many purposes, and is relatively easy to remember. I suspect (but don’t want to take the time to check) that 22/7 is the best fractional approximation that doesn’t require 3+ digit numbers to write.
Pi is an irrational number, which means that it cannot be exactly represented by the quotient of two integers, no matter how large you’re willing to try to make them. This also means that it can’t be exactly represented in decimal form, either: the digits just keep on going and going, never terminating and never repeating.
So, you could come up with a more accurate fraction than 22/7, but you will not find one that’s exactly right. For example, 2749/875 = 3.14171…, which is a bit closer to pi (3.14159…) than 22/7 (=3.14285…). However, it’s still not right, and for simple calculations I’ll stick with remembering 22/7, myself.
Actually, I find it easier to remember the first few decimal places: 3.14159. If I need to know it more accurately than that, I look it up.
Pi is an irrational number (a number that can not be expressed as the ratio of two positive intergers, no matter how large they are). Therefore no fraction will be exactly equal to Pi, but 22/7[sup]ths[/sup] is the nearest convenient set of numbers that comes pretty damn close. In olden times when many people actually preferred to work with fractions instead of decimals (can you believe it?), 22/7 was convenient & reasonably precise. Would you prefer to use this:
355 ¶ = --- 113
No? I thought so.
Damn you, brad. Way to steal my thunder on the irrational number thing.
[sub]:Writes brad’s name in little black book:[/sub]
Damn. I could recite 3.14159 in my sleep, I have it that deeply etched into the patters of my brain.
Pi is irrational and transcendental.
Irrational: Not the result of dividing any two numbers. It shares that property with Euler’s number (e) and the square root of two.
Transcendental: Not the root of a polynomial equation of the form
a[sub]n[/sub]x[sup]n[/sup] + a[sub]n-1[/sub]x[sup]x-1[/sup] + … + a[sub]2[/sub]x[sup]2[/sup] + a[sub]1[/sub]x + a[sub]0[/sub] = 0
, a property that proves you cannot ‘square the circle’, or produce a square with the exact same area as a given circle using the compass and the unmarked straightedge.
Dammit, the equation should read thusly:
a[sub]n[/sub]x[sup]n[/sup] + a[sub]n-1[/sub]x[sup]n-1[/sup] + … + a[sub]2[/sub]x[sup]2[/sup] + a[sub]1[/sub]x + a[sub]0[/sub] = 0
Just so it’s clear, it should be mentioned that the coefficient a’s in Derleth’s above polynomial are rational. So a transcendental number is one that is not a root of any polynomial with rational coefficients.
please take pity on an innumerate who wandered into this extremely learned and informative discussion:
Why in the world would you need to know it even to 5 decimal places?
I mean, I make do counting on my toes in a mathematical emergency, so I have trouble understanding who, aside from a couple of pencil-necked academic geeks, would ever need greater accuracy?
(What’s that? brad_d is a math prof, and on the thin side? oh… sorry [grins weakly])
I know it to five because I’ve seen it so often the five places I know are etched into my brain. I’ve never made a conscious effort to memorize any of it, it’s just that the first five places have a rhythm. Just say ‘three point one four one five nine’ and see if you don’t turn it into a chant after a while. Neat, no?
Here’s an absolutely incredible “mnemonic device” for the digits of pi. (Not really a mnemonic device, since its length would prohibit almost anyone from memorizing it, but still worth a look).
In so many words, 7 is not the diameter of a circle with a circumference of 22. A circle with a diameter of 7 has a circumference of: 21.991148575128552…; a circle with a circumference of 22 has a diameter of 7.002817496043395…
Now on the other hand, if your teacher said 7 over 21.991148575128552… was Pi, he would be correct, but unfortunately in this universe we cannot recite an infinite amount of numbers in a finite amount of time.
Approximate value of pi: 22/7,
Even more precise value of pi: 355/113 (I don’t know where you found that out, Attrayant, but I actually “discovered” this using a computer program I wrote, as well as many other numbers that come really close to Pi when you divide them.)
Even more precise value of pi: 3.141592653589793238462643383. Yes, folks, that is from memory. With Pi to that accuracy, you could theoretically make (or at least measure) an object the size of the known universe with a margin of error or no less than the width of 1 hydrogen atom. (Of course, if it were real life, differences in the space-time continuum would distort the measurements far beyond measurability, but we don’t need to worry about that.)
Exact value of pi: ¶
bah… thats 21.9911… over 7, not vice versa.
Well, come to think of it, I don’t think I ever have. I’m not a mathematician, but rather an experimental fluid dynamicist, and we never know anything to six significant figures!
I remember 3.14159 from some old college “cheer” that seems to have made the round at geek schools across the country. Sadly, I can’t recall much of it, the pi part of it just rolls off the tongue so sweetly…ah!
Or, maybe not. ::runs and hides in shame (and from Attrayant) :)::
> Even more precise value of pi: 355/113 (I don’t know
> where you found that out, Attrayant, but I
> actually “discovered” this using a computer program I
> wrote, as well as many other numbers that come really
> close to Pi when you divide them.)
Attrayant probably knows the fraction 355/113 because it’s a fairly well-known fact about pi. There’s a mathematical method called “continuing fractions.” It allows one to calculate what the best fractional approximations are to pi (where the numerator and the denominator are both integers and less than a given number). One can use the method to calculate fractional approximations much quicker than using your computer to just try every fraction up to that size. I’m sorry, but I can’t find any resources quickly on the subject of continuing fractions.
Knowing that pi is an irrational number is a very basic fact. In fact, it’s much more important to know that fact than what the decimal approximation of pi is. My immediate reaction is that any high school math teacher who teaches that pi is exactly 22/7 should be fired on the spot for not knowing their subject.
pi = 22/7
2 + 2 = 5 for very large values of 2
[sub]2.49 + 2.49 = 4.98[/sub]
Now that I and several others have pontificated on the irrationality of pi, I’m going to step down off the pedestal and ask a very humbling follow-up question to those that remain there:
So what? Why, in normal usage, do I care that pi is irrational?
I realize that this sounds like another of those depressing “Why the hell do I need to know X?” rants, and I want to avoid that so I’ll rephrase it a bit:
Since I’m interested, what significance does pi’s irrationality have?
Other than being aware that any decimal/fraction representation I use in my calculations won’t be precisely correct, I don’t give it a second thought most of the time. I’ve heard some people speak of it (Wendell Wagner’s post reminded me of this) in such a glowing fashion that I wonder if my worldview is somehow being restricted.