0.333~
pi/4=sum [ -1^(N+1)/(2N-1)] from 1 to infinity. That’s one way. It converges very slowly though. I don’t know what algorithms are used in practice.
First of all, get over any aversion you might have to imaginary numbers. Sure, i doesn’t exist, but then, neither does 1, or 2. They’re all just symbols that we can shuffle around on paper, and which fortuitously seem to correspond to some things in the real world.
Now, the next step is to realize that the exponential function (y = e[sup]x[/sup]) is closely related to the trigonometric functions like y = sin(x). If you take the derivative of e[sup]x[/sup], then you get back e[sup]x[/sup], and if you take the derivative of a trig function, you get another trig function.
The relation between the two types of functions manifests in the fact that e[sup]ix[/sup] = cos(x) + isin(x), where x is some real number. From here, it’s just a matter of plugging in pi: e[sup]ipi[/sup] = cos(pi) + isin(pi). And cos(pi) = -1, while sin(pi) = 0, so this reduces to e[sup]i*pi[/sup] = -1.
I’ve always wondered, is pi irrational only in base ten? Is there any base in which it works out as a whole number?
It’s pretty even in base Pi.
An irrational number is a number that can’t be represented as the ratio of two integers. The fact that pi != m/n, for integers m and n, is true regardless of what symbols we use to write the integers m and n.
But, I have very little understanding of what it means to have base pi…
That makes way more sense than the Wikipedia entry. I’ll have to do a bit o’ research on why e[sup]ix[/sup] = cos(x) + isin(x), as I don’t recall that relationship from any of the math courses I have taken or am taking, but everything else you’ve said, I get. Thanks!
And it wasn’t really all that long ago, relatively speaking. It was the great 18th-century mathematician Leonhard Euler who popularized the use of the letter pi to refer to this constant. I find it a little surprising that humankind had known of the concept for so long, yet took so long to come up with a name or symbol for it.
Pi would just have to be either the base unit of count or a multiple thereof. “How many pi dollors to buy 3pi pies?”
Why any sane person would do this is beyond me, though.
Agreed; this is a good read, largely because of Beckman’s opinionated crankiness.
Imagine my surprise when I did a web search on “state legislature pi = 3” only to find Did a state legislature once pass a law saying pi equals 3? - The Straight Dope as the first hit.
Which is the part you get stuck on: That irrational numbers (numbers which can’t be written as a ratio of integers) exist; that their decimal expansions go on forever; or that pi is such a number?
The first of these has been known since the time of the ancient Greeks, though it supposedly shook them up quite a bit when they first discovered it. I could point you to a proof that the square root of 2 is irrational, but it might be more enlightening to think of it this way:
For the square root of 2 to be rational (that is, to = m/n, where m and n are both whole numbers), m[sup]2[/sup]/n[sup]2[/sup] would have to equal 2. But if you take the fraction m[sup]2[/sup]/n[sup]2[/sup] and start reducing it by canceling common factors, you can’t be left with a single factor of 2; the factors of both m[sup]2[/sup] and n[sup]2[/sup] occur in pairs.
For the second of these: If the decimal version of a number did terminate, it would be easy to write it as a fraction (whose denominator is a power of 10). For example, 0.2871 = 2871/10000. So the numbers that can’t be written as fractions (i.e. the irrational numbers) can’t be writable as terminating decimals. (Some fractions, like the aforementioned 1/3, show up as repeating decimals. Irrational numbers are non-terminating and non-repeating).
For the third, don’t feel bad; it’s far from obvious. It wasn’t until the 18th century that anyone managed to prove that pi is irrational (and not for lack of trying). Wikipedia has a page on the proof that pi is irrational, but it’s certainly not the kind of thing that would make you go, “Of course!”
Ok, but, in base 10, you need only 10 symbols to represent any real number (provided you accept infinitely long strings of these symbols). In base 8, you need 8 symbols, and in base 16, you need 16 symbols. How many symbols do you need for base pi?
I don’t know…maybe the first and maybe the third. It’s kind of hard to explain. I guess part of it is it’s hard to understand how you can have something that can’t be measured exactly. Let’s say you have a string of length pi, like cmyk made when he was explaining it to his daughter. You then pull out a ruler and try to measure it. So, you measure it, and it’s more than 3 inches, for instance, so you get out a better ruler that measures half inches, and find out it’s between 3 and 3 and a half inches, so you get out a ruler that measure quarter inches, and so on. I mean, eventually, you’re going to get to some smallest point, it would seem…you get down to the atomic level, or something.
Beyond that, it just seems, I don’t know, disorderly. It doesn’t seemlike things like irrational numbers should be happening in a well ordered universe.
Chronos’s response to “i does not exist” was pretty much on the money (alas, the name “real number” has poisoned so much of this discussion with fossilized ignorance). But let’s try and make it even clearer in what sense “i” exists. As an added bonus, once we do this, it will also become pretty easy to see why e^(iπ) = -1, and more generally why e^(ix) = cos(x) + i*sin(x) [in radians].
Let’s think about sticks. Sticks have lengths. And we can scale these lengths; we can make a stick twice as big, or three times as big, or half as big, or 5.8 times as big, and so on. And it’s in terms of these length ratios that we actually give our measurements; we say “John is 5.8 feet tall” to mean “John is 5.8 times as big as a ruler; i.e., if you scaled a ruler by a factor of 5.8, it’d be as large as John”. And this shows us how to interpret certain numbers as actually about real-world quantities, and life is good. We might even say this shows that 5.8 “exists”, if you want to talk that way, but I’d really rather you didn’t.
And we can interpret addition and multiplication within this framework as well: multiplication means “chain the scalings one after another”: 7 * 5 = 35 because making something 7 times as large, and then making the result 5 times as large has the net effect of making what you started with 35 times as large. Addition means “carry out both scalings, then place the one stick after the other and see where you end up”" 7 + 5 = 12 because something 7 times as large as a ruler laid end to end with something 5 times as large as a ruler ends up at the same place as something 12 times as large as a ruler. So life is really good. We know perfectly well what arithmetic means now.
But wait… we’re missing something. We haven’t accounted for negative numbers. It wouldn’t seem like it means something to scale by a negative factor, so how can we make sense of them? Well, as you are probably familiar, there is a natural convention to adopt. Instead of focusing solely on lengths, we’ll now look at what direction our sticks are pointing in as well; in addition to scaling sticks up or down in size, we’ll also talk about flipping them 180 degrees around to point the other way. So, for example, -1 will mean “Turn your stick 180 degrees”, and -5 will mean “Make your stick 5 times as big and turn it 180 degrees”. But we’ll interpret addition and multiplication exactly the same way as before: -7 * 5 = -35 because “Make it 7 times as large and turn it 180 degrees” followed by “Make it 5 times as large” has the same net effect as “Make it 35 times as large and turn it 180 degrees”. And -7 + 5 = -2 because if I make two copies of my ruler, one 7 times as large but turned around, and the other 5 times as large and unturned, and place the one after the other, the ending point’s location is the same as if I’d just made a copy of my ruler which was twice as large and turned around. So life is super. Looks like negative numbers “exist” as well (but, please, don’t talk that way).
But, hell, once we’ve started talking about turning sticks, why limit ourselves to full half-circle turns? Why not look at quarter-turns, eight-turns, 23.4 degree turns, and so on?
Why not indeed. Once we toss these in, we get… the complex numbers. All that mysterious i means is “Make a 90 degree turn”. We still interpret addition and multiplication exactly the same way as before; multiplication is still “Do these in sequence” and addition is still “Do these in parallel, lay the results one after another, and see where you end up.” In particular, as far as multiplication goes, since “Turn your stick 90 degrees. Now turn it 90 degrees again.” has the same net effect as “Turn your stick a full 180 degrees”, we see that i * i = -1. That’s it; it’s extraordinarily simple. Life is fantastic. Complex numbers “exist” every bit as much as real numbers; it’s just that the complex numbers express scaling with arbitrary rotation, while real numbers are limited to scaling with half-turn-increment rotation. [And non-negative real numbers express scaling with no rotation at all.]
Does this help clear anything up so far as complex numbers go? Next, I’ll try to deflate the mystery from e^(iπ).
When you reason like that, then when you get down to the atomic level, there are no perfect circles, only jagged figures. But π isn’t about jagged figures; it’s about perfect circles. Math isn’t necessarily about our atom-filled universe; it’s about whatever you want it to be about. When you study the abstract concept of perfect circles, you end up discovering that the ratio of their circumference to their radius cannot be a ratio of integers. The mathematically discovered claim isn’t that irrational quantities exist in some particular physical system of measurement; the discovery is that they arise in certain abstract investigations. Whether or not any of this actually directly corresponds to the physical space we live in is a separate matter.
Right, and that’s my problem with it (and probably why I did so bad in math at school.) I’m not good with abstraction.
If it were up to me, I’d have given the nice name to what we now call 2π (the ratio between a circle’s circumference and its radius). That’s really the much more fundamental quantity. [Which makes all the π-worship seem particularly silly to me, something like hypothetical people fixating an awful lot on the “magical” number 5*sqrt(2).]
The bolded part just blew my mind. Everything else was basic vector addition (I’m in Calc III atm, I’m not a complete ignoramus, just a partial one) but then I got to that sentence and my brain reverted by default to i-j-k notation and suddenly the square root of negative one became k. Or something. I’m not exactly sure. Furthermore, if i = a 90 degree rotation of our hypothetical stick, I literally have no idea what circumstances would generate the other angles you mentioned, i.e. 23.4 degrees et al. I was always under the impression that i = sqrt(-1) essentially defined complex numbers but this is clearly not the case or else our hypothetical stick would only be able to rotate in 90 degree increments.
That, however, is another discussion in and of itself. I can acknowledge mathematical operations involving i as being the equivalent of rotating our stick 90 degrees and then scaling it, or adding sticks, or what have you. I’m not entirely sure how that corresponds to traditional “real” numbers we’re used to dealing with, the ones that define our “real” universe, because it I go outside and set a ruler down and then suddenly turn it 90 degrees, I have not changed its length - it’s still 12 inches long, only now it’s pointing that way points.
I do know i exists - I wish I had not said “doesn’t exist” to mean “is not a real number”, and this being the SDMB I really should have known better - even if I can’t relate it to the world I live in (if there’s any physical entities in the universe that can be measured in terms of i, excluding wiseass reamples along the lines of “a ruler is 12*(i[sup]4[/sup]) inches long”, I would very much be interested in seeing them). My main question, at this point, is the math behind e^(ix) = cos(x) + i*sin(x); I looked it up on Wikipedia and have a decent understanding of it, as all the proofs I saw only require up to Calc II - level material, but in order to really comprehend it I’d have to work through the math on my own a couple times.
Right now, though, I’m busy; I gotta do Thermodynamics homework. :3
It actually makes more sense with the definition of sin(x) and cos(x) with respect to the complex plane
e[sup]ix[/sup] - e[sup]-ix[/sup]
2i
is one definition of sin(x), cos is the same thing except you add the e terms instead of subtract them and there’s no i in the denominator. If you want to “prove” it to yourself just write down the definitions in place of sin(x) and cos(x) above and do some… I was going to say algebra but after you cancel the i’s from the sine term it’s pretty much just fraction addition.
If those formulas look familiar you probably are remembering the definitions you learned in Calculus (most likely) for the hyperbolic trig functions, which are the same thing without the imaginary units.
Actually, you are good at abstraction! Using language/words to communicate is a type of abstraction.
You’ve seen hundreds (thousands) of dogs to comfortably use “dog” as a placeholder in conversation. When someone says “dog” or an author puts “dog” in text for you to read, you “get” what a “dog” is. This works even though you haven’t seen every dog in existence in the entire world since the beginning of time.
It’s just abstraction with particular symbols (math notation) that some people don’t get the knack of right away. Some of it may be due to repitition – most people get repeated exposure to dogs so the “dog” equals dog is easier.
[the above example is basically Aristotle reworded…]