Only for those who don't understand pi

[Indistinguishable]

@Alastair_Moonsong

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.

Right. So, as I said, once we even start talking about negative numbers, we’re not talking about just lengths anymore:

@Indistinguishable

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

Bolding mine… er, mine now rather than then, I mean.

If you only want to talk about lengths, then you stick with non-negative real numbers.
If you want to talk about lengths and orientations, but limited to “points the same way” and “points opposite ways”, you go with real numbers.
And if you want to talk about lengths and arbitrary rotations in some fixed direction, you go with full-on complex numbers.

@Alastair_Moonsong

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.

Well, a 23.4 degree rotation would be the circumstances which did it… But, of course, what you’re asking is “How do I express that in the familiar form a + b * i?”. I’m going to give you an answer, but it may make things seem complicated in contrast to the conceptual simplicity I am trying to illustrate. So I want to stress that there is absolutely no need to specify complex numbers in any other format; that “a 23.4 degree rotation” is already a perfectly good description of a complex number.

Ok, disclaimers out of the way: suppose we were to rotate the vector V by angle x. We’ll take the result and think of it as the hypotenuse of a right triangle, one leg of which is parallel to V (the other leg, therefore, being 90 degrees rotated from this). How long are the legs? Well, per basic trigonometry definitions, the ratios of these leg-lengths to the hypotenuse-length are cos(x) and sin(x), respectively. But the hypotenuse is exactly as long as V, so the leg parallel to V is just V scaled by cos(x), and the other leg is just V scaled by sin(x) and then rotated 90 degrees. Thus, we see that, in general, a rotation of angle x is given by cos(x) + sin(x) * i.

And so, to answer your question, a 23.4 degree rotation is cos(23.4 degrees) + sin(23.4 degrees) * i.

[Coming up: why rotation by x radians is also equal to e^(ix). This is often presented as a sophisticated calculus result, but it really shouldn’t be.]

[Xema]

@Captain_Amazing

For what it’s worth, I feel the same way about a lot of quantum physics, though…

Feynman is rumored to have said: “If you think you understand quantum theory - you don’t understand quantum theory.”

[Thudlow_Boink]

@Captain_Amazing

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.

If I understand correctly, this is pretty close to how the ancient Greeks who discovered irrational numbers felt about them.

[eleanorigby]

@Blue_Mood

Inspired by the fifth post (and others) of this thread, a non-mathematical explanation of pi. To really play along you’ll need some string (a shoelace will do), a ruler, and a few circular objects such as CDs, coins, a mug, etc.

That was me! I’m SOMEBODY now!

This is very useful information because, as you have just seen, it is a lot easier to measure across a circle than around it. So if you need to know how big a circle is all the way around [circumference], you can cheat by just measuring across it[diameter], then multiplying that by 3 and rounding up a bit.

This is a very good explanation and makes sense. I will attempt to remember this, but (sadly) I know I will forget it.
Then WTF is x=pir[sup]2[/sup]

(That’s supposed to be Pi r squared. I don’t know how to make a pi on my keyboard. Is is the area of a circle?)

I do have to say this: this thread was for peoples who don’t get Pi. It seems to be full of math type peoples. It’s nice to see all of you, but I don’t understand most of what you’re saying. I’m willing to just stay over in the corner and look cute, ok?

[Ruminator]

@eleanorigby

Then WTF is x=pir[sup]2[/sup]

(That’s supposed to be Pi r squared. I don’t know how to make a pi on my keyboard. Is is the area of a circle?)

Yes it’s area of a circle. However, a verbal explanation won’t help as much as a visual one.

If you’re already familiar with how to calculate area of a rectangle, then the πr[sup]2[/sup] is the same thing. You take a circle and chop them up into pizza triangles and rearrange them to approximate a rectangle (parallelogram). You’ll notice that the area is base multiplied by height which is another way of saying πr[sup]2[/sup]. As I said before, a series of pictures showing this concept is easier to understand than a text description.

I don’t have a good link showing such pics handy at the moment. A google search shows Archimedes method but the pictures there isn’t quite as intuitive as what I was thinking.

[Indistinguishable]

@eleanorigby

Is is the area of a circle?

Yes.

As for how to derive this fact (well, you might not care, but why bother mentioning the formula without seeing where it comes from?): suppose you have a circular pizza, and you take some really tiny slice of it. The slice takes up some fraction of the whole circle; this will be the ratio between the area of the slice and that of the whole circle, and will also be the ratio between the length of the crust on the slice and the circumference of the whole circle. Thus, area of circle = circumference of circle * area of slice/length of crust on the slice.

Well, the slice is a rounded triangle, but since it’s so tiny, we can think of it as actually a triangle, so that the ratio of its area to its base is half its height (think about how two triangles combine into a rectangle). Thus, area of slice/length of crust on the slice = half the radius, and we can conclude, area of circle = circumference of circle * half the radius = π * twice the radius * half the radius = π * radius * radius.

[Ruminator]

@eleanorigby

I don’t know how to make a pi on my keyboard.

Btw, I don’t know what other people do but I always use the charmap program. It’s built into Microsoft Windows. It has all the Greek symbols and other wingdings like ♀♂ and accented foreign language characters like ȩ Ȇ ȍ. I also have a Mac but don’t know the equivalent steps on Mac OSX.

It seems to be full of math type peoples. It’s nice to see all of you, but I don’t understand most of what you’re saying.

I know how you feel. Aptitude and talent in math does not necessarily translate into ability to actually teach it in clear language so that folks understand. My first calculus teacher was very smart but horrible at explaining it.

For example, Albert Einstein had the talent to invent the Theory of General Relativity but I don’t think you’d want to learn it from him. All the newer books written by later physicists with visuals and metaphors would be easier to understand.

Lastly, if Indistinguishable and I both have a math explanation, take his. He teaches it every day. I just use math to estimate my electricity bill.

[Indistinguishable]

@Ruminator

Lastly, if Indistinguishable and I both have a math explanation, take his.

Thanks, though, in this case, I’d say we have the same explanation (and I think I actually prefer your presentation). As for that illustration you were looking for, how about this?

[Ruminator]

@Indistinguishable

As for that illustration you were looking for, how about this?

Yes, that’s the pic I was thinking of. And it was clever of the illustrator to use alternating colors so it’s easier to see the “transformation” from circle to parallelogram.

And here’s the crazy thing… if one can visualize chopping up those pizza wedges into smaller and smaller triangles, you’re doing calculus – or at least the introductory intuitive thinking behind calculus.

[Jragon]

@Ruminator

Btw, I don’t know what other people do but I always use the charmap program. It’s built into Microsoft Windows. It has all the Greek symbols and other wingdings like ♀♂ and accented foreign language characters like ȩ Ȇ ȍ. I also have a Mac but don’t know the equivalent steps on Mac OSX.

I just go to Wikipedia and copy+paste it from the article on “Pi.” If I need something more arcane I’ll go their list of Unicode characters and ctrl+f what I’m looking for (sometimes it takes a few tries since ctrl+f finds substrings as well).

[Indistinguishable]

Finally, returning to deliver on my promise:

@Indistinguishable

Coming up: why rotation by x radians is also equal to e^(ix). This is often presented as a sophisticated calculus result, but it really shouldn’t be.

Now, what I’m going to show is that, far from being some exotic coincidence, this actually follows quite directly from the defining properties of radians, i, and the function e^z. In fact, when we boil it down, all the claim is saying is this: “Suppose you keep facing the origin while moving to your side at a speed proportional to the distance between you and the origin. Then you’ll simply trace out a circle around the origin at the corresponding constant speed.”

How to see that this is the entire content of the claim? Well, let’s recall some definitions.

Radians are the units in which the the angle of a circular arc is the ratio of its arclength to its radius.

i, as we just saw, can be thought of as “Turn half of a full reversal [i.e., turn 90 degrees]”.

Finally, suppose some quantity’s rate of change is proportional to its value, so that the ratio between the two is always z per time unit. Then e^z is the amount by which the quantity multiplies per time unit.

Alright, so what?

[To be continued]

[Indistinguishable]

So applying the third definition, e^(ix) is the ratio by which a quantity multiplies per time unit, if the ratio of its rate of change to its value is always ix per time unit. Put another way, by rescaling units, e^(ix) is the ratio by which a quantity multiplies after x time units, if the ratio of its rate of change to its value is always i per time unit.

Now, thinking of the quantity as our position, and applying the second definition, we see that this is the ratio by which our position multiplies after x time units, if our velocity is always equal to our position rotated 90 degrees, per time unit. That is to say, e^(ix) is the cumulative effect on our position after x time units, if we constantly face the origin and move to our side at the speed equal to our distance from the origin per time unit.

But since we are always moving to the side, never in or out, our distance from the origin cannot be changing. Thus, we are simply tracing out some circle around the origin, at speed equal to our distance from the origin per time unit.

Finally, from the first definition, we see that this means we are tracing out a circle at a radian per time unit. Thus, we conclude, e^(ix) is rotation by x radians. And since π radians is a full reversal, we have that e^(iπ) = - 1. Q.E.D.

[Indistinguishable]

(I know, this is the thread for people who don’t understand things; what am I doing here? Well, I’m trying to help people understand things. This specific question was mentioned and asked about several times above, so I thought I’d write up my thoughts on how Euler’s Theorem can be perfectly well understood without knowing anything about Taylor series or the theory of differential equations or even any trigonometry; indeed, using nothing from calculus but the first-week idea of “rates of change”. Though, as always, seconds after submitting the post, I wish I’d worded some things differently. Ah well; if things are unclear, I can always answer questions.).

[Indistinguishable]

(For those who are comfortable with calculus terminology, I think it might be best to recast the argument in that form: e^x is defined as its own derivative; therefore, e^(ix) has a derivative which is i * itself. But since (i *) is 90 degree rotation, this means e^(ix) has a derivative which is always perpendicular to it, and thus e^(ix) isn’t changing in in magnitude (over real inputs, anyway). Therefore, e^(ix) must trace out a circle; since its derivative has the same magnitude as it, it is furthermore tracing out this circle at 1 radian per input unit. Finally, as the starting value is e^0 = 1 = rotate by nothing, we can conclude that e^(ix) is simply rotation by x radians)

[MindWanderer]

I’ve studied differential equations and used Euler’s Theorem for breaking apart certain kinds of Diff Eq’s but it was all a bit of a mystery to me before. The textbook I had covered Euler’s Theorem by saying it made sense from the infinite series, but the way they covered it always seemed like a kind of fudge, or just something that was defined to be the case.

That always seemed silly to me, so I’m glad to read your explanation of the whole thing Indistinguishable. It clears complex numbers up a lot for me. I knew about the idea of complex numbers as vectors, but I never thought about them as “rotations”. I think that is the key thing with i.

The similarities between hyperbolic and regular trigonometric functions were always a bit mysterious, but now when I’ve looked at the exponential form of trig functions again it makes a lot more sense, because those functions look similar in exponential form.

[eleanorigby]

Is this chopping up of a circle into wedges and then rearranging them to find the area a NEW (within the last 15 years or so) technique in schools? Because I will tell you this: it makes a LOT more sense to me than how I was taught (graduated HS in 1980), which was brute memorization or endless droning proofs. When I think of the area of a circle (not something I do much in everyday life), I tend to SEE the circle in my head as being overlayed on a grid–if one could count the squares, one would have the area. (I think–maybe not the actual number, but the concept–am I crazy?)
My kids started learning algebra in elementary school (along with some geometry). It still blows my mind to see a 10 year old solve for x–but it makes sense. Afterall, every story problem is really just solving for x, as is every math problem, really…
I don’t have to time to read all the posts this morning, but I wanted to say it’s fun to read the enthusiasm and intelligence displayed here. I can tell indistinguishable, for just one, loves to teach and loves his (her?) subject. It’s very refreshing.

[Dr.Love]

Regarding the relationship between the area and perimeter of a circle, I found a very nice article (which is meant as a primer for calculus) that has a good explanation with pictures, but doesn’t deal with ratios of slices directly.

[Napier]

Pi goes way beyond its endless representation and its irrationality. It is actually transcendental, too. It can’t be a root of any polynomial constructed of integers.

Also, because of Einsteinian general relativity, it is only the ratio of circumference to diameter if the circle encloses zero mass.

[Nicolas_Bourbaki]

@eleanorigby

Is this chopping up of a circle into wedges and then rearranging them to find the area a NEW (within the last 15 years or so) technique in schools? Because I will tell you this: it makes a LOT more sense to me than how I was taught (graduated HS in 1980), which was brute memorization or endless droning proofs.

Well, Archimedes is 22 centuries old, so ‘new’ is kind of a depressing word for it. Not your fault though, I didn’t see that picture in a classroom until college, so I can push your upper bound down from 15 years to 2. I think high school is always gonna suck; read the essay “Lockhart’s Lament” from the other thread if you haven’t already. If I was king of everybody, I’d toss all of HS math (save geometry) and make that required reading.

@eleanorigby

When I think of the area of a circle (not something I do much in everyday life), I tend to SEE the circle in my head as being overlayed on a grid–if one could count the squares, one would have the area. (I think–maybe not the actual number, but the concept–am I crazy?)

Exactly exactly exactly. The only hitch is – like you were getting at – is that there’s some squares that’re straddling the border. Just keep shrinking your squares and shrinking your squares for more and more accuracy, and then use MATHTM to prove that you’re gonna get somewhere with that. As you can imagine, that can get really fucking onerous. That’s the reason geometry is still so popular after all these years; sometimes pie pieces make a really pretty proof where boxes would produce a mess, or maybe some crazy-ass shape has some hidden property that requires it to have the same area (or boundary length, or whatever, depending) as some shape you already know about, and you get to root around and find something no one else has found before.
Interesting aside; for fractals, though, box-counting arguments are often the only thing we know of that reliably works.

[Alastair_Moonsong]

@Indistinguishable

Ok, disclaimers out of the way: suppose we were to rotate the vector V by angle x. We’ll take the result and think of it as the hypotenuse of a right triangle, one leg of which is parallel to V (the other leg, therefore, being 90 degrees rotated from this). How long are the legs? Well, per basic trigonometry definitions, the ratios of these leg-lengths to the hypotenuse-length are cos(x) and sin(x), respectively. But the hypotenuse is exactly as long as V, so the leg parallel to V is just V scaled by cos(x), and the other leg is just V scaled by sin(x) and then rotated 90 degrees. Thus, we see that, in general, a rotation of angle x is given by cos(x) + sin(x) * i.
And so, to answer your question, a 23.4 degree rotation is cos(23.4 degrees) + sin(23.4 degrees) * i.

[Coming up: why rotation by x radians is also equal to e^(ix). This is often presented as a sophisticated calculus result, but it really shouldn’t be.]

I was reading this, and it had me confused at first - I was looking for some Squared’s or Square Roots, thinking in terms of the pythagorean theorem and good ol’ fashioned geometry. Then I had one of those lightbulb moments and realized it was just vector addition. Now it makes perfect sense. Damn, I love math!

Will work on the whole “why rotation by x radians is also equal to e^(ix)” bit when I have more time; class starts in 15 minutes. I just wanted to let you know your efforts are not going to waste, I really am reading all your posts and thinking through the processes you outline.