OK, I’m missing something here. Why is e^z the amount by which the quantity multiplies per time unit? How did you go from
to
I’m probably just misinterpreting something here. Could you give an example with values, or at least variables? i.e. an example where Rate of Change = f(x)/t, Quantity = x, Ratio = (some number), etc, to show the connection between the ratio and e?
Here’s another pi oddity.
Toss a stick 1 inch long randomly over stripes 1 inch apart. The probability the stick contacts a line (contacts divided by tosses) = 2/pi! Do this for many tosses and keep track of the numbers, and you can calculate pi = 2*tosses/contacts.
This works for any length, of course, as long as the length of the stick = the width of the stripes. Pi gets in there because the stick can land at any angle to the stripes.
What do imaginary units amount to physically? I know they’re used in electrical engineering and have some relation to wave motion but I don’t entirely grasp what it amounts to. Is it an accounting method where nothing really “happens” but it makes the numbers work out pretty? Do some waves literally travel through an imaginary spacial dimension? Can waves travel in there and then “appear” in our concretely perceived dimensions if they collide with something and get rotated?
I know they exist like any other number, but since they generally are graphed with another axis appended I guess I may just be interpreting how they work concretely in a strange way.
I’m not sure what kind of answer you are looking for; perhaps it would be easiest if you started by giving the kind of answer you would expect for “What do real number units amount to physically?”. [Damn, I hate those names “real” and “imaginary”. If no one ever heard them, I imagine there’d be far less confusion today concerning the concepts]
Is the interpretation of imaginary quantities as rotations not a sufficiently concrete physical interpretation? Like I said, you don’t have to wait until electrical engineering and the study of waves to see applications of complex numbers; anywhere where you might be willing to talk about scaling and rotation [i.e., magnitude and two-dimensional orientation], you can phrase that talk in terms of complex numbers.
Yeah, that was something I wanted to reword after I wrote it; what I mean isn’t e^z * time = rate of change, but I didn’t write it at all clearly.
Anyway, the important thing is that what I was doing there wasn’t solving an equation. I was giving a definition. Specifically, what I was mentioning was a definition of some function of z, which we could refer to with a pedestrian name like “exp(z)”, but which we usually write with the special notation “e^z” instead. Now, this latter notation suggests that the function should be defined by first giving a general definition of what it means to raise one number to another power, and then defining a very special number e to serve as a base. But this isn’t the way I’m doing things; in fact, at the moment, I’m not going to bother thinking of e^z as raising a number to a power at all. Why not? Well, there are some technical reasons not to, but my main motivation is this: what makes e important, and what give it its most natural definition, are the properties of this function e^z. So, making the definition of e prerequisite to that of e^z is somewhat putting the cart before the horse. Instead, it’s the function e^z which we will define directly, and then, later, if anyone cares, we can show that this function does act like taking powers with some fixed base, and define e to be the value of this function when z = 1.
Fine. So I’m going to define some function exp(z), aka e^z. What’s the idea I am trying to capture with it?
Well, sometimes, we say that a quantity grows exponentially, and what we mean by this is that its rate of change is proportional to its value. That is to say, (rate of change of the quantity)/(value of the quantity) is constant. For example, if I had a bank account which was continuously earning interest at a rate of 5%/year, then, at any moment, the rate at which I was earning money would be my current amount of money * 5%/year.
Note that this doesn’t mean that, if I have 100 dollars now, then I’ll have 105 dollars in a year. In fact, I’ll end up having even more than that. Why? Because over the course of the year, my money keeps increasing, so my earnings rate keeps increasing as well; it doesn’t stay put at the rate $5/year, because my money doesn’t stay put at $100. It’s important to note that difference between “My money is currently increasing at the rate $5/year (equivalently, $10/2 years, $2.50/six months, etc.)” and “My money will go up by $5 in a year (or by $10 in two years, or by $2.50 in six months, etc.)”. This difference is what’s meant by the “compounding” nature of interest.
So, getting back to the point, let’s now define our function exp(z) [aka, e^z]. We want to use it in order to determine the cumulative effect of all this exponential growth. So, here is the definition: e^z is the amount my money will have multiplied by after time t, if it’s being stored at an interest rate of z/t.
To put it another way, if we pick such units as that my current balance is 1, and my interest rate is z, then after 1 unit of time, my amount of money will be e^z. This isn’t something for us to deduce; this is just a definition we are giving. So, for example, however many dollars I’ll have in a year with an interest rate of 5%/year if I start with 1 dollar, we’ll call that e^(0.05).
Now, I worded that in terms of “money” and “interest rate”, but it really applies to any situation of exponential growth; just replace “money” by whatever quantity, and “interest rate” by the fixed ratio between the quantity’s rate of change and its value.
Thus, for example, instead of talking about “money”, we can talk about positions (relative to some origin). In that context, e^(ix) is the amount my position will have multiplied by after 1 unit of time if my velocity is always ix * my position (in corresponding units). That is to say, if my velocity is always x times as large as my position, but rotated 90 degrees. Which is to say, if I am always facing the origin, but running to the side at a speed x times as large as my distance from it.
Well, since I am always running to my side, neither forwards nor backwards, I cannot be getting any closer to or further from the origin. Thus, I am travelling in a circle around it. And so, e^(ix) is what will happen to my position after 1 unit of time if I am travelling in a circle around the origin at a speed x times as large as its radius. Recalling the definition of radians, what will happen is that I will rotate by x radians. Thus, we can conclude, e^(ix) is rotation by x radians.
Well, perhaps the makers of HS (and Jr high) curriculums should have paid more attention to Archimedes…
Really? You mean I said something intelligent about math? :eek: Growing up, math was such a struggle, but looking back, I see that I did not learn the way math was commonly taught. Much like a dyslexic person struggles in English. I also will never forget the student teacher in 4th grade telling me 2 things: 1. that since I was so bright in “language arts”, I “must” be able to do math as well (ha!) and 2. that it didn’t matter so much that I wasn’t good at math, because I was a girl and wouldn’t need it. True story. That was math for girls in 1971.
We’ve come a long way, baby. Or so I hope…
I never got why people could not grasp pi…Pi is the number of times the diameter of a circle can go around the circumference. In other words the circle is 3.1415~ times longer around that it is across. The number never ends and thats just how the universe works.
To me, the interesting question is why anyone would think the decimal expansion of π should eventually end. Why would it? What’s so shocking about it not doing so? Decimal representation is just one particular style of notational system, not the end-all, be-all arbiter against which all numeric quantity is to be judged.
In general, I find the amount of attention paid to the decimal representation of π (memorization and pattern-hunts and optimized calculation algorithms and the like) rather surprising; seems pretty boring to me. But then, perhaps the problem isn’t really overfocus on decimal representations, as such, but rather the overall π-fetishization process.
It’s such a shame, really. The interesting stuff about π is in its geometric properties, how circles relate to differential equations, and through these to probability and so on… Stuff that actually makes us of this object’s nature as a ratio of continuous magnitudes. But the sequence of remainders modulo 10 arising from the largest integers whose ratio to subsequent powers of 10 remains below half that of a circle’s circumference to its radius? What a contrived, convoluted object of study, and yet this is the mascot we’ve let be chosen for the subject, to be shot mutely fascinated glances and “Oh, the mysteries” coos.
“Stuff that actually makes use of”, I of course meant to say.
I know Indistinguishable answered this, but to be very specific, the answer is that it physically amounts to a 90 degree rotation. It’s really no different than asking what do the degrees (of an angle) physically amount to. It’s just a different way of expressing it mathematically with rules that allow for easier calculation under certain circumstances.
I once killed a whole work day by proving pi. I didn’t realize I was doing it at the time.
Take a circle with radius 1. Fit a triangle in it that has all three sides equal. Calculate the area of the triangle, and you get 1.73.
Do the same thing with a perfect square. It turns out to be 2.
Now use a perfect hexagon. You get 2.56.
An octagon gets you 2.83. A dodecagon (12 sides) gets you 3. Double the number of sides to 24, and you get 3.1056.
The more sides you put in, the more the area approaches the value of pi. The area outside the figure and inside the circle approaches 0.
This pattern gets you a bunch of triangle with 2 sides that are length 1. Calculate the angle between by dividing 360 degrees by the number of sides, then use this calculator to determine each triangle’s area. Multiply that by the number of sides to get the figure’s area. Say you use a 60-sided figure. It’s going to be 60 triangles that have two sides equal to 1, and the angle between them equal to 360/60 = 6. Plug these numbers into the linked calculator and you get .052264. Multiply that by 60 and you get 3.13584. Getting close to pi.
So somebody put in 10 zillion sides, calculated the area, and got really really close to pi.
There’s an applet at this Web page that demonstrates this sort of thing.
That’s interesting. Why do you suppose the outside polygons converge on pi so much more quickly than the inside polygons?
Is it too late to pop in and say 4445.178 cubic centimeters of pie? And then run away?
(Although how you are going to fit chunky rhubarb pie into the tiny nozzle of a dodgeball is a totally different issue…)
That’s utter coolness. Makes me wish I had the programming skillz to put my Timecubelike thoughts in visual form like that.
People struggling to grasp Pi are likely struggling with the many intriguing ways it behaves, the places it turns up, the clues about its meaning, even the ways it can be calculated. If your interest in Pi ends with “that’s just how the universe works”, what’s there to try to grasp?
I think the reason that people can’t grasp pi is because pi is constantly referred to as a number. It ain’t, in the sense that most people understand numbers. The way we naturally approach numbers is as units, and pi just ain’t a unit, it’s a constant that expresses a certain ratio.
The gym class example posted above contributes to that. You can use units to see that the ratio being expressed isn’t a whole one, or to give you a rough approximation in units of what pi might be, but that just cements the issue of trying to conceive of a relationship of units as another unit, when that’s not necessarily the case.
Saying pi is 3.14… is helpful in the development of engineers, whose practical application of math often ignores the foundation on which it’s based, but it’s sucky when trying to teach people what mathematics actually is.
You don’t understand pi, and want to? Read Euclid.
What do you mean by “pi just ain’t a unit…”? What’s a “unit”?
If you’re just noting that it isn’t a ratio of integers, well, yeah, sure, but I’m not convinced this is the only way we naturally approach numbers. Do we not also have some conception of “numbers” representing (ratios of) continuous quantities such as lengths? Is there anyone who would have the dialogue: “How long did you run, in miles?” “Eh, sorry, I don’t think it came out to be a number”?
I agree that saying pi is 3.14… is totally conceptually unhelpful, though, which is perhaps your main point.
I had a statistics teacher explain probability and confidence in a graphical way that similarly made a light bulb go off in my head.
Conveniently enough, I found a very good example of the sort of diagram she did right right here. The diagrams on this page are about the Senate race in '06, and have other features (like the effect of time since last survey data) that were not present in my instructor’s diagram, but the sense of a “penumbra” fading out from the sampled number and the statement that 95% certainty was just a convention whereas we could visualize it fading out smoothly from low confidence / highly precise value to sloppier range of value / more confidence… suddently all those arcane digits and signs and terms and crap got OUT of my way and I could see WTF they were actually talking about.
All math should be taught this way. Get the numbers and formulas off screen until i’ve got the bloody concept THEN give me the recipe, willya?
While searching for something else, I ran across this Straight Dope column:
I’m not sure whether it adds anything new to the discussion at this point, but for anyone interested in seeing what Cecil had to say on the subject, there it is.