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Old 01-20-2010, 05:08 PM
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Metaphor for imaginary numbers?


Does anyone know a metaphor to use for imaginary numbers so I can understand them a little better?

Last edited by petew83; 01-20-2010 at 05:08 PM.
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Old 01-20-2010, 05:34 PM
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Here's the simplest, which other posters can expand further on:

First of all, we need an analogy for negative numbers. If you're looking at a number line, then multiplying something by -1 is equivalent to flipping it around to the other side of the line. If you make two 180 degree turns, you're back to where you started, which is why -1 squared equals 1.

But now we need something that, if you multiply it by itself, gives you -1. In our analogy, then, we want something that, if you do it twice, has the result of turning you around 180 degrees. Well, then, how about a 90 degree turn? If I turn 90 degrees left, and then turn 90 degrees left again, then I've turned around 180 degrees. So we have to turn our number line into a number plane, because now we've got two axes.

But wait, you might say, there are two different ways you can turn 90 degrees. You could turn left or right. But that's OK, because i and -i both square to -1, so we can just say that turning in one direction is multiplying by i, and turning in the other direction is multiplying by -i.

This analogy is actually remarkably robust, and can be extended to explain just about everything about imaginary numbers and complex numbers (combinations of imaginary and real).
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Old 01-20-2010, 05:56 PM
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The above is pretty much exactly how electrical engineers use complex (real+imaginary) numbers.

j amounts to a 90 degree phase shift, and all the math works from there. Oh, yeah, EE's use j =sqrt(-1) instead of i, because i is used for current.
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Old 01-20-2010, 05:59 PM
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Originally Posted by Chronos View Post
Here's the simplest, which other posters can expand further on:

First of all, we need an analogy for negative numbers. If you're looking at a number line, then multiplying something by -1 is equivalent to flipping it around to the other side of the line. If you make two 180 degree turns, you're back to where you started, which is why -1 squared equals 1.

...

But wait, you might say, there are two different ways you can turn 90 degrees. You could turn left or right. But that's OK, because i and -i both square to -1, so we can just say that turning in one direction is multiplying by i, and turning in the other direction is multiplying by -i.
Actually, the analogy I heard was the unit circle - imagine a circle of diameter 1 on the real-imaginary plane as described above. Turning 90deg is going imagineary, 180 deg is minus 1. Then -i is 270 degrees. Twice 20 is 180(+360)
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Old 01-20-2010, 06:12 PM
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To expand further on what Chronos said: Behind his analogy is the fact that we can (and mathematicians do) think of imaginary and complex numbers not only as operating on the plane, but as points in the plane itself. Traditionally, the real numbers are thought of as living on the horizontal axis and the imaginary numbers on the vertical axis, with i one unit up and -i one unit down from 0, which lives at the origin. The complex number a + bi is then the point with coordinates (a,b).

Multiplying a + bi times i gives -b + ai, and (-b,a) is the point you get by rotating (a,b) 90 degrees counterclockwise. If you multiply instead by -i, you get b - ai, and (b,-a) is the point you get by rotating 90 degrees clockwise.

Regarding the unit circle mentioned by md2000, those are the points in the plane at a distance of 1 from the origin, including 1, i, -1, and -i, but, of course, many others as well. If you have a complex number z on the unit circle, you can measure the angle from the positive real axis to z; multiplying by z rotates the whole plane by that angle.

If you multiply by a complex number not on the unit circle, you get not only a rotation of the plane but an expansion (if the number is outside the circle) or contraction (if the number is inside).

This is all more than an analogy. To many (most?) mathematicians, complex numbers are the points in a plane.
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Old 01-20-2010, 07:30 PM
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Here's the simplest, which other posters can expand further on:
The only major point I'd emphasize on top of this is that every combination of scaling and rotation by some angle amounts to a complex number; in fact, complex numbers are simply the same thing as scaling and rotation. "get five times larger and rotate 37 degrees" is just as good a description of a complex number as "5 * cos(37 degrees) + 5 * sin(37 degrees) * i".

(I've written a slightly more expository account of this on the boards before, which may be of some help in grasping the arithmetic details.)

Last edited by Indistinguishable; 01-20-2010 at 07:33 PM.
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Old 01-20-2010, 08:55 PM
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Mathematicians bend the rules of math when it comes to taking the square root of a negative number. I guess they love numbers so much, they weren't gonna let their own rules stop them from finding a solution. So, the solution to a negative square root yields an imaginary number. That's the simple explanation. They go on to do a lot of weird things with imaginary numbers, so let's leave it at that.

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Old 01-20-2010, 09:30 PM
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You need to understand that mathematics (at least as practiced now) is about playing around in a logical way. Invent some new object with well defined rules, then start proving theorems. If you come up with some easy to state, but hard to prove theorems, mathematicians will say, "Cool". If you figure out that these theorems are useful in electrical engineering or physics, the mathematicians may not be very interested.

So with imaginary numbers, let's invent an object, call it "i" which can be multiplied and added with other imaginary numbers or "real" numbers. For fun, let's suppose that i*i = -1. Can we define the operations so there are no logical difficulties? Yes! Can we prove some cool theorems as defined above? Absolutely! Is it useful? Tremendously.

Since you asked for an analogy, let me give you one that is a bit complicated, but displays some of the spirit of this intellectual game called mathematics.

Imagine the Rubik's Cube. Let's invent the object L, which corresponds to twisting the left hand side 90 degrees clockwise. We'll also invent objects R, T, B, F, and K, which correspond to twisting the right, top, bottom, front, or bacK by 90 degrees. Now let's define multiplication of L by T to mean, first do T (twist the top), then do L (twist the left). We'll write this as LT.

Now you can start playing and proving theorems, like L^4 = 1 (four twists brings you back to the start), or L^3*L = 1, which implies that L^3 is the multiplicative inverse of L, which we can write as L^3 = L^(-1). We can now start implementing complicated moves like LTL^(-1) and proving cool theorems. The mathematicians are now happy because they have invented some objects and proved some cool theorems. Rubik's cube devotees might not be interested in the math, except when the math guys use those theorems to show them how to solve the Rubik's cube.

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Old 01-20-2010, 09:53 PM
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So with imaginary numbers, let's invent an object, call it "i" which can be multiplied and added with other imaginary numbers or "real" numbers. For fun, let's suppose that i*i = -1. Can we define the operations so there are no logical difficulties? Yes! Can we prove some cool theorems as defined above? Absolutely! Is it useful? Tremendously.
What's interesting is that complex numbers actually have fewer logical difficulties than the real numbers that you're familiar with. With just real numbers, there are a bunch of questions that you can't give a meaningful answer to, like "What's the square root of -1", or "What number can I take the sine of and get 2", or "What's the logarithm of -1". Complex numbers were invented to answer the first of those questions, but it turns out that once you have complex numbers, you can also answer the second and third questions without adding any more complexity to the system. You can even take square roots, logarithms, or trig functions of complex numbers, and you still just get complex numbers.
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Old 01-20-2010, 10:00 PM
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Although...

You do begin to add a little hassle, in that logarithms (even logarithms of positive real numbers) are no longer unique (indeed, except for 0 which has no logarithm, every complex number has infinitely many complex logarithms), and this non-uniqueness seeps all over the place, giving non-uniqueness of exponentiation (e.g., there are infinitely many complex-valued exponential functions with base 2), non-uniqueness of taking roots (e.g., there are three complex numbers which are cube roots of 2), and myriad other things.

But, of course, what counts as hassle and what doesn't is in the eye of the beholder.

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Old 01-20-2010, 10:17 PM
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Originally Posted by Jinx View Post
Mathematicians bend the rules of math when it comes to taking the square root of a negative number. I guess they love numbers so much, they weren't gonna let their own rules stop them from finding a solution. So, the solution to a negative square root yields an imaginary number. That's the simple explanation. They go on to do a lot of weird things with imaginary numbers, so let's leave it at that.
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You need to understand that mathematics (at least as practiced now) is about playing around in a logical way. Invent some new object with well defined rules, then start proving theorems. If you come up with some easy to state, but hard to prove theorems, mathematicians will say, "Cool". If you figure out that these theorems are useful in electrical engineering or physics, the mathematicians may not be very interested.
While this is all true, it's not at all necessary to think of complex numbers as a purely formal abstraction. They did historically first get discovered this way, but history gets everything wrong.

As Chronos essentially pointed out, "complex number" is just another way of saying "combination of rotation and scaling" (just as "(possibly negative) real number" is just another way of saying "combination of scaling and 180 degree increment rotation"); this is a simple, fairly concrete geometric notion which is as much a part of everyday experience as any of the other basic mathematical concepts people consider mundane rather than exotic (e.g., natural numbers or negative numbers or non-whole ratios or vectors or what have you). Even small children have an intuitive understanding of complex numbers; they just don't realize that that's what their understanding of rotations and scalings is.

Complex numbers are no more exotic or removed from everyday experience than the integers. One doesn't have to resort to electrical engineering to see them in the world around us; every time one takes two lefts to achieve a U-turn, they're putting into action their knowledge of i^2 = -1.

Last edited by Indistinguishable; 01-20-2010 at 10:21 PM.
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Old 01-21-2010, 09:10 AM
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Complex numbers are more than just playing games. You cannot solve cubic or fourth degree equations without them (and that was the first use of them). Ask any electrical engineer what he would do without them.

As for every number having infinitely many logarithms, that is not qualitatively different from the fact that every number has two square roots. Both 2 and -2 are square roots of 4 and both i and -i are square roots of -1.

While most mathematicians think of complex numbers as being the points of the plane, perhaps it is better, as an answer to the OP to think of them as rigid motions of the plane that preserve orientation. The latter means that clockwise remains clockwise.
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Old 01-21-2010, 09:19 AM
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... You cannot solve cubic or fourth degree equations without them (and that was the first use of them). ...
You can't solve many quadratic equations without them, e.g., x^2 + 1 = 0
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Old 01-21-2010, 10:00 AM
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What always seemed amazing about complex numbers, to me, is this. You can define this number "i" to be the square root of -1. Combinations of this and real numbers give you a two-dimensional number set, represented by the complex plane.

Now, what about if someone asks for the cube root of -1? My first, instinctive thought would be that this number would be some other abstraction, not able to be represented by a point on the plane defined by complex numbers with i. But, amazingly, the cube root of -1 can be formed by a combination of normal, real numbers, and that square root of -1 that we defined earlier.

Who'da thunk it?
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Old 01-21-2010, 10:59 AM
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You can't solve many quadratic equations without them, e.g., x^2 + 1 = 0
Right, but the point is that in many cases solving cubic equations involves imaginary numbers, even if their solution is real. That's why mathematicians first came up with them.
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Old 01-21-2010, 12:39 PM
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As for every number having infinitely many logarithms, that is not qualitatively different from the fact that every number has two square roots. Both 2 and -2 are square roots of 4 and both i and -i are square roots of -1.
Or for a closer analogy, it's even less qualitatively different than the fact that any given number has infinitely many arcsines.
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Old 01-21-2010, 12:44 PM
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As for every number having infinitely many logarithms, that is not qualitatively different from the fact that every number has two square roots. Both 2 and -2 are square roots of 4 and both i and -i are square roots of -1.
Also potentially considered a new hassle, in comparison to the context of just the nonnegative real numbers (scaling with no rotation).

(Hell, in my experience TAing, I'd say this has to be one of the top five hassles for students (well, I suppose it's not much of a hassle if you just constantly ignore it)...)
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Or for a closer analogy, it's even less qualitatively different than the fact that any given number has infinitely many arcsines.
Indeed. As you say, it's essentially just the same fact. But it does now (in the complex context) bleed through into a lot more situations than it did before, whereas previously, one could ignore this for logarithms, exponentiation, cube roots, etc.

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Old 01-21-2010, 12:56 PM
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What always seemed amazing about complex numbers, to me, is this. You can define this number "i" to be the square root of -1. Combinations of this and real numbers give you a two-dimensional number set, represented by the complex plane.

Now, what about if someone asks for the cube root of -1? My first, instinctive thought would be that this number would be some other abstraction, not able to be represented by a point on the plane defined by complex numbers with i. But, amazingly, the cube root of -1 can be formed by a combination of normal, real numbers, and that square root of -1 that we defined earlier.

Who'da thunk it?
Yes, this is nice. But let's take the mystery out of it for anyone who's unfamiliar by showing why it happens:

As noted before, complex numbers are just combinations of rotation and scaling, with multiplication being given by sequencing these; it's easy to see that these already are closed under arbitrary nth roots (just divide the rotation angle by n, and take the nth root of the scaling factor).

So the interesting thing being pointed out is really that we can get arbitrary combinations of rotation and scaling for free, once we've introduced 90 degree rotation. That is, every combination of rotation and scaling is of the form (some scaling with no rotation) + (some scaling with 90 degree rotation) [more conventionally written a + b * i].

And why is this? Why, this is just the familiar fact that in two-dimensional space, given some reference vector, any other vector can be decomposed into the sum of components parallel and perpendicular to the reference vector (the parallel component being a scaling of the reference vector, and the perpendicular component being a scaling of the reference vector rotated 90 degrees). The basic and familiar geometric fact that lets us describe points in the plane with x and y coordinates. That's all there is to it.
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Old 01-21-2010, 02:05 PM
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As noted before, complex numbers are just combinations of rotation and scaling,
I think it's easy to trivialize this connection now. However, it's interesting to note that the rules for imaginary numbers were created 1572 but the idea for 2-dimensional complex plane to represent imaginary numbers geometrically wasn't published until 1799. For more than 200 years the average person had to think about complex numbers without the use of geometric intuition.

Also, my high school math classes did not teach the complex plane even though we did endless drills with algebra equations with imaginary numbers.
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Old 01-21-2010, 02:09 PM
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Yeah. Like I said, history gets everything wrong and it takes forever to clean up the cruft and damage (e.g., names like "real" vs. "imaginary" (ironically, from the same Descartes who invented analytic geometry with the realization that pairs of coordinates describe points in the plane)).

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Old 01-21-2010, 02:20 PM
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Before anyone else calls me on it, I just noticed that my example wasn't the best. The cube root of -1 has a perfectly obvious, real value.

But there are also two complex ones, so please pretend I was talking about that.
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Old 01-21-2010, 02:26 PM
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, history gets everything wrong and it takes forever to clean up the cruft and damage (e.g., names like "real" vs. "imaginary"
Being called "imaginary" was my NUMBER ONE problem learning imaginary numbers. If these are "imaginary", why are we wasting time with them? What's the point? (Of course, I could learn the rules in manipulating i^2=-1 like any other monkey, but I didn't truly learn their usefulness until much later. Later in life, I thought it was cool that MS Excel had functions for magnitudes of complex numbers without having to use Pythagorean theorem. Well, if MS Excel has them, they must not be so "imaginary" after all.)

It would have been much easier if my math teacher on day 1 forcefully explained, "today we're going to learn about imaginary numbers but let me emphasize that 'imaginary' is a terrible horrible stupid name for them."

Oddly, I never had this speed bump with "irrational" numbers. I never wondered if 3.1459 had a psychotic neurosis disorder that the rational number like 3.5 (7/2) did not.

I guess the word "imaginary" triggers more confusion than the word "irrational."
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Old 01-21-2010, 02:36 PM
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Perhaps because "imaginary" in the context of imaginary numbers was indeed coined to indicate imaginary in precisely the unfortunate sense people take it, while "irrational" in the context of irrational numbers was only ever meant to indicate "not given by a ratio" rather than "not conforming to reason", the conflation between the two being an unintentional pun. You can easily see "Oh, yeah, irrational just means 'not a ratio'", but you can't easily explain away the term "imaginary" as similarly devoid of derision.

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Old 01-21-2010, 08:55 PM
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You can't solve many quadratic equations without them, e.g., x^2 + 1 = 0
I should have been more precise. Even a cubic with all real roots cannot be solved without the use of complex numbers. There is a theorem to that effect, proved using Galois theory.
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Old 01-21-2010, 09:01 PM
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I think it's easy to trivialize this connection now. However, it's interesting to note that the rules for imaginary numbers were created 1572 but the idea for 2-dimensional complex plane to represent imaginary numbers geometrically wasn't published until 1799. For more than 200 years the average person had to think about complex numbers without the use of geometric intuition.
The average person couldn't do long division in those days. Come to think of it, I suspect that is still true. But the point is correct. Even sophisticated mathematicians of that era did not have the so-called Argand plane picture to work with.
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Old 01-21-2010, 09:12 PM
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A real answer to the question, mathgodz (kidding, ya'll are cool as hell)


A DOUBLE HELIX. Seriously, think about it as far as the number line definition and the abstract, metaphorical angle that the question is predicated on.
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Old 01-21-2010, 09:34 PM
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"irrational" in the context of irrational numbers was only ever meant to indicate "not given by a ratio" rather than "not conforming to reason", the conflation between the two being an unintentional pun. You can easily see "Oh, yeah, irrational just means 'not a ratio'"
No, it wasn't originally an unintentional pun. The Greek word a-logos, which was translated into Latin as "irrational", did have negative connotations of senselessness and illogicality ("illogicality" is in fact a cognate of alogos), just as "imaginary" has negative connotations in English.

The fact that some quantities like the square root of 2 obviously could exist as magnitudes (for instance, sqrt-2 is the length of the diagonal of a square of side 1), but could not be articulated with a reasonable (rational) numerical expression designating an integer or a ratio of integers, definitely weirded out the early Greek number theorists. The name alogos applied to such numbers conveyed a certain philosophical disparagement, not just a neutral technical designation.
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Old 01-21-2010, 09:34 PM
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Huh? What you just said is harder to understand than complex numbers, not easier.

EDIT: That was in reply to Sped. Teacher.

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Old 01-21-2010, 09:36 PM
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Shucks, and I was hoping I'd said something really profound.
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Old 01-21-2010, 10:19 PM
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No, it wasn't originally an unintentional pun. The Greek word a-logos, which was translated into Latin as "irrational", did have negative connotations of senselessness and illogicality ("illogicality" is in fact a cognate of alogos), just as "imaginary" has negative connotations in English.
A) I'm afraid I'm responsible for the quote you attributed to Ruminator

B) In its defense: While there were famously many among the Greeks who were greatly uncomfortable with irrational numbers, and while "ratio" as in fraction is indeed etymologically cognate to "reason" and does indeed derive from "logos" which is indeed cognate to "logical" and so on, I can't find any evidence that the use of "logos" for fraction (as apparently coined by Euclid) was meant to explicitly carry with it the meaning of "reasonable"/"logical" as well; all I can find is that it indicated something like "the result of a reasoning (i.e., a reckoning, a calculation)". But I have no expertise in the matter and I'd be interested to see whatever evidence there is that the pun (or even full-on identification of the two senses) was intentional in the coinage.

C) Regardless, to a modern speaker, it remains true that one readily distinguishes "ratio" from "reason" and thus can explain away "irrational" and "rational" in a way which one cannot so easily do with "imaginary" and "real"

D) Nothing to do with you, but to Sped. Teacher: Huh?

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Old 01-21-2010, 10:41 PM
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please sirs!


I think pete83 deserves a metaphor, so lets give a metaphor instead of endless description. Plots on a line, when flipped, etc., 3D, please! Visualize, as you would metaphors that explain aspects of shapes we dont fully understand. Think about planets and their similarities to atoms. Come on, I'm not making this stuff up. Does he not want a metaphor?
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Old 01-21-2010, 10:50 PM
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I think pete83 deserves a metaphor, so lets give a metaphor instead of endless description. Plots on a line, when flipped, etc., 3D, please!
Not everything abstract can be restated as a 3-dimensional physical.

What is infinity? What is the 3-D representation of infinity? How would you draw such a thing on a piece of paper?

Maybe a 3D picture involving imaginary numbers would be quaternion. However, that makes things harder, not easier.
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Old 01-21-2010, 11:02 PM
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I can't find any evidence that the use of "logos" for fraction (as apparently coined by Euclid) was meant to explicitly carry with it the meaning of "reasonable"/"logical" as well; all I can find is that it indicated something like "the result of a reasoning (i.e., a reckoning, a calculation)".
Yes, I agree that Euclid's sense of "logos" to mean "fraction" is a strictly technical term divorced from psychic connotations of reasonableness or logicality. But the use of "alogos" to characterize irrational numbers predates Euclid. As far as we can tell, it seems to have been introduced by the Pythagoreans, for whom "alogos" definitely did have a somewhat pejorative sense. If you want to see some better-informed discussion of the issue than I can provide, try here.

And apologies to you and Ruminator for my misattribution---oops!

D) Yeah, me too.
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Old 01-21-2010, 11:19 PM
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Oh, "alogos" predates "logos"? I had no idea. That's really quite interesting. (And it makes me now suspect that perhaps Euclid was aware of the pun he was creating(?). I mean, if "alogos" was already around and had been used as a description of precisely the opposite of the sort of thing he was about to make "logos" mean, how could he not be? And yet, I see no remarks that suggest he was. I... need to read more.)

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Old 01-22-2010, 11:06 AM
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And yet, I see no remarks that suggest he was.
Well, remember that we don't know much about Euclid, and we know even less about most of the pre-Euclidean Hellenistic mathematicians whose work was swept into historical oblivion by the fantastic popularity of Euclid's work. IMO it's likely that the word "logos" for fraction was kicking around for a while before Euclid himself---long enough to gain familiarity as a narrow technical term with no philosophical baggage about rationality---but Euclid's work happens to be the first surviving example of its use.
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Old 01-22-2010, 04:28 PM
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What is infinity? What is the 3-D representation of infinity? How would you draw such a thing on a piece of paper?
The north pole of a Riemann Sphere?
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Old 01-23-2010, 05:16 AM
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Before anyone else calls me on it, I just noticed that my example wasn't the best. The cube root of -1 has a perfectly obvious, real value.

But there are also two complex ones, so please pretend I was talking about that.
Sure. And also, once you've defined i such that i2 = -1, what about some number that, when squared, yields i? Intuitively we would think "Oh no, now we need some even more imaginary numbers, and so on ad infinitum", and it ain't so. It turns out that the square root of an imaginary number is itself partly imaginary and partly real, and so we only needed to add one level of weirdness, not an infinitely recursive stack of it.
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Old 01-23-2010, 08:38 AM
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I like when the metaphors are more complicated than the referent. Here's my answer: imaginary numbers are like your crazy uncle...there's internal consistency to the things he's mumbling, but it may not make much sense to outsiders.
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Old 01-23-2010, 08:39 AM
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Since nobody has mentioned the term yet, I'll jump in: To say that adding i gives solutions to all these other equations is to say that the field of complex numbers is algebraically closed. Precisely, every polynomial with complex coefficients has a complex root. This is not true of the real numbers --- the polynomial x2 + 1 has no real root. The amazing thing, as several people have pointed out, is that creating a root for this one polynomial creates roots for all other polynomials. This result is often called the Fundamental Theorem of Algebra. One interesting thing, tangential to the main discussion here, is that it's very difficult to prove this using only algebra. My favorite proof, obviously, involves some topology. Another uses complex analysis. (That's a technical term, not a descriptor. Here's another somewhat unfortunate term: "Complex" here means something built out of simpler parts, not "complicated.")

Last edited by Topologist; 01-23-2010 at 08:42 AM.
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Old 01-23-2010, 03:54 PM
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Eh, it's impressive enough that introducing complex numbers completely closes the system with respect to polynomials, but I think it's even more impressive that it also closes it with respect to things like inverse trig functions and logarithms. I imagine that this stems from the fact that trig functions and the like can be expressed in terms of infinite sums, but that still adds another layer of nontriviality.

Quote:
(That's a technical term, not a descriptor. Here's another somewhat unfortunate term: "Complex" here means something built out of simpler parts, not "complicated.")
More specifically, "complex" in a mathematical context refers to numbers with both a real and imaginary part.


Just for fun, by the way: Several others have mentioned that complex numbers suffice to find the square root of i, but I think it's instructive to do the demonstration. Let x = 1/sqrt(2) + 1/sqrt(2)*i . Now let's take x*x. We can do this the same way we'd multiply anything of the form (a+b)*(c+d):

x*x = (1/sqrt(2) + 1/sqrt(2)*i) * (1/sqrt(2) + 1/sqrt(2)*i) = 1/2 + 1/2*i + i*1/2 + 1/2*i*i
Replacing the i*i in the last term by -1, and combining the two middle terms, we have
x*x = 1/2 + i - 1/2 = i.
So x*x = i, which means that x is the square root of i.
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Old 01-23-2010, 05:07 PM
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Or, as noted before: since i is a 90 degree turn, the sqrt(i) is something which, when repeated twice, gives a 90 degree turn; namely, a 45 degree turn (or a 235 degree turn). Much simpler this way, eh?

(But, if one insists on breaking such motions down into their parallel and perpendicular components, yes, "v turned 45 degrees" is the same as "(v scaled by a factor of 1/sqrt(2)) + (v scaled by a factor 1/sqrt(2) rotated 90 degrees)", in accord with the side ratios of 45-45-90 triangles, as determined by the Pythagorean theorem. As shown above by Chronos, this all works out, as it must; indeed, one can go the other way around and use complex numbers to give a simple derivation of the Pythagorean theorem. It all hangs together, as it must.)

Last edited by Indistinguishable; 01-23-2010 at 05:10 PM.
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Old 01-23-2010, 07:52 PM
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Originally Posted by Indistinguishable View Post
... indeed, one can go the other way around and use complex numbers to give a simple derivation of the Pythagorean theorem.
Is this true? Does it depend on the interpretation of the absolute value of a complex number being its distance from the origin? And doesn't that interpretation depend on the Pythagorean theorem? (I may be off base, but I can't think of how to do this without it becoming a circular argument.)
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Old 01-23-2010, 08:38 PM
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Like so:

Suppose given a right triangle in a plane, with vertices a and b on the hypotenuse and c opposite to the hypotenuse. Let the vectors A, B, and C denote the vectors c - b, a - c, and a - b respectively (thus A + B = C). We must show that ||A||^2 + ||B||^2 = ||C||^2.

Let u be a unit length vector pointed in A's direction. Since A is perpendicular to B, there must be a 90 degree rotation which causes u to be pointed in the same direction as B; let this be the rotation denoted by i. There must also be some rotation which causes u to be pointed in the same direction as C; call this rotation theta.

Thus, A = ||A|| * u, B = ||B|| * i * u, and C = ||C|| * theta * u. But also A + B = C, so we have that (||A|| + ||B|| * i) * u = ||C|| * theta * u.

Since u is a nonzero vector, we can conclude from this last equation that ||A|| + ||B|| * i = ||C|| * theta; essentially, we've given both "rectangular" and "polar" descriptions of the complex number sending u to C. Now, take the complex conjugate of both sides to obtain that ||A|| - ||B|| * i = ||C|| / theta. Finally, multiply these two equations together to see that (||A|| + ||B|| * i)(||A|| - ||B||*i) = ||C|| * theta * ||C|| / theta, which simplifies to ||A||^2 + ||B||^2 = ||C||^2.

Of course, we could just have written "90 degree rotation in the chosen direction" each time instead of "i", and this would be a straightforward geometric proof. But that's the point: working with complex numbers is doing the geometry of scaling and rotation of vectors; the two are exactly the same thing, whether you write it this way or that.

Last edited by Indistinguishable; 01-23-2010 at 08:41 PM.
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Old 01-23-2010, 08:55 PM
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Incidentally, in case anyone is wondering what prevents this proof from being applied to non-Euclidean geometries as well (where the Pythagorean theorem fails), it's the fact that such geometries do not form affine spaces (so one cannot perform vector arithmetic with the differences between points).
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Old 01-23-2010, 09:40 PM
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Quote:
Originally Posted by Chronos View Post
Eh, it's impressive enough that introducing complex numbers completely closes the system with respect to polynomials, but I think it's even more impressive that it also closes it with respect to things like inverse trig functions and logarithms. I imagine that this stems from the fact that trig functions and the like can be expressed in terms of infinite sums, but that still adds another layer of nontriviality.
You don't need to appeal all the way to infinite sums. Logarithms come for free from the ability to take logarithms of positive reals (since every nonzero complex number is a product of scaling by a positive real and rotation by some angle, with rotation being an exponential function of its angle). And once you have logarithms (and square roots), you can get all the inverse trig functions from them for free (just as exponential functions give the trig functions for free, again as a consequence of rotation being itself an exponential function).

Last edited by Indistinguishable; 01-23-2010 at 09:41 PM.
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Old 01-24-2010, 01:12 PM
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Indistinguishable: Thanks (for the proof of the Pythagorean theorem). I hadn't seen that one before. Nice.
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