When we think of numbers, our intuitive sense is to think of them in terms of
quantity: counting, measuring, or comparing quantities. And that's a good intuition for real numbers. But when you start working with more advanced math,
you find out that those numbers - the real numbers - are just a part of the picture. There's more to numbers than just quantity.
As soon as you start doing things like algebra, you start to realize that
there's more to numbers than just the reals. The reals are limited - they exist
in one dimension. And that just isn't enough.
In terms of algebra - we know that if you have a polynomial of power n,
then it has n solutions. But if you look at many polynomials, you find that if
you limit yourself to real numbers, you don't get n solutions. For
a trivial example, the equation "x2+1=0" should have two roots. But there are no real numbers that are solutions to that polynomial.
The solution to that polynomial is the square root of -1, commonly called i, or the imaginary number: it is a number which is not a member of the reals. It is a number - it is real. But it's not part of the set of real numbers. Once you actually grasp the idea that i exists, then you can start doing some really cool things.
Instead of the numbers described by algebraic equations being points on a line, suddenly they become points on a plane. Complex numbers are really two dimensional; and just like the integer "1" is the unit distance on the axis of the "real" numbers, "i" is the unit distance on the axis of the "imaginary" numbers. As a result numbers in general become what we call complex: they have two components, defining their position relative to those two axes. We generally write them as "a + bi" where "a" is the real component, and "b" is the imaginary component.
The complex fix more than just the problem of some polynomials not having enough roots. The real numbers are not closed algebraically under multiplication and addition. With the addition of i, multiplicative algebra becomes closed: every operation, every expression in algebra becomes meaningful: nothing escapes the system of the complex numbers.
Arithmetic and algebra work beautifully on complex numbers - you just treat them
as if they were polynomials, and follow the same procedures you would for doing
addition, subtraction, multiplication, and division on polynomials. For example,
(3+4i)×(4+2i)=12+6i+16i+8i2=12+22i+8(-1)=4+22i.
Complex numbers are real; but they're not part of the set that we call real numbers, which is endlessly frustrating to math geeks like me.
Why do I insist that they are real? Because there are real phenomena in the world that
behave in ways that can only be described using complex numbers. If you try to
avoid the use of the complex numbers, you'll only wind up re-inventing them under another name. They're real, they exist, and they describe real phenomena.
What's interesting about the complex numbers in an abstract mathematical sense is
that they can be treated as a superset of the reals; the reals are the set of abstract
numbers whose imaginary component is 0 - all numbers of the form a+0i. What the complex
numbers do is add a second dimension to the number. A complex number is a
number with two dimensions, which is a fascinating idea - numbers become more than just
a line - they became a plane when you use the complex numbers. We've expanded our horizons, and can talk about things that just wouldn't make sense described using real numbers.
There are numerous real phenomena which are described using real numbers. Alternating current - like the current that's probably powering your computer as you read this article - requires the use of complex numbers to describe it. To perform
computations describing most phenomena involving waves - sound, light, etc., you
inevitably wind up encountering complex numbers.
However, that second dimension comes at a cost. By adding the second dimension, we
wind up with a set of numbers that is still a field - but which does not have
a total ordering. All of our ordering properties are out the window - they no longer
make sense. (Is 1+0i greater than 0+1i? It's a meaningless comparison.)
You can take the idea of adding dimensions to numbers, and create 4 dimensional numbers. They're called quaternions, and they're quite real too. A quaternion has four components: a + bi + cj + dk. They're very useful for describing rotation. But with quaternions, by gaining those dimensions, you lose even more - they're not commutative. That is, given two quaternions X and Y, it's no longer
true that X×Y=Y×X. So quaternions are no longer a field - and a lot of
algebra gets tossed out the window.
You can keep going. There are also 8 dimensional numbers, called octonions. Octonions lose associativity: (A×(B×C)) is not equal to (A×B)×C.
John Baez (he of the n-category cafe) has described these families of multidimensional numbers with a great metaphor:
The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
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I think you must be slightly mistaken here. Unless I missed something crazy, 1/0 still has no solution in the complex plane. Is this right?
So, if you were to keep adding dimensions, you could easily imagine that you 'lose something' at each step - yes?
Does that mean there is a limit to multidimensionality?
Perhaps you can provide us a brief run-down on why the idea of multidimensional numbers in, say, 3-, 5-, and 6-dimensions is less attractive than numbers in 2-, 4-, and 8-dimensions.
Xanthir: technically, 0 is never invertible, so trying to divide by it isn't an algebraic expression.
Regarding division by zero, you can extend the complex plane by adding a "point at infinity" via a one-point compactification. This is a natural dual to zero that has similar problems (e.g., no unique polar representation).
This extension also puts circles and lines in the same class of objects - lines are just circles that go through the point at infinity. This class of "extended circles" is precisely the one that is invariant under conformal transformation - a sort of generalisation of affine transformations (rotations, scales, translations).
I'll cut this digression short now, the point being the complex numbers are, as Mark points out, a beautiful and powerful generalisation of the reals and also have some wonderful geometric properties.
CLICK!!!
I learned about complex numbers in my final year of high school, and got my head around how to use them okay but never really got past the "imaginary" tag.
At University I stduied computer systems engineering, which included a number of electrical engineering subjects. I always had trouble really understanding things like power distribution because they used complex numbers to describe it, and I couldn't figure out why "imaginary" numbers were used to describe real phenomena.
12 years later I finally get it!
Thanks Mark.
It's interesting to realize that our idea that the complexes are 2-dimensional comes in part from fixing the reals as a basis of comparison. It's common to instead take the complexes as 1-dimensional and use that as a metric for measuring dimension.
The complex fix more than just the problem of some polynomials not having enough roots. The real numbers are not closed algebraically under multiplication and addition.
What? The real numbers are certainly closed under both multiplication and addition. "Algebraic closure" refers precisely to the problem of polynomials not having enough roots.
So, if you were to keep adding dimensions, you could easily imagine that you 'lose something' at each step - yes?
The process by which you get the complexes from the reals, the quaternions from the complexes and the octonions from the quaternions can be generalized and repeated ad nauseum. You can check out the Cayley-Dickson construction on Wikipedia for more about this. However, after the octonions we understand what's going on so little that we don't even know what we have, let alone what might be being lost.
Perhaps you can provide us a brief run-down on why the idea of multidimensional numbers in, say, 3-, 5-, and 6-dimensions is less attractive than numbers in 2-, 4-, and 8-dimensions.
It's not that it's less attractive; it's that it's impossible. You can't put an algebra on 3 dimensions.
In 3, 5, or 6 dimensions, you can't always divide.
The problem with the Cayley-Dickson construction is that after 8, you get "zero divisors": the product of two nonzero members can be zero. There's also a theorem in topology that implies that things are nice only in dimensions 1,2,4, and 8 (where nice means "division algebra").
You can put an algebra on 3 dimensions -- the problem is that you can't make it a division algebra. (You always have zero divisors, for example.)
What about 16 dimensional maths? Is it even theoretically possible?
There's another thing you lose by going to complex numbers.
Integers have the property that they can be uniquely decomposed (ignoring sign) into a product of prime factors. Gaussian integers (complex numbers with integer coefficients) can also be decomposed into a product of prime numbers, but the decomposition is no longer unique.
Pseudonym, Gaussian integers are a unique factorization domain, just like the integers. Decomposition into product of primes is still unique, up to units. The difference is that the integers have only two units while the Gaussian integers have four units. No loss there.
Mark, adding one point at infinity to compactify makes sense if you want to add geometry or topology, but not if you want to do algebra. Redefining 1/0 as infinity is useless because you cannot continue doing operations with it (unless you lose all sensible properties). And do not even get me started on that stupidity called the transreals.
"What's interesting about the complex numbers in an abstract mathematical sense is that they can be treated as a superset of the reals."
Don't you mean this the other way round?
Anon,
Complex is superset of Real
Real is subset of Complex
"The complex fix more than just the problem of some polynomials not having enough roots. The real numbers are not closed algebraically under multiplication and addition. With the addition of i, multiplicative algebra becomes closed:"
That's plain wrong, isn't it? The sum of any two real numnbers is real. The product of any two reals is real. Please can you give an example of how the real number system is not closed under either addition or multiplication?
I've always liked the term 'transverse numbers', myself. (Due to Gauss, I think.)
Also, there's a nice overview of octonions here.
Strabo, 16(any) dimensional "math" is certainly possible - it's just not the same sort of math that's happening with complex numbers - depending on what we want "same" to mean, it's either unique at 2, or generalizable to 2, 4 or 8 only.
The fact, mentioned by Walt and several others above, that division algebras are only possible in 1, 2, 4 and 8 real dimensions, ties in with mathematics from all over the place, and in the end has a lot to do with Hopf fibrations, topology and the existence of some very weird stuff in dimensions 0, 1, 3 and 7.
Mark: Could you be prodded to throw out a topology post on this kind of magic, in case you haven't already? It's a bit too far from my own field for myself to be able to do a really good job of it myself...
What the EE crowd does is describe a specific kind of wave phenomena in terms of amplitude and phase, and one natural way of describing that is using complex numbers. Every representation is isomorphic to complex numbers, so yes, you are right, complex numbers are required for that. But then again, they lean towards "phasors" which is nothing but complex numbers in exponential form using funny notational conventions to disguise the fact that these are complex numbers.
There are other multidimensional structure that we can do basic algebra on. Tensors and matrix come to mind. In fact any structures that form a ring. To me it seem that rings may be more like numbers then an octonions. But that may be that I have never seen any use for octonions.
Engineers uses different representations and transforms for different problems. Complex analysis can be useful in many cases in classic EM theory, conformal maps for geometries for example.
IIRC phasors are convenient in sparse AC network analysis since it makes linear loads easy to handle. (For larger nets and nonlinear loads other techniques are better.) And yes, the convention is simplified. I believe its use is somewhat cultural contingent, since I can't remember seeing the simplest (ie most obtuse :-) convention much around here.
Another fun thing is that the EE imaginary unit is "j", since "i" is already taken for current.
"...complex numbers in exponential form using funny notational conventions to disguise the fact that these are complex numbers."
Poppycock! EE's use complex numbers all the time and recognize them as such and do not try to "disguise" them. That "funny notation" is just a convenience, not a disguise. Phasors are hardly used anyway except maybe in power systems engineering, signal processing almost never. The only "disguising" EE's do is to substitute "j" for "i" so as not to confuse it with current. and BTW phasors represent a signal (function) not a number and even then, the phasor is producing a real number as a function of time.
Just a comment on a detail. You write: the complex numbers do not have a total ordering. But they can be totally ordered: e.g. first compare the real component, and when it's equal, the imaginary component. Your point is that such a total ordering can never be *the* total ordering (it cannot be preserved by a continuous mapping from the complex numbers to the reals).
it cannot be preserved by a continuous mapping from the complex numbers to the reals
Sure it can. Projection is a continuous map that preserves lexicographic ordering. The real problem is that this ordering doesn't play nice with the field operations:
0 <_lex i, and -i <_lex 1, but -i*i >_lex 1*i
Okay, that didn't come out right. Trying again:
0 <lex i
-i <lex 1
-i*i >lex 1*i
0 <lex i
-i <lex 1
-i*i >lex 1*i
In teaching math methods for physicists, I always point out how inappropriate the term 'imaginary number' is in desribing operations that involve the number 'i'. Once you start working in the complex domain to solve physical problems, you realize how much easier it gets and how some problems become trivial which could not be solved at all, or explained, simply working in the real space.
Three examples (outside the 'basics' category, though; sorry): 1. convergence of Taylor series. For functions of a real variable, it is not clear why some series converge everywhere and some only in an extremely localized region. Extend the function to the complex domain, and you find that the radius of convergence is related to the location of singularities in the complex plane. 2. Evaluation of integrals. Using the residue theorem of complex analysis, integrals which would be impossible to solve become a matter of some simple algebra and/or derivatives. 3. de Moivre's formula, which allows complicated trig multi-angle identities to be solved very easily.
During my teaching carrer I discovered that if I started talking about complex numbers from the imaginary unit as a solution to x^2 = -1, I was loosing most of the class. However, if I started by saying that complex numbers are points on a plane just like real numbers are points on a line, then defined addition and multiplication and then mentioned that if we multiply the point (0,1) by itself we get (-1,0), the results on the tests were much better. It seems that saying that the equation x^2 = -1 has a solution is so contradictory to the established intuition that it disturbs understanding of any further explanation of how that might happen.
Slawekk reminds me that it's also worth pointing out that all the algebraic behaviors and operations of complex numbers can also be defined by introducing vectors that satisfy certain multiplicative properties, without the need to introduce the idea of sqrt(-1) explicitly. This introduction of vector solutions to a seemingly one-dimensional equation x^2+1=0 might seem like a cheat, but it is very similar to what Dirac did to solve the relativistic Schrodinger equation, and that solution resulted in the theoretical prediction of both spin and antimatter.
The problem with trying to order the complex numbers is that in an ordered field, you want the product of positive numbers to be positive. Also, you want a total order, so that i has to be either positive or negative. Say it's positive. Then i2=â1 will also be positive, and you're already in big trouble. Same thing if i is negative: Just square âi instead, with the same result.
This is a wonderful site. Thank you, Mark.
If you know someone who would like to look at a slightly different (better?) interpretation of imaginary numbers, I would refer them to this paper
Imaginary Numbers are not Real - the Geometric Algebra of Spacetime.
By the way, thanks also for introducing me to the Dirty Three.
"The complex fix more than just the problem of some polynomials not having enough roots. The real numbers are not closed algebraically under multiplication and addition. With the addition of i, multiplicative algebra becomes closed:"
"That's plain wrong, isn't it? The sum of any two real numnbers is real. The product of any two reals is real. Please can you give an example of how the real number system is not closed under either addition or multiplication?"
Posted by: JoeSoap |
Joe or may I call you Soapy? The word that you are overseeing is algebraically. An algebraically closed or complete field is a field F which has the property that every polynomial equation with coeffcients in F has a root in F. Xsquared + 1 = 0 is a polynomial with real coeffcients but without real roots so the field of real numbers is not algebraically closed.O.K.?
sqrt(-1) is real?!
We conclude that there's no such thing as "n/0" because there is no number x such that x*0==n. Using i for sqrt(-1) might be a useful hack, and we can make use of the fact that this hack's linear independence from any number means that we can treat numbers "x + iy" as isomorphic to matrices "[x y]" then treat them like 2-dimensional numbers and such, but I fail to see how on earth you can call sqrt(-1) as anything other than a figment.
To date I've never gotten to teach anything more than the absolute basics of complex numbers (certainly no complex analysis yet), but I've always thought it would be interesting to try teaching them as matrices -- the complex number a+bi is considered shorthand for the matrix
[a -b]
[b a],
whereby |a+bi| is simply the determinant. On the one hand, this may be less counterintuitive, with the added bonus of highlighting the arbitrary choice between what we call i and -i. However, I think many students know about complex numbers before they do much linear algebra, so this may not be overly useful.
As opposed to such obviously "real" numbers as 2e, or the 11th root of pi? If you look deeply at the so-called Real numbers, there's much there which is far stranger than sqrt(-1).
Your use of "real" versus "figment" demonstrates a philosophy of math quite a bit different from that of most people who work in the field. I'd be curious to know what you mean when you say a number is "real."
You can put an algebra on 3 dimensions -- the problem is that you can't make it a division algebra. (You always have zero divisors, for example.)
Yes, Walt. That is actually what I was looking for. I've seen some proposals for three-dimensional numbers floated around, but they don't have the typical properties of a division algebra.
I'm curious as to whether 32-dimensional numbers have ever been constructed. Like sedenions, they wouldn't constitute a division algebra, of course, but perhaps they would obey similar but weaker properties. Can we go on like this ad infinitum with multidimensional numbers of the form 2n ?
I have a pretty good understanding of what the complex number (1 + i) / sqrt(2) means: I face forward, turn one-eighth of a circle, and take one step. But show me a jar full of jelly beans in a shop window, and I probably won't be able to guess how many there are inside. Without doing a little calculation, I couldn't tell you how big a pile of 50,000 jelly beans would be, or if it would look noticeably different than a pile of 60,000. The number of jelly beans is an integer, and (1 + i) / sqrt(2) is a complex number with irrational components, yet the latter feels much more comfortable.
See also the Isaac Asimov story I related here.
Davis, (pardon me for not quote-texting -- I don't know how to do it) what I mean by a number is "real" is that the rules for constructing it aren't a contradiction in terms. I can't enumerate pi, but I haven't defined it in a way that violates the terms which I use to define it. You can't a negative number of jelly-beans on a shelf, but you can "account" for the debt. Negative numbers in this way makes sense to me. Complex numbers as a way of encoding "x times right/left plus y times up/down" 2D positions makes sense. Even using them as zero-normed vectors makes sense. Just look at the number of examples above where two things bound together are "represented" as complex numbers. Fine, no problem. But sqrt(-1) itself doesn't make any more sense than n/0 -- it's oxymoronic.
But sqrt(-1) itself doesn't make any more sense than n/0 -- it's oxymoronic.
In what way?
n/0 and sqrt(-1) are fundamentally different: the field axioms prove that n/0 can't exist. They don't prove that sqrt(-1) can't.
How so? I see nothing oxymoronic in its definition. If there was a contradiction there, I assure you that mathematicians would have been the first to jump all over it.
If I had to guess, this is what I think your difficulty may be: you think of sqrt(x) as the number that, when you square it, gives you x (which is correct, with the added specification that sqrt(x) is positive). And of course, any number is positive when you square it (or alternatively, no negative number has a square root). However, this last statement is question-begging -- it's only true if by "any number" you mean "any number that is a member of the Reals".
(By the way -- the HTML tag you want is "blockquote".)
Well, I don't know from field axioms (not a mathematician am I), but.... oh, I get it. You're extending your domain to an escape the fact that nothing in the real domain can fit the bill, the same way that subtracting two natural numbers to get a negative number is valid.
Ok, that raises the question as to when inventing your way into a larger codomain is valid, but I probably wouldn't understand the answer anyway.
Well, it's valid pretty much because there's no reason for it not to be. Conceptually, it's no different than the steps from naturals to integers, from integers to rationals, and from rationals to reals. If you can give a rigorous, consistent definition of what you're doing, then it's allowable. (Consistency is the big headache for defining x/0, and results in the need to abandon some field axioms; this makes it somewhat undesirable.)
History of complex numbers: A complex story, part 1.
Mike: You can construct something for 2^n. (Really, if you don't want to suppose any properties, you can construct something of every dimension. You can even keep associativity and commutativity, if you're willing to give up division.) There's the Cayley-Dickson construction, which gives you something nonassociative of dimension 2^n where every nonzero element has a multiplicative inverse, but for 16, 32, etc. you get zero divisors. (How can you have inverses but still have zero divisors, you may wonder? It's because nonassociativity is bad juju.)
Is this how you'd do it?
Take an n-dimensional vector space over F with a basis {v0,...,vn-1)}.
Define 1=v0.
Define vi*vj = vi+j if i+ji*vj = -vi+j-n if i+j>=n. (This inherits associativity and commutativity from integer addition, if I'm not vastly mistaken.)
Extend so that multiplication distributes over addition in the usual way.
The subspace spanned by v0 is isomorphic to F, and a quick check leads me to believe that if F=R and n=2 you get a field isomorphic to C. (I'll have to wait after I've gotten some sleep to see if it's interesting for larger n.)
Define vi*vj=vi+j if i+j<n, and vi*vj=-vi+j-n if i+j>=n.
The preview eats < and >.
I always found it quite interesting when talking about complex numbers that, even though the use of complex numbers in physics is ubiquitous, all physical observables are real.
Not really about complex numbers, but regarding the above: although "the field axioms prove that n/0 can't exist", they only prove that n/0 does not exist in a field, whence n/0 could be in a larger, more closed structure that contained a field as a sub-structure, e.g. a pitch (defined in my 'To Continue with Continuity' on the pages that my "meaning"-full pseudonym links to), which has yet to be criticized by mathematicians (hint hint).
"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete."
Nonsense. The complex numbers are the important ones.
In algebra or analysis, working over the reals (instead of complex) is like working with one hand tied behind your back.
We should call the complex numbers the total numbers and the Real numbers the partial numbers.
I agree with JoeSoap -- I find it highly misleading the statement that "The real numbers are not closed algebraically under multiplication and addition." at least without some attempt to define "algebraically". Especially in a "Basics" posting. My first thought on seeing that is that of COURSE the real numbers are closed under addition and under multiplication, so what's he talking about.
One more thought: any creative thinker reading this will immediately be left with one final thought: If you lose ordering when going to a two-component thingy, and lose commutivity when going to a 4-component thingy, and then lose associativity going to the 8-component octonions, then what's lost when you go to 16 components? It wouldn't hurt to include a link where readers could discover sedenions and beyond through further reading.
Not quite, depending on the meaning.
In quantum mechanics the observables are self-adjoint operators, which seems to ultimately come only from the requirement of conserving probabilities. (Stone's theorem; self-adjoint operators are precisely the infinitesimal generators of unitary (probability conserving) groups of time evolution operators.)
Quantum mechanics not surprisingly use probability measures, and applying self-adjoint operators must end up producing real quantities, that is correct. I'm not sure if it is coincidental or not.
On the other hand, quantum phases are observable too. One can see its effects in SQUID's, for example.(Aharonov-Bohm effect)
Michael Chermside: For sedenions, and higher, you end up getting very little of the interesting algebraic structure (as in - you won't get a division algebra in any way at all any longer); so you'll end up with things that are about as insightful as algebras on for instance a 3-dimensional vector space or such. You can still impose algebra structures for any vector space, but it won't necessary have the kind of neat structures you could get in dimensions 1,2,4 and 8.
Note that the fact that the preview eats < and > points out a type 1 XSS attack that scienceblogs.com is vulnerable to.
(Anonymous because I'm hoping that this comment will get stuck in moderation and Mark will see it and delete it, but then complain loudly to the scienceblogs.com techies. Demo exploit at http://xrl.us/ujx8 which just pops an alert box)
See http://www.sixapart.com/movabletype/docs/3.3/h_changelog/3_32.html for a cryptic changelog comment that may explain how to patch this hole.
Of course, all MT versions less than 3.34 are vulnerable to a (possibly IE-specific) type 2 XSS bug detailed at http://www.securityfocus.com/archive/1/458196/30/0/threaded
(And yes, I'll confess: I'm mainly pointing out these security vulnerabilities because I'm annoyed that preview eats < and >, and hope that exposing these security vulnerabilities fixes that annoyance as a side effect)
Walt said, parenthetically:
Well, one could construct a vector with any number of components, yes? They're even useful in some circumstances, but a certain amount of algebraic interest is lost.
"all physical observables are real"
If you want to get right down to it, all physical *measurements* ever taken are *rational*. Think about it.
The complex numbers are an extended domain to cope with the weaknesses of the reals. The reals are an extended domain to cope with the weaknesses of the rationals. The rationals are an extended domain to cope with the weaknesses of the nonnegative rationals. Which were extended from the positive rationals to deal with the need for zero. And the positive rationals are an extended domain to cope with the weaknesses of the positive integers.
And you sacrifice some theorems every time you switch to a larger domain: the question is always whether you gain more than you lose for your particular field of study. In particular, going from integers to rational numbers causes you to lose prime factorization. Going from positive rationals to all rationals, you find that you can't define x^y meaningfully for negative x. (But you can when going from positive integers to all integers -- so....)
Who was it who said "God created the integers, all else is the work of man" ?
The connotations of the words reflect that history. The closer you are to the "whole numbers", the more "wholesome", "natural", "rational", "positive", and "real" the words are. As you go away from them, you get "negative", "irrational", "imaginary", and "complex" numbers. What a set of connotations! I'm surprised there aren't "illegal" numbers, "immoral" numbers, and "irresponsible" numbers!
On second thought, those are indirect observations. Forgedaboudit.
Kronecker.
Torbjörn Larsson:
Are they significantly less direct than observing, say, the electric and magnetic fields? (And off we go, sliding down into a quagmire-like argument over instrumentalism. . . .)
There are illegal primes, but being prime numbers, they are inevitably real, positive, rational, integral, natural and wholesome.
JimDesu:
Heh, you invented your way into a larger domain just a post ago, when you said that you can't really see a negative number of jelly beans, but you can 'account' for the lost ones.
That sort of operation wouldn't be allowed several thousand years ago. We used to think that only Z+ (the positive integers) were real. When we started doing equations, things such as X+2=1 simply had no solution. After all, you can't add 2 jellybeans to a pile and end up with only 1 - the smallest pile you can have is 0.
Eventually we realized that our intuition was wrong, and negative numbers are real. Or at least, real enough to use. An entire family of equations suddenly gained solutions.
There was a similar story behind the irrationals. Pythagoras famously didn't believe they existed, and one member of his school was reportedly murdered for revealing that sqrt(2) wasn't rational.
Nowadays we're running into the same issue with imaginary numbers. There's no such thing as a pile of jellybeans with i beans, just like there's no pile with -1 beans. But that doesn't mean that i isn't a useful number to have around. It solves a new family of equations, and is useful quite commonly in physics. Not just quantum physics, either - go take some basic electrical engineering courses and you'll run into imaginary numbers quite quickly. It's even useful for something as simple as rotations - give me a vector, and I can multiple it by i to rotate it 90o.
So, yeah, you'll never find i of something. But you'll never find -1 of something either. i is just as useful once you accept that it *does* correspond to real things - it's just doesn't help you count.
mj
As someone else said, nonsense! Clearly all physical observables are integers! If you claim that something is 4.5, I say that your units are twice as large as they need to be. Otherwise, you're not using real numbers, just made up 'fractions' that mathematicians like to use to confused people.
Seriously, though, yes, any value can be *measured* as a real (actually, a rational). You are essentially measuring the norm of the value. When you start actually *using* the value, though, it's pretty clear that rational numbers don't work right. In much the same way that you *could* use integers for everything (but you'd have to contort a lot of equations to keep things integral), you *can* use rationals for everything. But many, many equations become horrendously more complicated if they're restricted to the rationals, while allowing complexes simplifies them again. Take some basic electric engineering - anything involving the interaction of electricity and magnetism requires complex numbers to be comprehensible. I could measure every value as an integer if I wanted to be difficult, but most people find it easier to just call the values complex.
I've heard it said that irrational once meant only that a number could not be written as the ratio of two integers. The second, more disturbing meaning then arose out of Pythagorean desire for a universe whose natural order expressed itself only in simple combinations of whole numbers (the first "numbers racket"). In other words, the mathematical meaning came first and the other one followed.
I am not very sympathetic to the Kroneckerian thesis. After having studied the surreal numbers, one is led to speculate that God created the empty set and all else — to ω and beyond — is the work of man.
While driving to work today, I was considering how one coudl teach the imaginary numbers to school children. I decided it's best to throw the weirdness at them immediately, but in a slightly different context...
Today we're going to talk about a new type of number. In advanced math you deal with all sorts of numbers - imaginary, vectors, quaternions, matrices, etc. Many people say that these types of numbers aren't 'real' because you can't count with them. You can't make a pile with [3, 2, 5] jellybeans in it, for example. But they are very useful when doing certain things, so it's good to understand just how they relate to you. So today I'm going to discuss a very simple new type of number that mathematicians use called 'evil numbers'. Silly name, yes, but they're used in many equations and are quite useful in solving certain types of mathematical problems.
Evil numbers are pretty simple, really. The very first evil number is written as e. This is the basis of all the evil numbers, and you can get all the rest of the evil numbers just by multiplying a normal number (let's call them 'good' numbers) by e. So you've got 2*e, or 2e, and then 3e, .5e, sqrt(2)e, and so on.
Adding two evil numbers works exactly as it might look. e+e=2e, 2e+2e=4e, and so on. If you add together a good and an evil, you can't quite combine them, so we write it with the addition left in. 1+2e=(1+2e). That's about all we can do, though there is a way to simplify it that I'll go into later.
Notice, though, that 1+2e isn't a good or evil number - it's got parts of both. We'll call this a moral number. You might see that even good and evil numbers can be thought of as a type of moral number; 1 is the same as 1+0e, while 2e is the same as 0+2e. Adding moral numbers is really easy - just add their good and evil parts and put them back together. So 1+2e + 3+4e = (1+3)+(2+4)e = 4+6e.
That's it for addition. It's just like normal addition, except that when you add a good and evil number you get the slightly more complicated moral number. Multiplication is a tiny bit trickier.
You may have already noticed that when you multiple a good and an evil you get an evil number. You just multiply the good parts and stick the e back at the end. 2*e=2e. 3e*4=12e. That's simple. Multiplying two evil numbers works the same way, except now you have two e's at the end. 2e*3e=6e2. There's a nice trick you can do to eliminate the e2 that I'll show you in a second.
Multiplying moral numbers is a little bit more complicated - you multiply like they were binomials. So multiplying a good number by a moral number is just distributing: 3*(1+2e)=3+6e. Same with multiplying an evil and moral number together, though you get that e2 again. Two moral numbers are a little bit more complicated, because you end up with four terms:
(1+2e)*(3+4e)
= 1*3 + 1*4e + 2e*3 + 2e*4e
= 3 + 4e + 6e + 8e2
= 3 + 10e + 8e2
That's as good as we can get, and it's still pretty complicated. Multiplying a lot of moral numbers together would get pretty horrendous, since you'd just start building up those powers of e. But you can eliminate all of that with this simply identity: e2=1
So that number we just found is now equal to 3 + 10e + 8*1, or just 11+10e. That's a lot nicer - we're back to an ordinary moral number. We can do the same thing when we're multiplying two evil numbers together:
2e*5e = 10e2 = 10*1 = 10
You see, when we multiply two evils together we get a good.
So let's review:
good + good = good
evil + evil = evil
good + evil = moral
anything + moral = moral (just add up the good and evil parts separately)
good * good = good
evil * evil = good
good * evil = evil
anything * moral = moral (multiply like it was a binomial)
e2 = 1
Now, you may have heard about evil numbers before, actually. If you were young when you learned about them, they probably didn't call them 'evil', to keep you kids from thinking you shouldn't learn them. More than likely, you called them 'negative'.
...
^_^
Now we can learn about imaginary numbers. Imaginary numbers are exactly like evil numbers. Instead of moral numbers, we say complex numbers. Every single rule you just learned to use negative numbers works with imaginary numbers, except that i2=-1. Imaginary and 'evil' numbers are no different - neither can be used for counting things, but both are useful sometimes when we're talking about special subjects. So don't let anyone tell you that imaginary numbers are just a figment of mathematician's imaginations. They're just as real as negative numbers are; which is to say, only as real as we allow them to be.
I think that would blow the minds of most children, and really cement in their heads that even the math they take for granted is much more unusual than they give it credit for, and so they can't dismiss new math as not real just because it looks a little strange.
I agree with Blake, about it being nonsense to assert dogmatically that God created the integers. Surely the integers (i.e. the counting numbers) only fail to be a human creation by the dubious virtue of their being an innate conception. But we therefore have no more reason to expect them to be objectively real, prima facie, than we do to expect the green of the leaves to be objectively real (or indeed, space to be Euclidean, or objects to be classical), whereas it is paradoxically subjective. But the problem with God creating the empty set is, which one?
Oy, this will need a long reply. Sorry about that!
Answering the above could be helped by looking at the context.
We were discussing if all physical variables have a certain property, that of measurements resulting in real number quantities.
The observation I made about Aharonov-Bohm effect observations being indirect was related to that, since other, real, quantities are the primary observations resulting in values on phase differences. Quantum phase noise is also indirect observations. I believe I have seen that absolute quantum phases have been directly observed in other experiments, though I can't find references now which is why I dropped the argument.
Passing on to the issue of instrumentalism vs realism, I note that Nathanael reminds us that measurements, whether in fundamental or derived units, are made against a measurement standard.
I don't see how the process of measurement fully decides the nature of either the measured quantity or the properties of the real object however. Our standards and how we use them still seems coincidental to me.
For example, we approximate reals, not rational quotients, by measure error and repeated measurements. And we are used to convert multidimensional measurements into separate real-valued quantities. I'm not sure we have to - we can observe measurements of amplitude and phase by convenient plots, on an oscilloscope for example.
I certainly think that realism is preferable over instrumentalism, even if that takes me into a quagmire. We need to specify models in axiomatics, we need to specify semantics in languages, and we need to specify interpretations in physics. This seems to me show a need for, and realism in, these specifications. And there seem to be no difference in parsimony here, since physicists like to take the math object as identical to the physical object.
But to go further perhaps we need a model of measurement that makes predictions.
n-category café discusses measurements from time to time. John Baez had a post where he claimed that we (or at least mathematicians) often study objects by coordinate charts. With the help of the Yoneda lemma we can extract knowledge about an object from presheafs.
Now, the cool thing for me is that this does is to study objects by mapping known quantities into it. By mapping all possible quantities in all possible ways we can know the object well. "We know an object completely if we know all coordinate charts on that object, together with how these coordinates transform." That seems to me to suggest that our possibility to know about an object isn't really hindered by the nature of our measurements. ( http://golem.ph.utexas.edu/category/2006/09/dimensional_analysis_and_co… )
But I wonder if Baez description of measurements is correct as he didn't discuss realizations. Nor do I know how far this model takes us - does it give predictions? Perhaps you have any suggestions here?
as a high school teacher, i am always looking for basic, easy-to-state facts that motivate a need for complex numbers. my favorite is here on my blog:
http://polymathematics.typepad.com/polymath/2006/07/proof_of_cmfe.html
i call it the coolest math fact ever (CMFE): divide the circumference of the unit circle into n equal arcs with n points. pick one of those points, and draw the n-1 chords to all the other points. then the product of the lengths of those n-1 chords is always n.
i know of no proof that doesn't use complex numbers, because this is really a fact about polynomials.
I noticed just now, while reviewing the stuff to write something similar in Italian, that any try to use a system to represent rotations in 3d with just three unit vectors was doomed to failure, since they are not commutative. I wonder if Hamilton realized this while walking on that bridge in Dublin...
Hi,
If complex numbers are multidimensional then do they have a surface area?
or if 3 dimensional do they have a volume?
can complex numbers therefore be thought of as objects?
Hi,
Isn't saying that complex numbers are multidimensional confusing to students like myself who are new to this?
It's my understanding that the set of complex numbers form the two dimensional plane and that the numbers themselves represent a certain position on this plane and have no dimension.
Saying that the complex numbers themselves are multidimensional suggests that they have a length and height etc, which I don't believe is correct?
If I am wrong about this please let me know.
Aidan: If you consider a specific complex number z = a + bi then you are free to call the real part (a) the width (graphed as left or right along the horizontal x-axis) and the the imaginary part (b) the height (graphed as up or down along the vertical y-axis).
Area? Well, we can lump a and b together with the traditional:
|z| = |a + bi| = a^2 + b^2.
Also, the complex conjugate z-with-a-bar-on-top = a - bi.
Then |z| = z times z-with-a-bar-on-top.
Quaternions a + bi + cj + dk where i, j, k are different (noncommutative) square roots of -1, are by the same analogy 4-dimensional -- yet were invented to analyze 3-dimensional rotations. I've used quaternion algebra in the attitude control, and guidance and navigation, or jet planes, missiles, and spacecraft.
This doesn't fully answer your question, and is not clear by itself, but I am reassuring you that you asked a GOOD question. Mark Chu-Carroll: care to agree or disagree or explain further?
By the way, yesterday's New York Times had a good obituary of the great French mathematician Cartan (son of Cartan) who died at age 104. I posted the obit on a thread about Cartan at the n-Category Cafe.
Geometric (Clifford) algebra incorporates complex and quaternion (among other things) plus gives i a natural meaning.