Complex numbers: how "imaginary" are they?

[Sequent]

I’m aware of the historical context of complex numbers, and I’m aware that the term “imaginary” is an ironic holdover from the days when the validity of complex numbers was in doubt. Now, I’ve worked with complex numbers, particularly with fractals, so I’m familiar with the basics of complex numbers and the complex number system. It’s simply been one of those things, like quantum superposition, that I’ve just had to take for granted and not think about too much. Well, I’m back to thinking about it, prompted in part by a question from a friend to whom I wanted to provide a better explanation that I received, which was, basically: there’s a number i that when squared is equal to -1. Believe it.

Well, it was a bit more expansive than that. A good summary of the argument I’m familiar with can be found here . The problem with this argument–for me–is that it uses a four-part definition of a number system to show that complex numbers are a valid system, just like fractions, and therefore, really exist. Well, I don’t have a problem understanding they’re a valid number system and therefore exist as mathematical objects–but I’ve never been able to make the leap from “exists as a valid number system” to “exists in the real world.”

Another argument starts with the premise that all numbers, as we know them, are symbols, not real world objects. Thus i[sup]2[/sup] is no less valid than 2[sup]2[/sup] or even just 2, for that matter–they’re all just symbols. The number two doesn’t occur naturally, it doesn’t grow on trees–it’s a symbol. I can understand that. But the symbols I’m used to correspond to quantities which can found in the real world.

2: you can have two apples
3[sup]2[/sup]: you can have a quantity of three apples and multiply them by three
sqrt(4): you can have four apples and deduce the number of apples that, when multiplied by itself, will result in 4 apples
1/2: you can cut an apple in half; you can even imagine cutting it perfectly in half.
3/2: you can have three halves of apples
1.435435435…: you can have one and slightly less than one half of an apple such that it corresponds to the decimal fraction

Even negative numbers don’t require much of a conceptual leap. I can view a group of five apples next to a group of two apples and observe that the former has three more apples than the latter and the latter has three less than the former; we’re still talking about the number three, we’ve just invented negative numbers to obviate the need for the qualifiers “more than” or “less than.” So you can argue that negative numbers don’t really exist in the same way, but I could show you how real world relationships imply their existence. The best you could do is argue that negative numbers only exist when one is comparing quantities and therefore don’t really exist.

But I can’t show you i apples. I don’t know how I’d tell the difference between 2 apples and 2 + 2i apples. It’s easier for me to think of complex numbers as pairs of real numbers, using a separate dimension to track the complex part. But I still can’t visualize it.
Questions:

1. Can complex numbers be shown to represent real-world quantities or are they simply intellectual constructs? Note that by the latter, I’m not implying they’re not valid, just that there’s no way to quantify them.

2. Are all numbers really complex numbers? In other words, the natural number 2 is a complex number with complex part 0, correct? Once you’ve opened up the complex door, you’ve got a new system that includes all the familiar numbers already, right? Really, you can’t go back. You can, of course, go back to just dealing with the familiar numbers, but you’re simply ignoring the zero complex part.

3. Are complex numbers all there is? Essentially, when we expand our number system to include complex numbers, we’re essentially saying that every number has two parts–real and imaginary. Can we be certain we won’t someday discover a third dimension we’ve been overlooking all along?

4. How many more complex numbers are there than familiar numbers?

5. I understand that without complex numbers, much of our current technology would not have been possible, that their applications are ubiquitous in modern engineering and other fields. But I’ve also heard that everything that can be done with complex numbers can also be done with familiar numbers–it’s just much, much, much more difficult. Is that true–that complex numbers don’t actually bring anything new to the mathematical table, they essentially just greatly facilitate calculations which would otherwise be inordinately complex?

[tim314]

@Sequent

1. Can complex numbers be shown to represent real-world quantities or are they simply intellectual constructs? Note that by the latter, I’m not implying they’re not valid, just that there’s no way to quantify them.

If by “real world quantities” you mean can you count a bunch of apples and find out that you have 2+i of them, then no. However, complex numbers are useful for calculating a lot of real world things. As a simple example, the sine and cosine functions (used to describe harmonic motion, for instance), can be expressed in terms of complex exponentials, and sometimes it is advantageous to do so.

sin [symbol]q[/symbol] = (e[sup]i[symbol]q[/symbol][/sup]-e[sup]-i[symbol]q[/symbol][/sup])/2i
cos [symbol]q[/symbol] = (e[sup]i[symbol]q[/symbol][/sup]+e[sup]-i[symbol]q[/symbol][/sup])/2

2. Are all numbers really complex numbers? In other words, the natural number 2 is a complex number with complex part 0, correct? Once you’ve opened up the complex door, you’ve got a new system that includes all the familiar numbers already, right? Really, you can’t go back. You can, of course, go back to just dealing with the familiar numbers, but you’re simply ignoring the zero complex part.

Well, you can work with different systems of numbers. The real numbers is one system, and the complex numbers are another system. It’s true that all the numbers in the real number system also occur in the complex number system. But that doesn’t mean that when talking about these numbers in the real system one is ignoring the fact that they have a complex part. The number system wasn’t discovered, it was invented. (Invented because it’s useful in describing the real world, perhaps, but invented none the less.) So it’s not like we found out about numbers, and then found out they have complex parts. Rather, we (humans) created a system where all numbers are real, and there’s no such thing as complex numbers. Then we created another system where numbers are complex, having both a real part and an immaginary part. But creating that system doesn’t make the first system any less correct.

3. Are complex numbers all there is? Essentially, when we expand our number system to include complex numbers, we’re essentially saying that every number has two parts–real and imaginary. Can we be certain we won’t someday discover a third dimension we’ve been overlooking all along?

Again, we’re not discovering properties of numbers, we’re creating different systems of numbers in which they have different properties. Creating a new system doesn’t mean the old one was wrong, or that we were overlooking anything. But if you want to know whether you can extend the complex numbers to some other system in a way that’s analogous to extending the reals to the complex numbers, then the answer is yes. The Quaternions are an example of this.

4. How many more complex numbers are there than familiar numbers?
   It depends what you mean. On the one hand, both the real numbers and the complex numbers have the same Cardinality. That means that you can make a one-to-one mapping between the two. (For every real number, there’s a complex number, and vice versa.) On the other hand, if you take the complex numbers, and take out all the real numbers, you’re still left with an infinite set. The bottom line is that “how many more” doesn’t really have an obvious meaning when you’re comparing two infinite sets. (I’d expect that most mathematicians would probably use “how many” to refer to the cardinality, however.)

5. I understand that without complex numbers, much of our current technology would not have been possible, that their applications are ubiquitous in modern engineering and other fields. But I’ve also heard that everything that can be done with complex numbers can also be done with familiar numbers–it’s just much, much, much more difficult. Is that true–that complex numbers don’t actually bring anything new to the mathematical table, they essentially just greatly facilitate calculations which would otherwise be inordinately complex?

I’m not sure about that one. I suspect that might be true, but in doing things “without” complex numbers you might have to define something more-or-less equivalent along the way.

[DarrenS]

I’ll chime in with a few answers before the mathematical heavyweights have had time to drink their coffee and eat their donuts.

@Sequent

Questions:

1. Can complex numbers be shown to represent real-world quantities or are they simply intellectual constructs? Note that by the latter, I’m not implying they’re not valid, just that there’s no way to quantify them.

I’ve heard of complex numbers being used to represent quantities in electronic engineering (phase?) Some phenomena are more easily expressed with two variables, so complex numbers, being a ready-made and well-known theory involving two variables (real & complex parts) fits the bill. There is nothing inherently “complex” about the phenomena being described, it’s just that complex numbers are handy for describing them. Perhaps one of our resident electronic engineers could give examples.

@Sequent

2. Are all numbers really complex numbers? In other words, the natural number 2 is a complex number with complex part 0, correct? Once you’ve opened up the complex door, you’ve got a new system that includes all the familiar numbers already, right? Really, you can’t go back. You can, of course, go back to just dealing with the familiar numbers, but you’re simply ignoring the zero complex part.

Yes, in the same way that all integers are special cases of fractions. e.g. the integer 3 can be expressed as “6 / 2”. Math may seem to be about absolutes, but it’s a lot to do with what’s useful. If we’re dealing with complex numbers, then it’s useful to treat reals as “special cases” with the imaginary part = 0. If you’re working on number theory, e.g. prime numbers, it doesn’t really help to think of each number as a complex number with the imaginary part = 0.

@Sequent

3. Are complex numbers all there is? Essentially, when we expand our number system to include complex numbers, we’re essentially saying that every number has two parts–real and imaginary. Can we be certain we won’t someday discover a third dimension we’ve been overlooking all along?

Google “quaternions”. There are several ways you can generalize numbers; there’s also the “surreal” numbers (yes, really) and the transfinite numbers. Again, it comes down to what’s useful. I could invent a set of numbers that only contains “those numbers that appeal to Darren”, but it wouldn’t be much use to the mathematical community in general. So, no, complex numbers are not “all there is”, but as a subset of the infinite amount of possibilities, they are very useful for a variety of purposes.

@Sequent

4. How many more complex numbers are there than familiar numbers?

What do you mean by “familiar” numbers? Integers? reals? If it’s reals, I believe I saw a proof where they have the same cardinality - there exists a bijection between Z and C. I may be wrong about that. You do seem to realize that when dealing with infinite sets like Z and C, the concept of “how many” has to be made a little more rigorous: it becomes a question of mappings - if a bijection exists, they have the same cardinality. If not, they don’t. There’s probably something similar to Cantor’s diagonalization proof for this case.

@Sequent

5. I understand that without complex numbers, much of our current technology would not have been possible, that their applications are ubiquitous in modern engineering and other fields. But I’ve also heard that everything that can be done with complex numbers can also be done with familiar numbers–it’s just much, much, much more difficult. Is that true–that complex numbers don’t actually bring anything new to the mathematical table, they essentially just greatly facilitate calculations which would otherwise be inordinately complex?

Don’t know about this one.

Also, remember - it’s not so much that “there’s a number i that when squared, gives -1”. It’s more like, we define a symbol i to be the square root of -1 (or j if we’re engineers ); we then go on to do useful mathematics with that symbol. If you defined a symbol ‘#’ to be “a number that lies between 7 and 9, with the properties of an integer, but not equal to 8”, and you could use it to produce useful results in mathematics, you would soon have papers written about it and a whole branch of mathematics around it.

[Stranger_On_A_Train]

Permit me to introduce you to my buddy, Euler, for whom:

e[sup]iθ[/sup] = cos(θ) + i*sin(θ)

which, for θ=π gives the beguiling

e[sup]iπ[/sup]+1 = 0

Truely a thing of beauty, is it not?

i (or, as the EEs call it, j) is an intrinsic part of natural mathematics, fundamentally related to the natural number, e, and the circumferential ratio, π. Owing to the utility of e as pertaining to cyclic functions (like sinusoids), i tends to pop up a lot, especially any place where phases or two dimensional vectors are useful (planar mechanics, electrical circuits, controls theory, vibrations).

The complex domain is just another (and orthogonal) space in which to map operations. They are as valid (and numerous) as “real” numbers, as they include rational, irrational, negative, et cetera values.

If you really want to get freaking, consider quarternions, which give you the ability to handle four independant quantities in a similar fashion to imaginary numbers and two dimensions. They’re often used for representing time-variant space transformations.

We could live without complex numbers–indeed, one EE professor eschewed them, claiming that they were “too difficult to teach”–but mathematics would be much less elegant for the lack of them.

Stranger

[Stranger_On_A_Train]

Ufck…let me try that equation again:

Permit me to introduce you to my buddy, Euler, for whom:

e[sup]iθ[/sup] = cos(θ ) + i*sin(θ )

which, for θ=π gives the beguiling

e[sup]iπ[/sup]+1 = 0

Truely a thing of beauty, is it not? At least, it is when I preview instead of frantically submit. :smack:

Stranger

[MikeS]

It’s funny that you should mention quantum superposition in the OP, because one of the big “real world” uses of complex numbers is in quantum mechanics. As we currently understand quantum mechanics, you need to use complex numbers to predict experimental results (the double-slit experiment comes to mind), although things always work out such that you never see a complex number in the real world.

Then again, I’ve never really thought about what would happen if you tried to reformulate quantum mechanics in terms of real vector spaces instead of complex vector spaces. Perhaps my fellow SDMB physicists around here have some ideas.

[DarrenS]

I just remembered another situation where they pop up: if you plug a velocity > c into Einstein’s equations, you end up with an “imaginary” mass. I’m not sure what the real-world interpretation of that is - other than “don’t do that” - but it’s interesting.

The equation is something like (going from memory here):

m1 = m0 x sqrt ( 1 - v^2 / c^2 )

When v > c, this becomes the square root of a negative number, i.e. imaginary.

[Sequent]

I thought there would have to be many, many more complex numbers than real numbers. Think of a Cartesian coordinate system. Let the x-axis represent the real part of all numbers. Let the y-axis represent the complex part of all numbers. Obviously there are infinite points on both axes, but for every real number where the complex part is zero, there are an infinite set of numbers with the same real part and a non-zero complex part, right?

[Stranger_On_A_Train]

@DarrenS

if you plug a velocity > c into Einstein’s equations, you end up with an “imaginary” mass. I’m not sure what the real-world interpretation of that is - other than “don’t do that” - but it’s interesting.

I’m sure you could use this information to scam a large number of people into your “Lightspeed Diet Program” under the premise of converting some of their real mass into “imaginary” mass.

Stranger

[yerba_buena]

@Sequent

Can complex numbers be shown to represent real-world quantities or are they simply intellectual constructs? … I’ve also heard that everything that can be done with complex numbers can also be done with familiar numbers–it’s just much, much, much more difficult. Is that true

Yes, and yes.

Since you are looking for a physical example, I offer one: a RLC-circuit hooked up to an alternating current.

I’m sorry, I’m not as proficient with the superscripts and subscripts, so my post won’t include equations.

Given an inductor, a capacitor, and a resistor hooked up to an AC source in series, you can derive a differential equation for the current as a function of the voltage. Because the capacitor involves charge (the integral of current), the resistor involves the current directly, and the inductor involves changes in current (the differential of current), you have a 2nd order differential equation.

With some complex math, you can solve the equation.

However, if you assign an impedence to each component of the circuit, with the impedence for the inductor and capacitor being imaginary, you can solve for the current in mere seconds.

[CJJ]

1. Can complex numbers be shown to represent real-world quantities or are they simply intellectual constructs? Note that by the latter, I’m not implying they’re not valid, just that there’s no way to quantify them.

Unlike real numbers, the complex numbers do not have a natural ordering. They therefore do not have a natural property such that if a != b, then either a<b or a>b. This is a pretty basic requirement to the general concept of quantity, so on their face complex numbers do not represent quantities. However, quantitative real values can be derived from them (e.g. the norm or magnitude of a+ib = sqr(a^2 + b^2)), and the value of complex numbers is that these single numbers can contain two independent quantities (e.g. the same complex number a+ib has a phase = arctan(b/a)). This leads another of your questions:

5. I understand that without complex numbers, much of our current technology would not have been possible, that their applications are ubiquitous in modern engineering and other fields. But I’ve also heard that everything that can be done with complex numbers can also be done with familiar numbers–it’s just much, much, much more difficult. Is that true–that complex numbers don’t actually bring anything new to the mathematical table, they essentially just greatly facilitate calculations which would otherwise be inordinately complex?

Regarding many real-world applications, AC electrical circuits and fluid-flow problems can be explained without complex numbers if you stick with differential equations; I suspect the same is true for QM. Complex numbers make these easier to solve because certain dual quantities associated with (in these cases) electric voltage and flow vectors can be embedded in a single complex variable.

But there are other cases where complex numbers IMO seem essential; the exact evaluation of certain definite integrals is impossible unless you assume complex numbers.

3. Are complex numbers all there is? Essentially, when we expand our number system to include complex numbers, we’re essentially saying that every number has two parts–real and imaginary. Can we be certain we won’t someday discover a third dimension we’ve been overlooking all along?

Complex numbers arise out of the natural desire for the number system to be closed under certain common operations. Negative numbers were “discovered” when someone noted the whole numbers were not colsed under subtraction (e.g. if a>b, b-a is not a whole number). Fractions arose when someone noted the integers were not closed under division. Irrationals were found when the Pythagoreans noted some simple geometric lengths were not expressible in rational ratios.

Imaginary (and later complex) numbers developed out of a need to make the (square, cube, fourth, etc.) “rooting” operation closed. A natural question then is to ask whether some other combination of operators will force us to invent another class of numbers. The Fundamental Theorem of Algebra states that, at least for all algebraic operations (+, -, *, /, powers, roots), the answer is no.

In this sense, Hamilton’s quaternions aren’t “numbers” because they depend on non-commutative properties of their group; if you require that numbers behave such that ab=ba, quaternions don’t pass the test. Of course, if you also require numbers to be naturally orderable (i.e. have natural quantity), complex numbers also fail the test. Then again, if numbers must correspond to countable groups (number of chickens, length in inches), you probably would not consider fractions or negatives to be numbers. I guess its a matter of “to what degree do I define ‘numberness’”.

One final note on Stranger on a Train’s posting for Euler’s equation. This formula for e^it represents a desire to define raising a number to a complex power. Think about it: addition, subtraction, multiplication and division of complex numbers is straightforward to define, but it’s not so clear how to evaluate 2^i. Euler came upon this formula by requiring that complex powers should obey the familiar laws of powers for real numbers (e.g. a^m*a^n = a^(m+n)); he focused on e because he had proven (d/dt)e^at = ae^at, and wanted that fact to remain true even if the constant a were imaginary. The rest is just (tedious) math:-).

[tim314]

@Sequent

I thought there would have to be many, many more complex numbers than real numbers. Think of a Cartesian coordinate system. Let the x-axis represent the real part of all numbers. Let the y-axis represent the complex part of all numbers. Obviously there are infinite points on both axes, but for every real number where the complex part is zero, there are an infinite set of numbers with the same real part and a non-zero complex part, right?

Well, like I said, “more” doesn’t necessarily have an intuitive meaning when it comes to sets of infinite numbers. Another example is that there is (in some sense) the same number of positive integers as there are even positive integers. Intuitively, that makes no sense – you’d expect there to be twice as many. But consider this: I can give you a one-to-one correspondence between positive integers and even positive integers. Specifically:
1 <–> 2
2 <–> 4
3 <–> 6
4 <–> 8
etc.
So for every positive integer, there’s a corresponding even positive integer, and for every even positive integer, there’s a corresponding positive integer.

Normally, if the elements of two sets can be put into one-to-one correspondence, we say they have the same number of elements. So in that sense, there are the same number of positive integers as even positive integers.

Again, that seems intuitively wrong, but infinite numbers have a way of defying our intuition.

[Pleonast]

@Sequent

I thought there would have to be many, many more complex numbers than real numbers. Think of a Cartesian coordinate system. Let the x-axis represent the real part of all numbers. Let the y-axis represent the complex part of all numbers. Obviously there are infinite points on both axes, but for every real number where the complex part is zero, there are an infinite set of numbers with the same real part and a non-zero complex part, right?

It’s not that simple. I’m not sure of the case for real vs complex. However, there is the same number of rational numbers as integers, even though there’s an infinite number of rationals between each every pair of integers.

[Driver8]

@Sequent

I thought there would have to be many, many more complex numbers than real numbers. Think of a Cartesian coordinate system. Let the x-axis represent the real part of all numbers. Let the y-axis represent the complex part of all numbers. Obviously there are infinite points on both axes, but for every real number where the complex part is zero, there are an infinite set of numbers with the same real part and a non-zero complex part, right?

In a sense, yes.

From a less mathematical point of view, complex numbers are no less or no more real than standard real numbers. That is, both are just “intellectual constructs”. The fact that standard real numbers are easier to visualize doesn’t really change anything for me. It’s all a case of setting up definitions of items so that they work in a useful way.

I could be completely off the mark here though. It’s been a while since I studied maths

[CJJ]

@Sequent

I thought there would have to be many, many more complex numbers than real numbers. Think of a Cartesian coordinate system. Let the x-axis represent the real part of all numbers. Let the y-axis represent the complex part of all numbers. Obviously there are infinite points on both axes, but for every real number where the complex part is zero, there are an infinite set of numbers with the same real part and a non-zero complex part, right?

This is the problem in dealing with infinite sets. For example, you would think there are twice as many whole numbers as even numbers, since the whole numbers contain all the even numbers plus a corresponding odd number (one half the value of each even).

However, note that you can create the even numbers by taking each member of the infinte set of whole numbers and doubling its value. So these two sets must be the same size.

This is why the concept of cardinality has to be carefully defined in dealing with infinte sets

[Chronos]

Think of a Cartesian coordinate system. Let the x-axis represent the real part of all numbers. Let the y-axis represent the complex part of all numbers. Obviously there are infinite points on both axes, but for every real number where the complex part is zero, there are an infinite set of numbers with the same real part and a non-zero complex part, right?

Right. And yet, the cardinality of the two sets is the same. This is why we say that questions about “how much bigger” can be counterintuitive when dealing with infinite sets. You might take comfort in the fact that the measure of the complex plane is infinitely greater than the measure of the real number line, but “measure” is not the usual sense of “size” for a mathematician.

[billfritter]

It made my day when I discovered that complex numbers were usable for calculating things like magnetic fields.
But then late I became disgusted with them, as I found people trying to use them to justify things like string theory and hidden dimensions and time travel and space travel and Martians creating the pyramids.
So now I avoid them.

[MikeS]

@CJJ*

Regarding many real-world applications, AC electrical circuits and fluid-flow problems can be explained without complex numbers if you stick with differential equations; I suspect the same is true for QM.

Not easily. The Schrodinger equation, which is pretty fundamental to QM, is (using ket notation)

i H |psi> = d |psi>/dt,

where H is the Hamiltonian operator and |psi> is a vector in the complex Hilbert space you’re using. Notice that i is right there. The complex numbers are embedded even deeper than that in QM, actually: the usual method to “quantize” a classical system is to take the coordinates on phase space x and p and promote them to operators such that [x, p] = i*hbar. (hbar being Planck’s constant over 2 pi.)

That said, I can’t say for certain whether you couldn’t reformulate quantum mechanics without the use of complex numbers, merely that it would require a very deep-level shift in our understanding of it.

[ultrafilter]

@Sequent

1. Can complex numbers be shown to represent real-world quantities or are they simply intellectual constructs?

Yes and yes. Complex numbers have exactly the same ontological status as real numbers, and they are used to represent physical quantities.

2. Are all numbers really complex numbers?

Basically, it depends. For technical reasons you can’t necessarily say that the natural number 2 is identical to the complex number 2, but it’s entirely reasonable to say that they’re equal.

3. Are complex numbers all there is?

No. Hypercomplex numbers have been studied for over a century. They’re somewhat less interesting than your plain vanilla complexes, and you run into some complications, so they’re not as widely studied.

4. How many more complex numbers are there than familiar numbers?

As has been mentioned, the two sets are of the same size. In general, the set of pairs of elements of an infinite set is the same size as the original set.

5. …Is that true–that complex numbers don’t actually bring anything new to the mathematical table, they essentially just greatly facilitate calculations which would otherwise be inordinately complex?

No. Not even close. x[sup]2[/sup] + 1 has no roots over R, but it has two roots over C.

[Thudlow_Boink]

Real numbers (the “familiar numbers”) can be visualized as lengths or distances, or, correspondingly, as points on the real number line. You can picture a line segment sqrt(2) units long, or a point sqrt(2) units to the right of 0. You can picture -3 as a line segment 3 units long but pointing left, or a point 3 units to the left of 0. Essentially, the real numbers “look like” one-dimensional vectors.

The complex numbers “look like” two-dimensional vectors. Non-real complex numbers correspond to points above or below the real number line, or to line segments that point in some other direction than right or left. i is one unit above 0. i+1 is northeast of zero, sqrt(2) units away.

@Sequent

3. Are complex numbers all there is? Essentially, when we expand our number system to include complex numbers, we’re essentially saying that every number has two parts–real and imaginary. Can we be certain we won’t someday discover a third dimension we’ve been overlooking all along?

Depends on what you mean. Complex numbers arise because there are fairly innocent looking equations (like x[sup]2[/sup] + 1 = 0) that have no solutions in the set of real numbers, but do in the set of complex numbers. But once you have the set of complex numbers, you have all you’ll ever need to find solutions to algebraic equations whose coefficients come from the set of complex numbers (or even just the set of real numbers).