Rethinking Complex Numbers: A New Perspective on Teaching and Understanding

[meBigGuy]

I have a view of complex numbers and the way they are taught. I think the whole concept of i as the sqrt(-1) is a terrible place to start. And calling it "imaginary" is worse yet. They should be called blue numbers, or vertical numbers, or something. They are anything but imaginary. It is about simple 2 dimensional numbering systems. The sqrt(-1) is actually an advanced topic, not a starting point. Let me explain.

I view complex numbers as the general case and real numbers as a special (but simple) case.

Things in nature generally have a magnitude and a direction, like impedance and phase, or momentum and direction. If you draw a magnitude and phase "arrow" on a two dimensional graph, you can expresses it as a vertical and horiziontal component. That's the "complex" plane. What is so complex about that. It's pretty simple. Why not call them simple numbers? And what is imaginary about the vertical component? In some representations the horizontal component is just as "imaginary", or hard to visualize.

If you don't case about the phase or directionality you can just deal with 1 quantity, typically a real number. In reality there may be underlying 2-dimensional components but you can treat the whole as a real number.

The whole idea of 2 dimensional numbers can be extended to 3 and more dimensions. what do you call the 3rd dimension? The really imaginary number?

I'm not saying I have a cirriculum to teach complex math in Jr High, but sqrt(-1) is imaginary is not the place to start.

 

[Mentallic]

When I was first exposed to complex numbers, I - as well as my peers - had a good grasp of polynomials and their roots. The biggest amazement for me was when I learned about the fundamental theorem of algebra and seeing that the field of complex numbers are algebraically closed was very enlightening.

While seeing rotations with complex arithmetic was also pretty amazing in itself, I can't deny that since I had been working with polynomials for many years prior to taking that class, finding out that every n degree polynomial has n roots with this new set of numbers was just that much more impressive.

Now, since it makes no sense to relate vectors to the roots of polynomials, and since most students would be exposed to polynomials a lot more than vectors at that stage, $i=\sqrt{-1}$ which is clearly not found on the real (not imaginary) number line would suffice.

Challenge a student to find where the graph $y=x^2+1$ cuts the x-axis, and once they say that it never cuts the x-axis, try to convince them that it does in fact intersect, and that you aren't imagining it

 

[pwsnafu]

I have a view of complex numbers and the way they are taught. I think the whole concept of i as the sqrt(-1) is a terrible place to start. And calling it "imaginary" is worse yet. They should be called blue numbers, or vertical numbers, or something.

Bad naming is nothing new in mathematics. See "irrational numbers".

They are anything but imaginary.

All numbers are imaginary, i.e. a figment of our imagination. It's the name "reals" that's the problem, because it makes everything else look like not-real, so to speak.

It is about simple 2 dimensional numbering systems.

No, its a 2 dimensional algebraically closed field. That's why it's important and not just a 2D vector space.

The sqrt(-1) is actually an advanced topic, not a starting point.

I don't know about what education system you came from, but I learned 2d vectors before complex numbers. Multiplication (and hence polynomials) was the key separating point.

I view complex numbers as the general case and real numbers as a special (but simple) case.

General case and special case of what? Where do the quaternions fit in this? What about the p-adics?

Things in nature generally have a magnitude and a direction, like impedance and phase, or momentum and direction.

Phase is a great example for complex numbers, but momentum is a vector quantity.

What is so complex about that. It's pretty simple.

Complex does not mean "difficult to understand" rather "more than one part". Compare "complex carbohydrate".

The whole idea of 2 dimensional numbers can be extended to 3 and more dimensions. what do you call the 3rd dimension? The really imaginary number?

Does not exist due to Huritz's theorem. 3d vectors exist, but again, no multiplication (technically invertible multiplication).

 

[D H]

I have a view of complex numbers and the way they are taught. I think the whole concept of i as the sqrt(-1) is a terrible place to start. And calling it "imaginary" is worse yet.

The square root of -1 is *exactly* the place to start. It is fundamental.

As far as the name "imaginary numbers" is concerned, look no further than the negative numbers, zero, and the irrationals. You have no problem with -1, 0, or √2. That these are "numbers" just makes sense. That has not always been the case. The qualifier "negative" has a rather derogatory meaning. Negative number quite literally means a quantity that is not a number. Zero is perhaps just a name to you, but it has the same root as cipher. Zero was a very puzzling concept not all that long ago. Finally, the term irrational has an even more derogatory meaning: "numbers that don't make a bit of sense".

Imaginary is yet another of those apparently derogatory labels that no longer have a negative meaning. Just get over it.

 

[SteamKing]

I look at irrational numbers as those which cannot be expressed as the ratio of two integers, or 'non-rational' numbers, if you will.

 

[NascentOxygen]

Vulgar fractions are universally offensive.

 

[Tobias Funke]

I have a view of complex numbers and the way they are taught. I think the whole concept of i as the sqrt(-1) is a terrible place to start.

I don't necessarily agree with your entire post, but you're right on the money here; i^2=-1 is a better place to start. When students are introduced to the sqrt function, it's stressed that we take the positive root to get a single-valued function. So naturally, since both i and -i squared result in -1, and i=sqrt(-1), then i is positive and -i is negative. At least, that's what any reasonable person would think, but not only is it not true, it doesn't even make sense. I don't know why anyone would prefer i=sqrt(-1) over i^2=-1 (and I don't think the difference is as minor as it may at first seem). Maybe someone else has some thoughts?

I also think the correct history of complex numbers should be taught, or at least briefly mentioned. Mathematicians didn't just decide one day that they really wanted solutions to x^2+1=0, and therefore invented them. The real history is more interesting (and it's, you know, real, ie not a lie). I suppose one could argue that the books and teachers just say "well, suppose we want to solve this", or "we can actually define new numbers to solve more equations", and don't necessarily claim that that's how it happened historically, but they should not assume the students will grasp this (and why would they?).

Lastly, I wish the geometric view of complex number arithmetic was given more exposure early on. If they can't prove it, fine, tell them they will in trig, but it's helpful to know, it appeals to "geometric learners", and it can even give them an early view of linear transformations.

I'm sure that many high school books and teachers actually do these things, but from what I've seen it's not very common. Also, sorry if I hijacked your thread with my own musings (rants?) about complex numbers.

 

[HallsofIvy]

It's the distinction between "rational" and "irrational" mathematicians that we need to make!

 

[meBigGuy]

pwsnafu beat me up pretty bad (obviously I'm not a mathematition nor a physicist). But the overall point remains that i is simply part of the "No, its a 2 dimensional algebraically closed field." As I remember, the two dimensional aspect is not addressed initially (in high school) and that is the basic concept, not the significance of i^2 = -1 and its usefullness in solving x^2+1 = 0. Once you learn the 2 dimensional stuff, i^2 = -1 can be derived. Instead you first learn to blindly manipulate this imaginary "i" thing to find roots to equations you really don't care about.

Everybody had good stuff to say and I appreciate all the corrections.

 

[Curious3141]

Vulgar fractions are universally offensive.

Not as much as improper ones.

 

[Jorriss]

If your point is that one should start teaching complex numbers from a geometric point of view, that's not a bad idea.

 

[HallsofIvy]

pwsnafu beat me up pretty bad (obviously I'm not a mathematition nor a physicist). But the overall point remains that i is simply part of the "No, its a 2 dimensional algebraically closed field." As I remember, the two dimensional aspect is not addressed initially (in high school) and that is the basic concept, not the significance of i^2 = -1 and its usefullness in solving x^2+1 = 0. Once you learn the 2 dimensional stuff, i^2 = -1 can be derived.

This is simply not correct. "i^2= -1" does NOT follow from two-dimensionality. There exist other two-dimensional models in which that is not true.

Instead you first learn to blindly manipulate this imaginary "i" thing to find roots to equations you really don't care about.

Everybody had good stuff to say and I appreciate all the corrections.

 

[LCKurtz]

@MeBigGuy: Along the lines of your original post, you might be interested in my article "There's nothing imaginary about complex numbers" at:

http://math.la.asu.edu/~kurtz/complex.html

 

[Mandelbroth]

I have a view of complex numbers and the way they are taught. I think the whole concept of i as the sqrt(-1) is a terrible place to start. And calling it "imaginary" is worse yet.

Nah. The term "imaginary" gives us an excuse to make fun of electrical engineers.

They should be called blue numbers, or vertical numbers, or something. They are anything but imaginary. It is about simple 2 dimensional numbering systems. The sqrt(-1) is actually an advanced topic, not a starting point. Let me explain.

Actually, I'm pretty sure $i^2=-1$ is the starting point.

Things in nature generally have a magnitude and a direction, like impedance and phase, or momentum and direction. If you draw a magnitude and phase "arrow" on a two dimensional graph, you can expresses it as a vertical and horiziontal component. That's the "complex" plane. What is so complex about that.

cf. Definition 1.

 

[cosmik debris]

Nah. The term "imaginary" gives us an excuse to make fun of electrical engineers.
cf. Definition 1.

That's not very nice :-)

Actually for Electrical Engineers j = -i

 

[meBigGuy]

Yet another way to say it.

http://math.la.asu.edu/~kurtz/complex.html

"This article gives a pedagogical approach to introducing complex numbers to students who haven't seen them before. Arithmetic on the number line is generalized in a natural way to arithmetic on the plane. The terms "imaginary" and "complex" are not used during the development, and the "number" whose square is -1 arises naturally. At the end of the development, the standard complex number notation is given. "

Actually, I'm pretty sure $i^2=-1$ is the starting point.

I don't think so. Not any more than 1^2 = 1 is the starting point for real numbers. Can one derive all of complex analysis from just the statement $i^2=-1$?

I talked to a colleague today who learned complex numbers by the geometric approach, in High School. So I guess it depends on the teacher.

 

[pwsnafu]

This is simply not correct. "i^2= -1" does NOT follow from two-dimensionality. There exist other two-dimensional models in which that is not true.

Just to elaborate on this there are precisely three 2-dimensional algebra systems: the complex numbers (i²=-1), the split-complex numbers (j²=1) and dual numbers (ε²=0).
Proof is on Wikipedia.

I don't think so. Not any more than 1^2 = 1 is the starting point for real numbers. Can one derive all of complex analysis from just the statement $i^2=-1$?

Depends. If you already have the real numbers and its properties, then the complex numbers are constructed either as in the article that was linked, or (better) as polynomials with real coefficients. This allows one to rigorously define i, and then you discard the notation you had for a+bi with i²=-1.

And yes that's enough to derive all complex analysis, provided you have real analysis already defined.

I talked to a colleague today who learned complex numbers by the geometric approach, in High School. So I guess it depends on the teacher.

Definitely. A friend of mine was doing high school tutoring and he said the girls he taught understood algebraic concepts better, while boys understood geometric concepts better. So both should be taught.

 

[deiki]

Don't go with i = √(-1). It gives you awful stuff like the following :

i = √(-1) = √(-1/1) = √(1/-1) = √(1)/√(-1) = 1/i ... ouch.

i i = -1 works better as a starting point.

Baby Rudin starts this way : Let a, b, c, d be real numbers. By the definition, x = (a,b) and y = (c,d) are complex numbers if x+y = (a+c,b+d) and x y = (a c-b d,a d+b c). If e is a real number, e = (e,0). At last you define i = (0,1).

 

[Mandelbroth]

I don't think so. Not any more than 1^2 = 1 is the starting point for real numbers. Can one derive all of complex analysis from just the statement $i^2=-1$?

Well, if you give me some other basic-ish concepts (limits, etc.) and, for the sake of making things easy for me, the generalized Stokes' Theorem, I suppose I could.

The point is you'd need to recognize $i^2=-1$ in order to construct the complex plane, unless you just assumed that $i$ is "orthogonal" to the real numbers. In the latter case, you could just use vectors. You don't really gain anything special about complex numbers from a geometric point of view unless you can use it in tandem with $i^2=-1$. For example, vectors in $\mathbb{R}^2$ can all be represented by $\begin{bmatrix}x \\ y\end{bmatrix}=\sqrt{x^2+y^2}\begin{bmatrix}\cos\theta \\ \sin\theta\end{bmatrix}$, but complex numbers can be represented by $x+iy=\sqrt{x^2+y^2}e^{i\theta}$ because $i^2=-1$.

 

[meBigGuy]

I guess I'm saying i is orthogonal to the complex plane, and i^2 = -1 develops naturally from that. It feel better than "let's hypothesize something you can square to get -1" and from that you develop the complex plane.

 

[SteveL27]

I've always agreed with this point of view. If you told students that i represents a quarter counterclockwise turn of the plane, then i^2 = -1 would be an immediate and obvious consequence. Instead, we pull the "square root of -1" out of nowhere, call it imaginary, and simply bury the kids who are already confused by the quadratic formula.

i was discovered algebraically; but centuries later we understand that its true nature is geometric. If I'm facing east and I turn to the north, then turn to the west, I am now facing directly opposite of where I started. This is a very natural and intuitive starting point for teaching the complex numbers.

 

[Mentallic]

I've always agreed with this point of view. If you told students that i represents a quarter counterclockwise turn of the plane, then i^2 = -1 would be an immediate and obvious consequence.

However, why we consider i to be a rotation in a plane wouldn't be quite so obvious, and how this extends to finding all roots of polynomials will be even more confusing.

Instead, we pull the "square root of -1" out of nowhere, call it imaginary

It's not "out of nowhere". Many students by that point would have realized that you cannot square any number get a negative result, hence you cannot take the square root of a negative value.

and simply bury the kids who are already confused by the quadratic formula.

Any student that doesn't have a firm grasp on quadratics shouldn't be learning about complex numbers.

i was discovered algebraically; but centuries later we understand that its true nature is geometric. If I'm facing east and I turn to the north, then turn to the west, I am now facing directly opposite of where I started. This is a very natural and intuitive starting point for teaching the complex numbers.

I won't deny that the geometric aspect of complex numbers is very crucial and elegant, but it's not a necessary starting point. I believe that students should first be taught the algebra of complex numbers which includes adding, multiplying, dividing, conjugates, equating etc. and then move onto the argand diagram.

 

[lurflurf]

i= √-1
and
i²=-1
are almost the same thing and either can be used as a starting point. The troubling thing that arises in either case is that
(-i)²=-1
.
So we can always replace i by -i and have an equivalent expression. This is the complex conjugate.

 

[Mandelbroth]

i= √-1
and
i²=-1
are almost the same thing and either can be used as a starting point. The troubling thing that arises in either case is that
(-i)²=-1
.
So we can always replace i by -i and have an equivalent expression. This is the complex conjugate.

The problem is that complex exponentiation by non-integers is multivalued. $(-1)^{1/2}=e^{(\ln(-1)+2\pi in)(\frac{1}{2})}=\pm i$. Additionally, if we have monstrosities like quaternions, there are multiple other square roots of -1. It's better to just say $i^2=-1$. That way, we also incorporate that $(-i)^2=(-1)^2i^2=-1$.

Caveat: This discussion is well beyond the scope of the original post, so I'm not sure we should continue this discussion if we're catering to someone who has only recently been introduced to complex numbers. We risk further confusing him/her.

 

[HallsofIvy]

A much better method of handling complex numbers is to define them as pairs of real numbers, (a, b), with addition defined by (a, b)+ (c, d)= (a+ c, b+ d), and multiplication defined by (a, b)(c, d)= (ac- bd, ad+ bc). The rest of the properties follow from that.