direct interpretations of an imaginary number

There have been many questions about "applications" of complex numbers. I don't want to ask for examples of how complex numbers are used, but for a specific way to interpret one complex number in an example. Let me clarify:

Natural numbers have many possible interpretations (take nr 5:

• The amount of some collection of objects (there are 5 stones on the table)
• The index of something that identifies it and distinguishes it from other similar objects (go to house nr. 5, instead of house nr. 3)
• distance from some point (the mountain is 5 miles away from our town)
• etc.

Negative numbers have been unintuitive to mathematicians in the past, but they also have interpretations (take nr -100):

• The amount of money you owe someone (I have minus 100 dollars. I owe someone 100 dollars)
• The height compared to sea level, if you are under water (we are at negative 100m height. We are 100m below sea level)
• etc

Rational numbers also have clear interpretations:

• the part of a larger object (I have 1/3 of a pie)
• the length of an object measured in larger units than that length (this book is 1/3 of an inch thick)
• etc

I'll skip irrationals, but can anyone give an interpretion of the imaginary number $\sqrt{-x}$?

That is, can you form a sentence that makes sense, which points to the square root of a negative number?

NOTE: The examples don't have to be elementary-school-like, as the ones I gave above, but I am explicitly not just looking for a way to use complex number operations, because I already know many examples (scaled rotations). I am looking for a way to interpret an actual imaginary number. i.e. a sentence such as: "there is $\sqrt{-5} kJ$ of energy in this room" or something, except that it should make sense.

• Vectors on a plane – Yuriy S Sep 18 '16 at 11:48
• One viewpoint is that a complex number tells you both a magnitude and a phase. A Fourier coefficient tells you both how much of a pure frequency is present, and what the phase shift is for that frequency. – littleO Sep 18 '16 at 11:51
• msm you cannot possibly have read my question and given that response? Yuriy, that is not an interpretation in the way I explained it. @littleO, this is getting there. But still, it seems to me that the imaginary part of a fourier coefficient is not directly related to the square root of a negative number, the way that "debt" is related to the negative of a natural number. It seems to me that using imaginary numbers for that is mathematically convenient but it doesn't have a clear interpretation that relates it to what the imaginery number is (namely, the sqrt of a negative number). – user56834 Sep 18 '16 at 12:02
• If you want a complex number $z$ to have physical significance, perhaps it is better to think of $z$ as $r e^{i\theta}$ rather than as $x + iy$. I think $r$ and $\theta$ may have more direct physical meaning. In MRI, where Fourier coefficients are measured directly, I think Fourier coefficients are viewed as telling us both a magnitude and a phase. Someone else may be able to explain this in more detail. – littleO Sep 18 '16 at 12:08
• @littleO I appreciate the suggestion to interpret $z$ as $re^{i\theta}$, but that would be counterproductive, because I'm precisely interested in an interpretation of $\sqrt{-n}$. That is, I'm more interested in an interpretation of $i$ than of $z$. – user56834 Sep 18 '16 at 12:14

One possible direct interpretation of a complex number is certain linear transformation of a plane.

A real number $x$ can be considered as "scaling" transformation that converts each vector $\mathbf{v}$ to $x \mathbf{v}$. The number $x=-1$ then changes the direction of vectors. What is the "square root" of such transformation? Obviously, it's a rotation of plane by $\pm 90$ degrees.

So a complex number $r e^{i\varphi}$ may be interpreted as a transformation of the plane that to a vector $\mathbf{v}$ assigns $R_\varphi (r\mathbf{v})$ where $R_{\varphi}$ is the rotation by $\varphi$. Composition of such linear transformation corresponds to the multiplication of complex numbers. Note that $\pm i$ corresponds to a rotation by $\pm 90$ degrees and $i^2=-1$ means that if you rotate twice, you end up with a transformation that reverses the direction of vectors.

• That would not qualify as a "direct interpretation" the way I am referring to it. I understand that we can see a complex number as a linear transformation, e.g. as the matrix $$\left(\begin{matrix} a & -b \\ b & a \\ \end{matrix}\right)$$, but this does not "look into" the imaginary number $\sqrt{-n}$. That is, it is not an interpretation of the imaginary number itself, its just an interpretation of an operation on a complex number. – user56834 Sep 18 '16 at 12:19
• @Programmer2134 No, what I mean is that you may define a complex number as a transformation of the plane, not an element of the plane. The multiplication is then defined obviously (although the addition becomes then somehow artificial). Then the square root of $-1$ is interpreted really as the "complex number" $x$ such that $x\times x=-1$. Whether or not it is "direct" enough, I'm not sure :) – Peter Franek Sep 18 '16 at 12:24
• Because firstly, the rotation is an operation on the complex number, not a complex number itself. Secondly, because it does not relate specifically to the square root of a negative number, which is the definition of imaginary numbers. – user56834 Sep 18 '16 at 12:25
• Please don't seek too much in how "direct" is defined. What I am looking for is simply an interpretation of an imaginary number, such as "there is $\sqrt{-5}$ energy in this room", or "this system has $\sqrt{-5}$ hoolahoompah" (whatever hoolahoompah means). I am looking for a way that the sqrt of a negative number can be interpreted. Such examples would be very different from most applications that I've heard of, which simply have an interpretation that is akin to a matrix transformation, and has nothing to do with the fact that $i=\sqrt{-1}$ – user56834 Sep 18 '16 at 12:32
• if you define complex numbers as transformations, this completely neglects the thing I'm interested in, namely an interpretation of $\sqrt{-n}$ – user56834 Sep 18 '16 at 12:33

I'll skip irrationals, but can anyone give an interpretation of the imaginary number $\sqrt{-x}$? or of the complex number $a + \sqrt{-x}$?

That is, can you form a sentence that makes sense, which points to the square root of a negative number?

I am explicitly not just looking for a way to complex number operations

Sense is a very subjective thing. A familiar thing makes more sense than an unfamiliar one. So, perhaps, if my choice of examples does not strike you as sensible, I hope, you would have a bit of patience with my point of view.

Continuing with your analogy, let us first have a one for real numbers.

Real Numbers - Imagine an infinite line, with some point predefined as origin and some unit of measurement. A real number can be used to mark some position on the line.

Now, consider $x \epsilon R$ and $y \epsilon R$. What is a good way to understand a pair of real numbers?

Pair of real numbers - Imagine an infinite plane with some point predefined as origin and some unit of measurement. A pair of real numbers $(x, y)$ can be used to mark some position on the plane.

This is how I make sense of complex numbers. Normally, when I do this, most of my students cry foul! "This is not complex numbers, this is coordinate geometry".

So, let me elaborate. Another system which makes use of this representation of pair of real numbers is vectors. When I write a vector (2, 3), you would automatically assume an arrow on a plane or a point on a plane. But that is not enough for something to be a vector. A vector also needs to have some operations defined, namely '+', '$\times$' and '$\cdot$'. Without these operations, we won't be able to call $(x, y)$ to be a vector.

Vector - A vector is tuple of real numbers such that the operations '+', '$\times$' and '$\cdot$' follow some specific rules.

Some of these operations are more sensible than others. The '+' and '$\cdot$' operations are good, but the '$\times$' is a very difficult one to intuit.

The point being, after the basic type pair of real number, if we want to understand further specializations like vectors and complex numbers, we need to understand the operations which they entail.

With this in mind, let us define a new class -

Not yet complex number (NYCN) - A pair of real numbers $(a, b)$ with operations '+' and '$\cdot$' defined as follows -

$(a, b) + (c, d) = (a + c, b + d)$ and

$(a, b) \cdot (c, d) = (ac - bd, ad + bc)$

How do we interpret the '$\cdot$' operation? It has two functions, rotation and scaling. If you take an arrow $(x, y)$ and want to rotate it anti-clockwise by say $\theta$, you could get the new point by performing $(x, y) \cdot (\cos \theta, \sin \theta)$.

This rotation operation is what differentiates us from vectors. The problem with operation is in its pedagogy - it is so different from everything we have done yet (unless you count cross product in vectors).

So, how do we reconcile these operations with our previous versions of '$\cdot$'. What follows is cosmetic surgery -

Complex Numbers - $x + iy$ is a complex number for ($x, y \epsilon R$), if $(x, y)$ is a NYCN.

$i$ is a constant which does not 'interact' with a real number much like $\hat{i}$ in vectors. Now, we want complex numbers to have our real number like addition and multiplication rules. Addition - $a + ib + c + id = (a + c) + i(b + d)$. So the new NYCN is $(x_1 + x_2, y_1 + y_2)$ which is good for us. Multiplication - $(a + ib) \cdot (c + id) = ac + i^2bd + i(ad + bc)$. This is very close to our requirement of NYCN. The $y$ term is perfect - $(ad + bc)$. But the $x$ term has $i^2$ instead of -1.

So, for complex numbers to have same functionality as NYCN, we need to take $i^2 = -1$.

"But this does not make sense!! Why take $i^2 = -1$?" Don't. Work with NYCNs. You would be good. The $i^2 = -1$ is a tool which makes things cleaner and equations become more beautiful. If you don't like makeup, don't put it on. The thing is, a bit of makeup does wonders to one's looks.

• I think I completely understand your way of thinking. It is a very reasonable approach. I think underlying this is basically an attitude of: "don't think about what it means, just make sure you can use it". I don't think there is anything wrong with this intrinsically, but if means that your answer to my question is basically "you should not ask that question at all". However I won't do that, because I'm still interested if there is an interpretation of $\sqrt{-5}$. – user56834 Sep 18 '16 at 14:09
• Fair enough. I didn't mean, "You should not ask that question at all", it was more of an experience with young kids where they ask something but mean to ask something else. I hope somebody benefits from this. :) – Pratyush Rathore Sep 18 '16 at 15:35