Between Being and Non-Being: Imaginary Numbers

jonah

Imaginary numbers
are a fine and wonderful
refuge of the divine spirit
almost an amphibian between
being and non-being
~Gottfried Wilhelm Liebnitz

http://www.flickr.com/photos/hotelmidnight/7985421295/
This work, by Deborah McMillion Nering, is titled Imaginary Numbers Are Friends. And, as we will see, they certainly are!

One of the first things we learn how to do is multiply numbers. 2\times 3 = 6. That sort of thing. But what if we multiply a number by itself? This is the familiar operation, which we call squaring a number. 2^2 =2\times 2 = 4 and 3^2 = 3\times 3 = 9. That sort of thing. You can take a number to a power by multiplying it by itself some number of times equal to the power. So if you square a number, you’ve taken it to the second power. You can take it to the third power too, and this is called cubing. 2^3 = 2\times 2\times 2 = 8. And so on.

Now we can ask another question. Given any number, call it N. Is there some number (say “a“) such that N is the square of a? In other words, can we find a such that a^2 = a \times a = N?

In the case of positive real numbers, the answer is always yes, although you may not be able to write down the number as a fraction. (These numbers are called irrational,  and that is a story for another time.) We call a the square root  of N, and we denote it a=\sqrt{N}.

You’ll notice that I said positive real numbers. Why do I specify this? (For that matter, why do I specify the word real? You’ll see.)  Well, (-1)\times (-1) = 1 and this goes for every negative number. The square is always positive. The same goes for positive numbers. It’s impossible for a real number to square to a negative number. In other words, negative numbers do not have square roots.

Nevertheless, mathematicians and physicists take the square roots of negative numbers all the time. We call them imaginary numbers. Where do these come from? Why do we use them? Read on to find out!

Polynomials. Polynomials.

The story of imaginary numbers begins with the story of the roots of polynomials. Roughly, a polynomial is a function that takes some input, and constructs the output as some combination of powers of the input. For example, a quadratic polynomial takes the form

    \[f(x) = a x^2 + b x + c,\]

where x is the input, f is the output, and a, b, and c are real numbers. (Interesting side note: with the right choice of coefficients, a quadratic polynomial describes the trajectory of an object shot out of a cannon.) Specifically, we might have that

    \[f(x) = x^2 + x - 6 = (x+3)(x-2).\]

If you plot the output as a function of input, with the output on the vertical axis and the input on the horizontal axis, it looks something like this.

my quadratic friend
The plot of the input of a quadratic polynomial as a function of the output. The input is on the x-axis and the output is on the y-axis.

You’ll notice that when the input is x=-3 or x=2, the output is zero—the line crosses the horizontal axis. These are called the roots of the polynomial, and they’re important because finding the roots of a polynomial allow us to solve equations involving polynomials. For example, our graph tells us that the equation

    \[x^2 +x = 6\]

is solved by x = -3 and x = 2. We know this because we can rearrange the equation to become

    \[x^2+x-6 = 0.\]

In other words, we’re looking for the values of x, our input, such that the output is zero.

Notice that not all polynomials have roots. For example, the polynomial

    \[f(x) = x^2 + 4\]

does not have any roots. This is because, if you make a plot like above, the curve  never crosses the zero line:

rootless!
Sometimes, a quadratic polynomial has no real roots. This polynomial certainly doesn’t have any!

But it was not in the study of quadratic polynomials that imaginary numbers were discovered. Rather, it was in the study of cubic polynomials. A cubic polynomial is a polynomial of the form

    \[f(x) = a x^3 + b x^2 + c x + d,\]

where a, b, and c are real numbers as before. For example, if we choose a=1, b=0, c=-4, and d=0, we get the cubic polynomial

    \[f(x) = x^3 - 4x.\]

If we plot this polynomial as before, we get something at looks like this:

Cubic! Cubic!
This polynomial has three zero-crossings!

Notice how this cubic polynomial has more roots than the quadratic? This is a general feature of polynomial equations. The maximum number of roots a polynomial can have is equal to the highest power in the unknown of the polynomial. In this case, we have an x^3 term, so the the highest power is three and we can have three roots.

Of course, we can have _fewer_ roots. It’s very easy to construct a polynomial with only one root. Take our cubic from before, but set d=4. when we plot this, we get:

1 root! 1 root!
This cubic polynomial only has one root.

Can a cubic polynomial have zero roots? Well, technically yes. If we choose a=0, b=1, c=0, and d=4, we end up with

    \[f(x) = x^2+4,\]

which we know is a quadratic polynomial with zero roots. But what if we force a\neq 0? Well, cubing—multiplying a number by itself three times—preserves sign. (-2)^3 = (-2)\times (-2)\times (-2) = -8 and 2^3 = 2\times 2\times 2 = 8. In general, if a is x is a real number, (-x)^3 = -x^3. Furthermore, any positive real number can be represented as the cube of some smaller positive real number. Combined, these results mean that every real number can be represented as x^3 for some real number x. Mathematicians call this property surjectivity. What this means is that so long as we choose a\neq 0 in our cubic polynomial it will have at least one root.

The Impossible Case

In the mid 1500s, mathematician Gerolamo Cardano noticed that every cubic polynomial (with an x^3 term) has at least one root. Even better, he found an explicit formula to find at least one root of a polynomial if it had the form

    \[f(x) = x^3 -ax - b,\]

where a and b are positive real numbers. This is the formula:

    \[x = \sqrt[3]{\frac{b}{2} + \sqrt{\left(\frac{b}{2}\right)^2 - \left(\frac{a}{3}\right)^2} }+ \sqrt[3]{\frac{b}{2} - \sqrt{\left(\frac{b}{2}\right)^2 - \left(\frac{a}{3}\right)^2}},\]

where \sqrt[3]{n} means the cubed root of n. It’s the number which, if you multiply it by itself three times, gives n.

The formula is horrible, I know. We now call this formula Cardano’s Formula. Disturbingly, however, Cardano’s formula sometimes yields square roots of negative numbers, which we know don’t exist. For example, if our polynomial is

    \[f(x) = x^3 - 15x - 4,\]

then Cardano’s formula tells us that a root is

    \[x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}.\]

This doesn’t make any sense! But mathematician Rafael Bombelli noticed something. If we just pretend the square roots are okay but don’t evaluate them, things work out. Bombelli found that

    \[(2 + \sqrt{-11})^3 = 2 + \sqrt{-121}\text{ and }(2-\sqrt{-11})^3 = 2 - \sqrt{-121}).\]

Then, he just evaluated Cardano’s formula as if it was okay:

(1)   \begin{eqnarray*} x &=& \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}\nonumber\\ &=&\sqrt[3]{(2+\sqrt{-11})^3} + \sqrt[3]{(2-\sqrt{-11})^3}\nonumber\\ &=& 2 + \sqrt{-11} + 2 - \sqrt{-11}\nonumber\\ &=& 4.\nonumber \end{eqnarray*}

This disturbed a lot of people, of course. Cardano and Bombelli themselves were deeply uncomfortable with it. Nevertheless, Cardano used this trick regularly. In his magnum opus, Artis magnae sive de regulis algebraicis liber unus (Trans: The great Book of Art), Cardano wrote about a simpler example:

It is clear that this case is impossible. Nevertheless, we shall work thus: we divide 10 into two equal parts, making each 5. These we square, making 25. Subtract 40, if you will, from the 25 thus produced, as I showed you on the chapter on operations in the sixth book leaving a remainder of -15, the square root of which added to or subtracted from 5 gives parts of the product which is 40. These will be 5+\sqrt{-15} and 5-\sqrt{-15}.

Putting aside the mental tortures involved, multiply 5+\sqrt{-15} and 5-\sqrt{-15} making 25-(-15) which is +15. Hence the product is 40.

(Source: A History of Algebra: From al-Kwarizmi to Emmy Noether, by B.L. van der Waerden. Emphasis mine.)

The Impossible Possible

(At this point, I’m going to leave the chronological narrative behind and discuss ideas in a way that are conceptually easy for me. So the people I mention first may have been born years after the people I mention last died.)

At first, people just used the square roots of negative numbers to produce real numbers. Rene Descartes (of I think therefore I am fame) coined the term imaginary numbers, and the name stuck. Carl Friedrich Gauss, one of the finest mathematicians in history, noticed something else though. If we treat \sqrt{-1} as a number, every quadratic polynomial has two roots. Remember our rootless quadratic from before? It was

    \[f(x) = x^2 + 4.\]

But we can re-write this as

    \[f(x) = (x+2\sqrt{-1})(x - 2\sqrt{-1}).\]

Then the polynomial is zero exactly when x = 2\sqrt{-1] or x = -2\sqrt{-1}. Indeed, if you allow for imaginary numbers, every cubic polynomial has exactly 3 roots and every polynomial with x^4 in it has exactly 4 roots, etc. This is the fundamental theorem of algebra, and it’s an incredibly powerful theoretical tool.

The Complex Plane

What these imaginary numbers mean, however, requires additional tools. Leonhard Euler, another of the giants of mathematics, took the complex the complex numbers out of one dimension and into two. He also gave the fundamental complex number a name. He let

    \[\sqrt{-1} = i.\]

In Euler’s formalism, the numbers no longer lived on a number line, they lived in the complex plane. Multiples of the number 1 were on the horizontal axis, and multiples of i were on the vertical axis. Sums of real and imaginary numbers could be anywhere on the plane.

complex plane
Now numbers live in the complex plane, rather than on the real number line.

With the advent of the complex plane, the world of numbers grew dramatically.

It’s worth noting that you can always convert a complex number into a real number by taking its norm squared. I’ll explain by example. Say we have the number

    \[2 + 3 i.\]

If we square this number, the result will still be partly imaginary:

    \[(2+3 i) (2 + 3 i) = 4 + 12i - 9 = -5 + 12i.\]

But, if we flip the sign on the term with the i, we get a new number. If we multiply this new number by the original number, we get a real number:

    \[(2 + 3 i) (2 - 3i) = 4 + 6 i - 6 i -9 i^2 = 4 + 9 = 13.\]

And this holds true for any complex number

    \[a + b i.\]

The Most Beautiful Equation in All of Mathematics

Now that we have two dimensions to play around in, we can make circles. In the xy-plane, a plane made of just real numbers in two directions, the x and y coordinates of a point on a circle can be described by sine functions and cosine functions. You might have seen a diagram like this one in school.

Polar coordinates! Polar coordinates!
We can translate a description of a circle based on positions on the x- and y- axes into the amount you’ve traveled along the circle, (the angle) and the radius of the circle.

This description is called polar coordinates. Given a point, we’re relating the position of the point along the x- and y- axes of the plane to it’s position around a circle, which we call the angle \theta, and its of radius r.

This is all for planes made of real numbers. But is there an equivalent to polar coordinates in the complex plane? Euler found one. But now the y-axis is imaginary! Euler found that polar coordinates for the complex plane look like this.

Complex polar coordinates!
In the complex plane, the y is replaced by an i.

But Euler didn’t stop there, he found a very strange relationship between exponentiation (taking things to powers) and angles in the complex plane:

    \[e^{i\theta} = \cos(\theta) + i \sin(\theta),\]

where e\approx 2.718281828459... is known as Euler’s constant, an irrational number that is to calculus as \pi is to circles. We call this formula Euler’s formula.

At first glance, this formula looks really funny. What does it mean to multiply a number by itself an imaginary number of times? For that matter, what does it mean to multiply a number a fractional or irrational number of times? Really this is totally nonsensical. Generalizing to fractional exponentiation is easy. We’re just abusing notation a bit. You see, we can write the n^{th} root of a number a as

    \[\sqrt[n]{a} = a^{(1/n)}.\]

And then we can take fractional exponents by multiplying the number the correct number of times and then taking the correct root. For example,

    \[2^{(3/2)} = \sqrt{2^3} = \sqrt{2\times 2\times 2}.\]

This makes manipulating exponents extremely easy because now the n^{th} root of any number to the n^{th} power is obviously the number itself:

    \[\sqrt[n]{a^n} = a^{n/n} = a^1 = a.\]

But what does it mean to take an imaginary power? All I can say for now is that there is a straightforward way to generalize the operation I told you about—multiplying a number by itself some number of times—to this more abstract notion of exponentiation using a tool from calculus called a Taylor series, which I’ll describe another time.

Euler’s formula works, though… and if we choose \theta=\pi, the sines and cosines simplify and we get what many people call the most beautiful equation in mathematics:

    \[e^{i\pi} - 1 = 0.\]

e is fundamental to calculus. \pi is fundamental to geometry, i is the fundamental unit of imaginary numbers, and 0 and 1 are the building blocks of the counting numbers. In this one formula, we have the five most important numbers in all of mathematics, related to each other in an incredibly simple way.

Applications

That’s about all I wanted to say about what imaginary numbers are. As pure mathematical constructs, imaginary numbers are beautiful and powerful tools for solving algebraic equations. But are they useful in the real world? The word imaginary implies not so much.

But actually, imaginary numbers are everywhere in math, science, and engineering. The mathematician J. S. Hadamard perhaps put it best:

The shortest path between two truths in the real domain passes through the complex domain.

I’ll give a few examples here.

Waves

Imaginary numbers show up in electromagnetism and electrical engineering, where scientists take advantage of Euler’s formula. Electromagnetic waves are made out of electric and magnetic fields feeding into each other, and the fields change in both space and time. Really, this means we should use two functions to describe the waves, one describing the space evolution and one describing the time evolution. However, we can get away with using only one function with real and imaginary parts. This is because the waves are described by sine functions and cosine functions. We hide the extra function by using Euler’s formula to transform the sines and cosines in x and y into a single e^{i\theta} term.

Quantum Waves

From the start, physicists used imaginary numbers to formulate quantum mechanics. The quantum wavefunctions that describe the positions of particles live in the complex plane and it is their norm squared that determines the probability of finding a particle in a given position. In its full complex glory, the Schrodinger wave equation is:

    \[i\hbar \frac{\partial}{\partial t} \Psi(\vec{r},t) = \left[\frac{-\hbar^2}{2m}\nabla^2 + V(\vec{r},t)\right]\Psi(\vec{r},t).\]

The i is indeed the imaginary i.

The Feynman path integral I described last time also uses imaginary numbers. Indeed, it uses Euler’s formula. In the path integral, you sum over the directions of many arrows pointing in different directions as you travel along all possible paths between two places. Those arrows can be represented, using Euler’s formula, as an imaginary number. And, in the language of mathematic,s we write a Feynman integral as

    \[\mathcal{A}(\vec{p_1},\vec{p_2}) = \int_{\vec{p}_1}^{\vec{p}_2} e^{i S[\vec{x}]}\mathcal{D}[x],\]

where \vec{p}_1 and \vec{p}_2 are the initial and final points respectively and where S is the energy cost for the particle to travel along a given path. The big \int\mathcal{D}x tells us to sum over all paths connecting the initial and final points… and the big \mathcal{A} is equivalent to the quantum wavefunction in Schrodinger’s equation.

Special and General Relativity

In some of my previous articles, I’ve explained how the speed of light is constant and how this leads to special relativity. Later, I described an alternate formulation of special relativity called Minkowski Space. In Minkowski space, space and time are unified into a single spacetime. Furthermore, vectors (just think arrows—a direction and a length) pointing in the time direction have negative square length. In other words, vectors point in the time direction are imaginary! So in Minkowski space, time is an imaginary number! Then we can think Lorentz transformations—the mathematical transformations that generate length contraction and time dilation—as rotations by an imaginary angle.

Wick Rotations

There’s a beautiful connection between quantum mechanics, which I’ve talked a lot about and statistical mechanics, which I’ve only touched upon. We can connect these two disparate fields by a Wick rotation, which rotates a quantum system through the complex plane. It changes a Feynman path integral into a more classical sum over statistical states.

All of these applications are why I chose Deborah McMillion Nering’s picture. The image shows a number of applications of complex numbers. See if you can spot them!

Further Reading

That’s about all I have to say for applications for now. I promise there will be more about complex numbers in future posts. For now, I just wanted to list a couple of resources that I liked.

Questions? Comments? Insults?

That’s all for now. If you have any questions, comments, feedback, or corrections, please don’t hesitate to let me know!

3 thoughts on “Between Being and Non-Being: Imaginary Numbers”

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