Circular Math

Five years ago today I posted, Beautiful Math, which is about Euler’s Identity. In the first part of that post I explored why the Identity is so exquisitely beautiful (to mathematicians, anyway). In the second part, I showed that the Identity is a special case of Euler’s Formula, which relates trigonometry to the complex plane.

Since then I’ve learned how naïve that post was! It wasn’t wrong, but the relationship expressed in Euler’s Formula is fundamental and ubiquitous in science and engineering. It’s particularly important in quantum physics with regard to the infamous Schrödinger equation, but it shows up in many wave-based contexts.

It all hinges on the complex unit circle and the exp(i×π×a) function.

As you’ll see, this post brings together many previous posts:

• Beautiful Math, which introduces Euler’s Identity and Formula. (For this post, it’s important that you’ve read this one.)
• Many posts about complex numbers, and in particular the complex plane. (In this post you do need to know what the complex plane is.)
• Fourier Curves, which introduces some Fourier transform ideas, including that it’s used in JPEG compression. (Optional reading.)

An irony about that last post is that, except for the title and image captions, I don’t actually mention Fourier transforms at all. (It was mainly about the pretty pictures.) In the next two posts I plan to show you how Fourier transforms work.

That — and more — is based on the following:

One might see just an opaque bit of math. One might recognize the mathematical operation of exponentiation — something to the power of something. One might even recognize the constants e and i but not have any sense of their appearance, use, or implications, here.

The point is that the expression might look mysterious, but it doesn’t look surprising.

On the other hand, if one is more familiar with the constants and the operation, one might very well find the expression not just mysterious, but surprising. And weird! There may be a sense of WTF? or that it can’t work (or at least how does that work?).

The level of hey wait a minute may only increase with this:

What in the world can it possibly mean to raise the transcendental number, e, to the power of i times π? How is that even possible? And how can that be equal to minus one?

One clue is that we’re using the imaginary unit, i, which means the complex numbers, which means the complex plane, which means there is a geometric aspect to this (and it’s the geometry that’s cool and makes Fourier transforms work).

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To make sense of it, we must extend the definition of exponentiation from the basic one most of us first learned.

That definition saw the exponent as an integer specifying how many times to multiply the base number times itself. For instance:

We multiply the base number, N, times itself five times. Simple, but as we got a little deeper into math, we did encounter other definitions:

Note that all three of the above equalities can be mathematically justified using the hyper-important exponent rule:

There are also logarithms, which use real exponents, which gets us closer to the idea of something like π as an exponent.

(Regarding pi, 10^π is just the number 1,385.455731… — as with the logarithm above, pi is just another real exponent with a value a bit more than three. It’s using i as an exponent that’s the weird addition here.)

The point here is that, even on the outskirts of math, exponents are part of a larger picture, some of which you’ve probably already seen.

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With that introduction, here’s a very good (and very short) video that introduces the ideas behind what at first seems an impossible notion:

(Note that 3.14 minutes is not the same as 3:14 minutes!)

The Executive Summary: In the extended definition, exponentiation with i becomes rotation on the complex plane. When we choose to use e as the base, a full rotation of 360° is exactly 2π radians, which is the circumference of a circle and which therefore allows a direct mapping to trigonometry.

(As I’ve posted about not long ago, multiplication by i rotates points by 90° on the complex plane. See also: Matrix Rotation)

This mapping to trig is where Euler’s Formula comes in:

e^ia = cos(a) + i sin(a)

Both sides of the expression are ways to denote a complex number that lies on the complex plane at angle a. As you might imagine, treating a complex number as a single exponential value offers some calculation advantages.

(Euler’s Identity, e^iπ = -1, is the specific case where a=π.)

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But why e? What’s so special about that number? This (14-minute) video explains:

The short version is that the function e^x is its own derivative, which tames certain aspects of the math.

As the video gets into, derivatives are important throughout physics (and other areas of life, such infection rates and your bank account). We encounter derivatives everywhere; velocity, for example, is the derivative of distance over time (as seen in the first video).

In general, the function a^x (for some value of a) is proportional to its derivative, but only when a=e is the derivative equal. (Remember that e is just a transcendental constant with the value 2.7182818284590…)

The upshot is that, when e is the base, there are useful manipulations of formulas containing it (as seen in both videos). That it’s the base of the natural logarithm makes it a natural choice in many situations (as, for instance, the unit circle on the complex plane).

As far as how the calculation is actually carried out, the short form is:

The exp(x) function is the same thing as raising e to the power of x. It makes the expression easier to write and typeset. It also makes it more clear that “e-to-the-x” is a function.

That short form expands to:

For however large we make n. (The larger it is, the more accurate the calculation.) Note that x can easily be a complex number in this expression.

Just remember: e^x = exp(x)

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Figure 1. Various a^x curves.

Let’s start with the general exponential function.

Figure 1 shows a set of exponential curves.

Each curve plots a^x for some value of a.

The blue curves have values for a above one, and the purple curves have values below one.

The red curve (flat at one) is a=1.0. (One to the power of anything is just one.)

The green curve is a=e.

The dark-blue curve is a=1.5. I included it to show how the flat a=1.0 curve starts to deflect upwards for positive values of x (and downwards for negative values).

The three blue curves, from outside-in, are a=2.0, a=4.0 & a=8.0. (Notice the green a=e curve is between the 2.0 and 4.0 curves — the value of e is 2.71828…)

The purple curves, from outside-in, are a=0.50, a=0.25 & a=0.125. When values for a increase above 1.0, the curve deflects more and more sharply upwards (after passing through 1.0 at x=0.0). For values of a less than 1.0, the curve deflects more sharply as the value decreases towards zero.

All curves pass through 1.0 at x=0.0, because a⁰ is always 1.0.

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There is a bit of magic we can apply regarding the base value, a:

In other words, a^x is the same as e^xk for some constant k — specifically the natural log of a. So for example, if a=3:

This means we can draw an identical set of curves as Figure 1 using just the exp(x) function (which is exactly how I did it).

We can also leverage that nice derivative property and unify our mathematical approach. So it’s the right-hand version that usually appears in physics formulas. That’s part of why it shows up so often.

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The exp(x) function does much more than create “exponentially rising” curves. Here is another set of curves drawn with the exp(x) function:

Figure 2. Various Gaussian curves calculated with the exp function.

You may recognize these as “bell curves” — their formal name is Gaussian (after mathematician Carl Gauss). They are described by the Gaussian function:

Which, remember, is the same thing as:

The constants A, B & C, control the shape of the curve. The value of A determines the height of the peak; the value of B controls where it is centered, and the value of C (which must be greater than zero) controls its width.

In Figure 2, the value of A is 1.0, the value of B is 0.0, and the various curves reflect different values of C.

(Gaussian curves show up a lot in quantum mechanics as the localization of momentum, energy, or position, to name a few. This is closely tied to what comes next.)

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Now consider the following double-chart, which shows two ways of representing the same thing:

Figure 3. A waveform (upper) and its Fourier transform (lower).

Both show a combination (or superposition) of three sine waves at three different frequencies: 13, 27, and 42 cycles per second. The upper chart shows the energy at given times, and the lower chart shows energy at given frequencies.

The upper chart averages three copies of the exponential function, each generating a different frequency sine wave (note the use of i here):

The lower chart averages three copies of the Gaussian function, each centered on one of the frequencies:

Two ways of looking at the same thing. The lower chart represents a Fourier Transform of the waveform. The upper chart can be constructed from frequency information via a Inverse Fourier Transform.

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The Fourier transform is a fundamental tool in wave physics.

Audio Spectrum Analyzer

As one example, audio spectrum analyzers are common among audio engineers and enthusiasts, because they display, in real time, how much energy (sound) exists at different frequencies. (They’re also fun to watch.)

After good old VU meters, they are one of the more common audio displays (see an example image to the right).

Such analyzers use a Fourier transform to break an incoming signal — which is energy varying in time (the upper chart in Figure 3) — into its component frequencies. (The image here is essentially the same thing as the lower chart in Figure 3.)

In quantum mechanics, Fourier transforms are involved in solutions to the Schrödinger equation, and they underpin the Heisenberg Uncertainty Principle.

That’s where I’ll pick up next time.

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For this and many other posts I’m deeply indebted to Grant Sanderson and his YouTube channel, 3Blue1Brown. If you have any interest in math, you should to subscribe to this channel. It’s by far the best math illumination channel I’ve encountered.

I owe a lot of my recent progress to this guy and his videos!

Stay exponential, my friends. Go forth and spread light and beauty.

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