This post may seem a little out there, but that might be the point.

Last week in differential
equations we learned about a process our textbook called complexification. (You
can go ahead and google that, but near as I can tell what you’ll find is only vaguely
related to what my textbook is talking about.) Complexification is a way to
take a differential equation that looks like it’s about sines and cosines and
instead make it about complex exponentials. What does that mean?

Well, I think most people
know a little bit about sine and cosine functions. At the very least, I think
most people know what a sine wave looks like.

Shout out to Wikipedia. |

Such a wave is produced by a
function that looks something like f(x) = sin(x). Sine and cosine come from
relationships between triangles and circles, but they can be used to model periodic,
fluctuating motion. For example, the way in which alternating current goes back
and forth between positive and negative is sinusoidal.

On the other hand,
exponential functions don’t seem at all related. Exponential functions look
something like f(x) = e

^{x}, and their graphs have shapes such as this:Thanks again, Wikipedia. |

Exponential functions are
used to model systems such as population growth or the spread of a disease.
These are systems where growth starts out small, but as the quantity being
measured grows larger, so too does the rate of growth.

Now, at first blush there
doesn’t appear to be a lot of common ground between sine functions and
exponential functions. But it turns out there is, if you throw in complex
numbers. What’s a complex number? It’s a number that includes

*i*, the imaginary unit, which is defined to be the square root of -1. You may have heard of this before, or you may have only heard that you can’t take the square root of a negative number. Well, you can: you just call it*i*.
So what’s the connection? The
connection is Euler’s formula, which looks like this:

e

^{i}^{x}= cos(x) +*i*sin(x).
Explaining why this formula is true turns out to be very
complicated and a bit beyond what I can do. So just trust me on this one. (Or
look it up yourself and try to figure it out.) Regardless, by complexifying,
you have found a connection between exponentials and sinusoids.

How does that help with
differential equations? The answer is that complexifying your differential equation
can often make it simpler to solve.

Take the following differential
equation:

d

^{2}y/dt^{2}+ ky = cos(x).
This could be a model of an
undamped harmonic oscillator with a sinusoidal forcing function. It’s not
really important what that means, except to say you would guess (guessing
happens a lot in differential equations) that the solution to this equation
involves sinusoidal functions. The problem is, you don’t know if it will
involve sine, cosine, or some combination of the two. You can figure it out,
but it takes a lot of messy algebra.

A simpler way to do it is by
complexifying. You can guess instead that the solution will involve complex
exponentials, and you can justify this guess through Euler’s formula. After
all, there is a plain old cosine just sitting around in Euler’s formula,
implying that the solution to your equation could involve a term such as e

^{i}^{x}.
This idea of complexification
got me thinking about the topic of explaining things to people. You see, I
think I tend to do a bit of complexifying myself a lot of the time. Now, I don’t
mean I throw complex numbers into the mix when I don’t technically have to;
rather, I think I complexify by adding more than is necessary to my
explanations of things. I do this instead of simplifying.

Why would I do this? After
all, simplifying your explanation is going to make it easier for people to
understand. Complexifying, by comparison, should make things harder to understand.
But complexifying can also show connections that weren’t immediately obvious
beforehand. I mean, we just saw that complexifying shows a connection between
exponential functions and sinusoidal functions. Another example is
Euler’s identity, which can be arrived at by performing some algebra on Euler’s
formula. It looks like this:

e

^{i}^{π}+ 1 = 0
This is considered by some to
be one of the most astounding equations in all of mathematics. It elegantly
connects five of the most important numbers we’ve discovered. Stare at it for
awhile and take it in. Can that identity really be true? Can those numbers
really be connected like that? Yup.

That, I think, is the benefit
of complexifying: letting us see what is not immediately obvious.

It turns out last week was
also Carl Sagan’s birthday. This generated some hubbub, with some praising the
man and others wishing we would just stop talking about him already. Carl Sagan
was admittedly before my time, but he has had an impact on me nonetheless. No,
he didn’t inspire me to study science or pick up the telescope or anything like
that. But I am rather fond of his pale blue dot speech, to the extent that there’s
even a minor plot point about it in one of my half-finished novels.

Now, I read some rather interesting
criticism of Sagan and his pale blue dot stuff on a blog I frequent. A
commenter was of the opinion that Sagan always made science seem grandiose and
inaccessible. That’s an interesting take, but I happen to disagree. Instead, I
think we might be able to conclude that Sagan engaged in a bit of
complexifying. No, he certainly didn’t make his material more difficult to
understand than it had to be; he was a very gifted communicator. What he did
do, however, and this is especially apparent with the pale blue dot, is make his
material seem very big, very out there. You might say he added more than was
necessary.

In doing so, he showed
connections that were not immediately obvious. The whole point of his pale blue
dot speech is that we are very small fish in a very big pond, and that this
connects us to each other. The distances and differences between people are,
relatively speaking, absolutely miniscule. From the outer reaches of the solar
system, all of humanity is just a pixel.

But there are more
connections to be made. Not only are all us connected to each other; we’re also
connected to the universe itself. Because, you see, from the outer reaches of
the solar system, we’re just a pixel next to other pixels, and those other
pixels are planets, stars, and interstellar gases. We’re all stardust, as has
been said.

This idea that seeing the world
as a tiny speck is transformative has been called by some (or maybe just Frank
White) the overview effect. Many astronauts have reported experiencing euphoria
and awe as a result of this effect. But going to space is expensive, especially
compared to listening to Carl Sagan.

So yeah, maybe Sagan was a
bit grandiose in the way he doled out his science. But I don’t think that’s a
bad thing. I just think it shows the connection between Sagan and my
differential equations class.

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