Boo! Imaginary Numbers Exist
How “real” are imaginary numbers?
Nathan Ong
A friend had asked me what use an “imaginary” number had in the real world—as
someone with superficial knowledge of its applications in engineering and physics, I
wanted to give a better answer other than just “it works”. But to do that, I need to draw
from an analogy on something most people are familiar with: negative numbers.
The following is a point of philosophical contention, but it’s a central
metamathematical question to ask (and is relevant to this discussion on numbers)—is
math invented or discovered?
You could argue either way, but I would say both: symbols, syntax, and the rules that go
with them are invented—but innate, deeper meaning is discovered. Semantics and
structure arise out of the building blocks of symbols and logical rules of inference we set
in place.
More on that another time—I’m starting off with saying that while we invent
representations of numbers, their place and meaningfulness in the world is discovered.
Now, imagine the early humans who had only a primal notion of the natural (counting)
numbers: 0, 1, 2, 3, … etc. (Worth noting: early on, many cultures actually did not
concretize the idea of zero as a number—the first evidence of zero being used was
around 5,000 years ago with the Sumerians, and it developed in the mathematics of
other cultures throughout different points in history.)
The natural/counting numbers make sense as starting place for numbers; after all, we
can instinctually visualize one apple, two dogs, or five fingers. There exists perceivable,
physically-realized representations of natural numbers within our lived-in world.
At this time, negative numbers didn’t “exist” (yet). We teach it in elementary arithmetic
now, but back then when the concept of a number was in its infancy, how could we
expect early humans to grasp what -3 chickens meant, much less know that negative
numbers could be used for other things besides indicating debt?
Back then, people might have just added “down” or “back” to indicate loss or direction.
I started off with 0 coins, then I bought something on debt for 5 coins, so I have 5
down-coins. You took 2 steps forward/up and I took 2 steps back/down.
Of course, -5 coins and -2 steps don’t make sense physically. Yet we see the former in
our debit statements, and maps can have negative longitude and latitude to indicate
position relative to a central point on a 2-axis coordinate plane of earth.
How debt may make us feel—drowning in imaginary, negative water.
The point is, most of us have an intuitive grasp of negative numbers and why they’re
used, even though we know that they don’t physically “exist” in the way that positives do
when using numbers to represent perceivable quantities. But we’ve moved beyond what
we can see with our eyes and know how useful negative numbers are in describing and
navigating the world—we don’t need to touch (count) grass to know it.
Now, we don’t need negative numbers to indicate debt or position when we can just say
they’re their positive counterparts with a ‘down’ flavor, or just perform an axis shift. But
when we define negative numbers to be the additive inverse of positive numbers, we see
that all kinds of useful properties arise when we insert them into our equations—their
applications extend beyond just counting or direction, like quantifying different rates of
change, or connecting division and multiplication which are just a sign away in relation.
So, negative numbers were invented to satisfy the following (where ∀ means for all and
∃ means there exists):
∀x>0,∃y:x+y=0
In words: for any positive number x, there exists a number y such that x added to y is 0.
And from that definition, we discovered how negative numbers were able to extend our
mathematics to modelling something beyond what we can touch.
Let’s go back to imaginary numbers—truthfully, calling them “imaginary” rather than
teaching them as complex numbers with an additional component was a historical
mistake, one that lead to future generations questioning their validity as a number
type—a reasonable doubt, given mathematicians’ poor naming choice. (It’s too bad they
didn’t go with the original name, “lateral numbers”)
To recap, we formulated negative numbers as an answer to the following questions:
●​ Is there any number we can add to a positive number and get back 0?
●​ What happens if we take away a bigger number from a smaller number?
Those questions were answered because we defined a number type (negative) that
satisfied those properties in arithmetic. Then, more questions arose—let’s turn our
attention to other, but similar kinds of questions:
●​ Is there any number we can multiply to itself and get back a negative
number?
●​ What happens if we take an even root of a negative number?
The nature of squaring a number showed us that no “real” number defined so far,
whether positive or negative, can fulfill that. Hence, we define a new number type that
can satisfy this property, and call it i:
i:i2=−1;i=−1
Again, don’t let the fact that we call it “imaginary” put you off! Just like negative
numbers, imaginary numbers don’t exist in real life. An i number of balls isn’t any more
nonreal than -1 ball; neither has any physicality we can grasp.
But we saw how useful properties arose when we inserted negative numbers into our
equations: while the ‘negative essence’ is not something we can use to be able to
describe physical quantities, the ‘negative property’ very much models real-life systems
and behaviors! Take note of this particular distinction.
So let’s see what happens when we try using imaginary (complex) numbers in our
equations—in particular, what happens when we have an imaginary component in the
exponent of a number.
ni
I will digress a bit, but only because I want to address questions that make sense to ask
at this point—you may be wondering—how does it make any sense raising something to an
‘imaginary’ power/number of times?
Consider: How does it make sense raising a number to anything that’s not a
natural/whole number?
3−2= ? 41.5= ?
We know how to find answers to the above, but we don’t use our physical intuition to do
it. A number multiplied to itself a negative or fraction number of times is meaningless
to our senses. But if you recall how we worked around this, we extended the notion of
the exponent beyond something that would only make sense for positive whole
numbers:
3−2=132=19
41.5=432=(412)3=23=8
Reciprocals are multiplicative inverses and one standard of how division is applied, so
we don’t need to visualize a number multiplied to itself -2 or 1.5 times—we know now
that flipping the sign flips the operation; similarly, having a fraction in the exponent is
the equivalent of taking roots of a number.
But what about irrational exponents? you may also be wondering—
32= ?
Indeed, the aforementioned methods only work with rational exponents—we can’t
rewrite the square root of 2 as a fraction. Yet we can plug this expression into a
calculator and return a value, or graph any exponential function and see that it smoothly
takes in all points on the real number axis—so how do we do this?
It involves performing a Taylor expansion (rewriting the exponential as a series of
polynomials) and computing the result of the series (a different base other than e =
2.7182… would just add additional coefficients to our terms):
ex=1+x1!+x22!+x33!+x44!+ ...
e2=1+21+(2)22+(2)36+(2)424 ...
In fact, it’s the extending of exponentiation to include irrational numbers that enables
us to compute an expression with a complex exponent (!).
eix=1+ix1!+(ix)22!+(ix)33!+(ix)44!+ ...
eix=1+ix−x22−ix36+x424+ ...
Now, raising a number to an imaginary exponent is reduced to adding up a series of
complex numbers (and that we can do!).
I don’t want to make this too long on why that is, but you can see here for more details
on Taylor series (an excellent analogy made is that Taylor expansions form a unique
“math DNA” of a function—using only polynomials as building blocks, any
differentiable function can be constructed as a Taylor series).
My point to all of this: we simply establish methods that extend these operations beyond
what we can grasp with our senses and apply them (provided they follow consistently
from our original definitions).
So back to doing weird stuff with exponents: consider the following function, where e is
the natural logarithm base constant and i is the imaginary unit root:
f(x)=eix
In order to visualize this, to our familiar 2-D coordinate axes we introduce a third,
nonreal axis (call it Im) and plot it against x and Re(f(x))—the real component of f(x).
3-D graph of f(x)
It’s helpful to project this visual onto lower dimensions so it’s easier to see what’s going
on:
the real (left) and imaginary (right) part
Aha! Anyone who’s ever suffered through precalculus may be able to recognize what’s
happening here. It’s the delinquent duo, cosine and sine, hidden in our expression even
though they weren’t there to begin with—our little complex exponential was really a
trigonometric function in disguise!
Our boy Euler developed an equation to quantify this behavior we’re seeing—and it’s
used almost universally in any field where sinusoidal (cyclic/periodic) properties comes
into play:
eix=cos⁡(x)+i⋅sin⁡(x)
Side note: Multiple proofs for this famous equation exist, though a more common one involves
doing Taylor expansions and proving it that way. For the curious
Moreover, this equation has the very useful property of being able to easily separate the
real and imaginary components—and it isn’t necessarily indicative of measuring an
imaginary unit of something when we do use this equation in “real life”. If complex
numbers are a 2-dimensional analog/extension of real numbers, the imaginary
component is simply another dimension/variable—it’s useful for us to be able to deduce
multiple variables that may have very real quantities, but are intricately connected to
one another in this wave-like dance between time and output.
I’m no engineering bro, but that looks to me like imaginary numbers being used in
circuit analysis
😎
From circuits in electrical engineering to probabilities in quantum mechanics, having a
number type with the following property turns out to be very helpful in modeling real
behaviors: if you insert me into an exponent, you turn the exponential function I’m squatting in
into a trigonometric, parsable one.
Just like how negative numbers have the property: if you insert me into an exponent, you
turn the exponential (repeated multiplication) into a reciprocal (repeated division).
Another way to think of complex numbers is how they extend the number line, given
our starting positive number line:
Negative numbers just extended the positive number line into the other direction:
Then complex numbers again extended the real number line, which we can think of as
adding another dimension, making it a complex plane:
Whether or not this visualizing of the imaginariness of numbers is helpful, remember:
just like how intuition of the ‘negative property’ comes more easily than ‘negative
essence’, we don’t necessarily need to grasp the ‘imaginary essence’ of a complex
number to understand its place in mathematics—just think of it as another kind of
number: not negative, not zero, not positive, but a new kind that imbues equations and
other systems with additional special properties, just like how negative numbers
enabled us to extend our mathematics to beyond just modelling the physical and being
able to model states of change.
To add, complex numbers arise in areas beyond just modelling wave-like behavior. In
calculus, we can use complex numbers to evaluate integrals normally considered
difficult to solve for when you are otherwise limited to working in the real field. Or
transforming with polar coordinates to smoothly deal with calculating equations of
motions in many mechanical systems.
Sure, you don’t need any knowledge of complex numbers to be able to file your taxes or
balance a checking account when negative numbers suffice. But the device you’re
reading this on had complex numbers arise at some step in its creation, so it’s worth at
least acknowledging that—they are helpful, if not essential to the behind-the-scene
machinations of everyday life.
Imaginary numbers aren’t any weirder or more “fake” than negative numbers—all
numbers are (which, debate on this encroaches more towards math-philosophy than it
does mathematics). No numbers are truly real, unlike this Southwest chicken wrap I’m
eating that I got from Trader Joes, or a mother’s love for her children.
But I suppose the thought of a ghostly number “possessing” an equation in spirit and
making it do things you can’t normally do is pretty frightening.
And with that… BOO!