This episode is sponsored by Brilliant. Like most people, I had my first exposure
to imaginary numbers in highschool. And like most people, this was my response. “Ok.’ Fast forward a few years into my physics degree,
and we start using imaginary numbers in wave mechanics and quantum physics. Strange sentences start popping up like “the
wave phase is imaginary” and “the quantum states are complex”. It was here that I started to question what
this whole “imaginary number” thing actually meant. What were they? Did part of the wave not exist? Did quantum particles need therapy? It’s now been 3 years since I finished my
degree, and about 3 weeks ago I was taken by a sudden urge to get to the bottom of these
imaginary numbers once and for all. So let’s start with the name. Imaginary. In my opinion, it’s the worst name that
anyone could have possibly come up with in the history of anything ever. It makes things so confusing because...they’re
just as imaginary as any other number. Or as real, depending on how you look at things. You can’t point to anything in the real
world and say “that’s a 7” or “that’s a 2”. You can point to 7 puppies, or the symbols
that represent a 2, but the concepts of the numbers themselves aren’t tangible physical
things. Now does that mean they’re not real? Well, not necessarily, it depends what school
of thought you prefer. That’s right, there are whole schools of
thought around mathematical philosophy. If you’re a non-platonist, you think that
numbers are merely inventions by humans, used to help us keep track of situations. If you’re a platonist, you think that numbers
exist independent of humans and were discovered, not invented. Neither of these views is necessarily the
right one, but just different ways of thinking about what numbers are. Now the problem I have is that the name “imaginary
numbers” makes it seem like they’re somehow different or less natural than the other numbers,
but they’re not. To demonstrate this, let’s start with something
simpler. The negative numbers. We’re pretty used to the idea of them now,
but imagine the first time they were introduced. What does -50 physically represent? How can you have less than nothing? This little minus sign might seem pretty innocent,
but for people in the 1700’s, it was a big shift in mentality that took years to sink
in. The old English mathematician Francis Maseres
said they “darkened the very whole doctrine of the equations.” So why were they introduced then? Well, they’re very useful. A negative sign can represent things like
debt and direction effortlessly. Instead of always having to say “I owe $50”,
your bank account just says -$50 and it’s obvious what it means. It’s easy to keep track of and and simple
to do calculations with. We use them so much today that we don’t
even think to question them. The same story is true for the irrationals. When the ancient Greeks realised that the
hypotenuse of a right angle triangle with sides of length 1 couldn’t be expressed
as a ratio of 2 numbers, they were so upset they drowned the discoverer
in the Medditteranean Sea. But square roots of things are so useful. Any Pythagorean fan will tell you that. It would be very limiting in today’s world
if we didn’t have them. And it’s exactly the same thing with imaginary
numbers, they were introduced as a means to answer questions that would before be impossible
to answer without them, like, “what is the square root of a negative number?” This idea might seem outrageous You want the square root of less than nothing? but we perform operations on negative numbers
all the time. To represent a doubling in debt, you multiply
a negative number by 2 to get the new amount you owe. We just multiplied a quantity less than nothing. It’s not too much of a stretch of the imagination
that sometimes we’d need to take the square root of less than nothing. So just like with all the other numbers, humans
thought of a way to do just that. Now let’s get onto what this imaginary number,
i, really is. First, let’s think about what squaring something
means. Take the following equation: Now the left hand side of this equation can
actually be broken up into three components. X times x times 1. I’m including the 1 here for a reason so
just sit tight. So what does this literally mean? Well the most basic way to look at it is “what
number can we multiply 1 by twice to turn it into 25?” If you’ve done any basic high school math,
you’ll know that there are two answers, 5 and -5. 5 times 1 takes you to 5, and then times 5
again takes you to 25. -5 times 1 takes you to -5, and then times
by -5 again flips the sign back and takes you to 25. Pretty straight forward. But now what about this? What number when I multiply twice turns 1
into -1? Well it’s not 1, because we’d just end
up with 1 again, and it’s not -1, because the sign flips twice and we end up with 1. So here’s where the novel idea comes in
which I have no idea why they don’t tell you in school. What if we do a rotation? Who says we have to stick to the number line? What if we enter the netherworld above or
below? What if instead of only flipping, we can rotate. If we do two 90 degree rotations, we can turn
1 into -1! That’s what the number i means. A 90 degree rotation about the real number
axis. Hence, i done twice, or squared = -1. We’re used to thinking of numbers on a line,
but with the introduction of imaginary numbers, they actually exist on a two dimensional plane. Yeah, numbers are 2 dimensional. Who would’ve thought? They’re made up of the real axis and the
imaginary axis, and you can transit from one to the other by rotating. So what exactly are imaginary numbers good
for? Well, what are negative numbers good for? One thing is that they’re very good at keeping
track of the sign of alternating systems, like a light switch. If we keep multiplying by a negative number,
we get a pretty obvious pattern. 1, -1, 1, -1, 1, -1. So negative numbers can keep track of any
system that toggles. What about when we multiply by the imaginary
number i? We get 1, i, -1, -i, 1, i, -1, -i. There’s a pretty obvious pattern here too. Imaginary numbers are good for keeping track
of systems that rotate. We’re just not as familiar with these types
of systems because they’re introduced in higher physics like quantum systems and wave
mechanics. Now if you’ve heard of imaginary numbers
you’ve probably also heard of complex numbers. This is just a number that has real and imaginary
components. I once had a student who was having a hard
time with these and said “I can see why they’re called complex numbers”, but complex
numbers aren’t called that because they’re complicated. It’s the same kind of complex as in a housing
complex, in that one whole can consist of different parts. Numbers don’t need to be all real or all
imaginary, they can be a mix of both, hence, complex numbers. To be honest, this knowledge probably won’t
help you get better grades. But hopefully by peeking at the intuition
you’ll be able to have the appreciation that I never did. I’m a huge fan of intuition over memorization,
and a resource I use for that exact reason is brilliant.org. Brilliant is an interactive learning website
which focuses on problem solving and deep understanding rather than memorizing trivial
facts and rules. There are tonnes of courses to choose from
mainly on math, physics, and computer science. The latest thing I’ve become obsessed with
are these Daily Problems, which are a great way to stay sharp, and nothing gives you deeper
intuition than putting your problem solving skills to the test. See if you can solve what happens if you cut
a mobius strip in half, or how many squares in this grid contain the blue square. Signing up is free, but for a 20% discount
off their premium membership, be one of the first 200 to sign up using this link. Brilliant.org/upandatom. You can find it below in the description. Did you find this video helpful? What other concepts do you not really “get”
even though you use them every day? Let me know in the comments and maybe we can
tackle them together. Until next time, bye!
A minor point but it bugged me: at 2:52 not all square roots are irrationals, and not all irrationals are square roots.