When I was in school, I wondered, like many others: what happens in between the different forms of graphs of elementary functions? How does this, become this, become this, or become this?
And what’s the deal with the Pac-Man like behavior around division by zero? It made no sense to me
that these wildly different shapes and features have nothing to do with each other. There had to be some hidden metamorphosis between the integer powers. And there was! Extending the domain and
range of these functions to Complex numbers does give us a much deeper insight. But we no longer
have the luxury of graphing two values against each other in two dimensions. We’re stuck with either abstract algebraic answers, or slices of the story akin to taking x-ray images of a
cocooned caterpillar turning into a butterfly. I was never quite satisfied with that. So today’s video
will be taking a somewhat unorthodox approach to indulge the intuition of my younger self.
We’ll introduce complex arithmetic as it’s needed, and draw almost everything in glorious
uncomplicated 2D. The goal is to watch the wings of the butterfly form and grow. I hope you enjoy!
Let’s first just see what happens if we start with f(x) = x, the simplest possible
The Messy Powers
linear graph. Make the first power of x explicit, and vary that power up to 2 in discrete steps.
The negative side seems to be jumping up and down and disappearing in between. Let’s look at
a few specific examples. For x to the power of 1.5, we only get a graph in the first quadrant, where
both x and f(x) are positive. Both the domain and range of this function are Real numbers
greater than or equal to zero. This is because x to the power of 1.5 is actually x to 3/2, which is
a square root of x cubed. For negative values of x, x cubed will be negative and when
that negative ends up under a square root, there won’t be any real valued answers. So right off the bat,
half of our shape is missing. In this case we can still recover it, because it’s just the other
square root. There are always two. We can get both by treating our expression as an equation
instead of as a function. We’ll write y = √x³ and then square both sides. That gives us y² = x³
and the graph looks like this. It’s just the piece we’ve already had, reflected about the s axis.
Let’s just take that as it is and move on. What about x^1.4? Again, we make sense of this by
turning the exponent into a fraction. In this case 14/10, that simplifies to 7/5, which makes f(x)
the fifth root of x to the 7th power. The power under the root is odd, so it can yield a negative
but that’s okay since the degree of root is also odd and we just get negative results out. We also
don’t have to deal with multiple roots since there is only one odd degree root of a number in the Reals.
This shape does look like a slightly bent line, an intermediary step on the way to a parabola.
But the left side doesn’t look like it has any intention of bending upwards. Let’s keep exploring.
What happens at x^1.6? That’s 16/10 or 8/5, so same story with the odd degree root; negatives
are allowed. But this time it doesn’t really matter, since under the root we are raising all numbers
to an even power, 8, making them always positive. That gives us a shape similar to a parabola, but it
came out of the blue without following any pattern from the shapes we’ve seen before. So there isn’t
really a smooth transition between the first and the second power. The negative arm of the graph jumps erratically between the three remaining quadrants and we didn’t even try to plug in an
irrational power. That’s a whole other can of worms we won’t be opening today. Why is this so messy?
And why is all the mess happening when x takes on negative values? It turns out that negativeness
is a part of a much bigger story, where numbers can shoot off not just away from each other relative to zero, but in all other directions too. I have a whole video about that, a link should pop up in
the top right corner, and it’s also available in the description. Raising negative numbers to
non-integer powers can have the effect of “breaking down” negativeness and taking numbers away from the
Real line altogether. Let’s do a quick rundown of what these so-called Complex numbers are and how
they work. All the Real numbers can be arranged in an order on a line relative to zero. We get
About Complex Numbers
Complex numbers when we define another direction for numbers orthogonal to the Reals. This forms a
two-dimensional plane, and specific complex numbers are points on this plane. All the Reals are
of course included, but there’s also all these other numbers above and below the Real line. Numbers that
lie specifically on this new vertical line are called Imaginary, and they are the same as Reals in
every way except for the direction in which they are spanning, just like negatives are equivalent
to the positive, except for the direction. And just like a unit of movement for positive Reals is 1
and for negatives -1, imaginary numbers also have their own version of the number 1, called i,
that happens to point at a right angle away from the Real line. So any number on the Complex plane
can be expressed as a combination of Real and Imaginary, like x and y coordinates of a point
on a regular coordinate system. And we express this combination by adding them together. An example of
a complex number would be 2 + 3i, and you can think of it as walking away from zero, first by
2 in the positive direction, and then by 3 more in the Imaginary direction. This also works
as you would expect in the other quadrants. What can we do with this? Within the regular Reals, we
have this concept of absolute value, which gives us the magnitude of a number independent from
its direction. The same thing exists for Complex numbers, but this time we have a whole circle of
infinitely many complex numbers that are the same distance away from zero, and therefore all
share the same absolute value or magnitude. And this is what we we’ll need for our journey today.
Instead of considering just positive and negative numbers, we’ll be plugging in the entire range of
directions for each magnitude and tracking what happens to them as we vary the power we raise x to
in our expression. By doing that, we are extending the domain and range of our functions to Complex
numbers. Two coordinates go in, two coordinates come out. This is not as simple as drawing a line.
Basically all the points on the plane map to some other points, also on on a plane, and we can’t plot
them against each other because we can’t draw in four dimensions. This is where the unorthodox part
comes in. We’re going to compress the behavior of the Complex valued function onto a single Complex plane.
Let me walk you through it. We’ll start by taking the function f(x) = x², and importing it
Importing a Function into the Complex Plane
into the Complex plane. We do this by redefining it as all Complex number z of the form z = x + f(x)i,
so actually x + x²i, and for now we’ll just use Real inputs for x. This keeps the shape exactly
the same. Only now we have the advantage that we can also plug a Complex number in for x and see
where the value of z we get ends up. Let’s first see what this looks like, and then we can talk
about what’s going on behind the scene. First we’ll take all Complex numbers of magnitude 1. We’ll
start taking them from the Real value of -1 here, going around through the negative Imaginary
half of the circle to positive 1, and then back to -1 around through the positive Imaginary half.
All right, now we draw the z values in the same order. Watch the bigger Complex plane on the
right. The curve we get starts at the regular point of the parabola
for x = -1, travels to the regular point for positive 1,
and comes back to x = -1 another way. Let’s do the same for x of magnitude 1/2.
And now we zoom in a bit, and fill in magnitudes between 0 and 1.
You’ll notice that for magnitudes of 1 and 1/2, which were drawn twice, the curves appear a
bit brighter than the rest. In this visualization, curves drawn over each other enhance brightness.
Let me remove them and dim the axes so we can observe where else this is happening. We have
a bright spot right here, and it appears exactly where the curve for magnitude 1/2 formed a kink.
This is the focus of the parabola. We’ll mark it with a white dot. Now remember, we’re getting this
by plugging the function into our expression for z shown on the left and graphing the values of
z on the complex plane for chosen ranges of x. The main thicker shape that coincides with the
graph of the parabola we got from Real values of x, and the colored curves from plugging in a
series of Complex values of the same magnitude, ranging from 0 to 1. With this established, we
can play around with the function. I’m going to extend it to a general second degree polynomial.
By changing the B and C terms, we can move the parabola, so let’s vary them just slightly to see
what happens. Don’t worry about specific values we’re using for B and C, this doesn’t really matter.
All right, the shape formed by the colored curves is starting to turn, but the bright spot
we found still follows the focus. Flipping the parabola by changing the sign of the A
parameter also doesn’t disturb that property. This gives us an opportunity to vary the A parameter
gradually back from -1 up to 1 and watch the parabola transform between downward and upward
facing. In between it will hit zero, making the function linear, so we’ll get our first chance
to observe a parabola transforming into a line and vice versa. Let’s take the whole screen for
that. Okay, the focus gets launched out, the curves re-form into concentric circles, and then the focus
reappears from the other side, reassembling into a mirrored image of the whole setup.
It may appear as if the focus detaches from the the bright spot phenomenon formed by the curves, but this only happens because we’ve drawn the curves for a very short range. If we increase
the range, like this, they will follow along longer. If we were to extend them to Infinity, they would
keep following along. The focus travels along this straight line, aligned with this feature
that looks like a ray, formed where the curves intersect themselves. We’re seeing a hidden layer
of the behavior of the parabola pop out. And now that we’re a bit more familiar with this
way of graphing and how it behaves, we’re ready to go back to the experiment we started at the
beginning of the video: varying the power we raise a single instance of x to. We’ll start with a simple
f(x) = x, and for this regular line, the curves formed from Complex values of x would just be
concentric circles, like we’ve seen before. Let’s remove them and take the power up a notch to 1.1
Now we have two lines on the negative side. What’s going on here? Let’s draw in just one curve
for x of magnitude 1, again starting at -1, and going all the way back around to -1.
We started with -1 and ended with -1, but somehow our curve didn’t connect back to itself.
How is this possible? I’m sure you’ve heard of the famous Euler’s Identity, the one that states
Overshooting with Euler
that e^iπ = -1. This is actually a special case of representing a Complex number
in terms of magnitude and an angle. The full form would look like this. So a Complex number z given
by this expression would be M distance away from the origin, at a Theta angle in respect to the Real
axis, which means its Real and Imaginary parts could be obtained trigonometrically like this.
The famous special form of Euler’s Identity is just a Complex number of magnitude 1 and an angle of
π radians: halfway around around a circle. And here’s a catch: we can hit that same spot where
the number -1 lies with an angle of π, but also with an angle of 3π, or any odd number of
increments of π. And we can go the other way around too. Remember, I’ve shown you that when
we pick the Complex values of x we put into our function, we start at -1 and travel this
way around to get back to -1 again. The way these values are generated behind the scene
is that we’re ranging the angle from -π to π. The round bracket on the left just means that
we’re only including values strictly larger than -π, because this ensures that every
point is represented only once. Why this range in particular? It follows certain conventions in
Complex arithmetic, but long story short, you could say it’s the range that gives us the nearest of
the infinitely many representations of a Complex number we can get in this form. We reach all angles
with an overall minimum amount of travel from the angle of zero. Now let’s look at raising a Complex
number in this form to a power. The exponent is going to distribute. It’s going to affect the
magnitude normally, like it would any Real number, and it’s going to multiply the angle. Now we can
finally examine the case of our function. We were raising -1 to the 1.1 power. We’ll do it first
as the angle of -π. Okay, the magnitude we can get rid of, because 1 to the power of anything is just 1,
and we bring the 1.1 next to π so the angle is easier to comprehend. Now we do the same for
the angle of positive π. These two expressions end up in different places on the Complex plane
and escape our initial constrained range. One overshoots a bit into the positive Imaginary
side, and one into the negative. Neither of those two numbers are Real, they are both detached from
the Real line. If we were drawing the graph of the function in an ordinary way, these values, and
generally all the values for negative inputs of x, would simply not be shown. But we’re not
drawing the graph of the function. We are instead, remember, showing numbers of the form z = x + f(x)i
on the Complex plane. And these numbers exist for any input of x we can think of,
at least for functions we’ve considered so far. In the specific specific case of our function, this
results in lines on the negative side slightly deflected from each other. All right, enough
background. Let’s fill in some more curves and slowly crank the power the rest of the way up to 2.
Okay let’s look at that again, rolling back to 1… and again to 2. We can see the results
for negative x values rejoin each other as we reach the next integer power,
and in doing so, wrapping the span between the negative and positive sides over itself to
form the focal point. Let’s push this model a bit further. How does x² become x³?
Looks like the focus that already existed survives, and another one forms through the same wrapping back
onto itself motion. But there’s a problem: there is no such thing as a focus for the graph of x³,
let alone two such points. This is where the geometric definition of a focus and the behavior
of our system go their separate ways. But never mind, it looks interesting, so why don’t we crank
the power up some more. Let’s go to x to the fourth power, and then to the fifth. Right. It’s making less and
less sense to think of these points as foci. We get them in areas where the graph of the function
bends, but also elsewhere seemingly unrelated to what the graph is doing. But if we add lower power
terms to the polynomial, so that we express all the possible extrema, the points will travel to align
with them. I increased the range and density of the curves so you can better see these features.
Pause here if you’d like to take a closer look. This phenomenon of curves splitting between integer
powers and joining at them has to do with roots. Any non-integer power is really a combination
of a root and raising to a power, like we’ve seen earlier. Let’s write Real numbers a bit more
explicitly using Euler’s identity. Positive Real numbers can be represented by any angle that is
an even multiple of π, because all of those are on the Real line on the positive side of zero.
2π is one full revolution. Likewise, negative Real numbers can have any angle that is an
odd multiple of π. Multiplying those angles by a specific integer always yields one definitive
location on the circle, no matter what choice of K we make. If we multiply an odd number by 3 [i.e.],
it will always be odd, no matter which odd number we chose. And if we multiply an even number by any
integer at all, it will stay even, reflecting the fact that positive numbers stay positive under any
integer power. We have multiple representations of the same numbers, but we get the same result
under exponentiation with integers. However if we multiply those different angle representations
by a fraction, we get different results, the most familiar of which is the case of 1/2, the square root.
The 1/2 will make the period cycle just one π, which means it will sometimes be even and
positive and sometimes odd and negative. We get the two roots of 1, and the same happens for
any positive number. There’s quite a lot of nuance here, and we’ll explore it by driving our graphing
method further down, below the first power. Let’s again first just go down a notch to 0.9. Here we
see a similar separation happening like before, but this time, the lines are not crossing over
each other. They are instead opening up a gap between them. Let’s follow this down to 0.5. 1/2.
The square root, that gives us half of a parabola turned on its side. The lines for negative values
of x got compressed onto the negative side of the Real axis, with a bit sticking out on the positive side.
And that bit is exactly touching the focus of this half drawn sideways parabola. The proper focus
is back. How about we add the negated version of this function in as well, and complete the picture?
Right the parabola is now fully reconstructed, along with the same shape of curves we got for
x squared, and we also got this extra bit that looks like the handle of a pitchfork. Of course, this is
not the actual graph of the function you see on the screen. We’re doing this z-value shenanigans
and cherry-picking which parts of it to show. I’m going to leave it up to you to ponder on why we get thepPitchfork handle here. Meanwhile, we continue our descent down the powers, and things
get even more interesting at our next stop: 1/3 or the cube rout. Just doing what we’ve done before,
we get to this picture. But there’s something off here: this function does actually have Real outputs
for negative values of x. If we drew it the regular way, it would look like this: a perfect and complete
inverse of x cubed. That’s because within the Reals, there is only one cube root of a number, and it’s
always the same sign as the number itself. So why do we get this other picture instead? We have to
look at Complex exponentiation again to find that out. To keep it simple, we’ll again just consider
the general positive and negative cases for the number 1, expressed using Euler’s Identity.
It works the same for other magnitudes. First we have the positive case e the power of 2kπ. We raise that
to the 1/3 power, which gives us 2/3 K π. As we iterate K, we get the following angles; and then
they repeat. Only one of them, this one, is a Real number, and it’s positive. For the negative case,
we’ll get 2/3 K + 1/3, all times π. As we iterate K, that will give us these three angles, of which
again, only one is Real, this time negative. The way we draw our graph, the positive side gets
represented by an angle of zero, that stays zero, so we get positive outputs for positive inputs and
everything just looks normal. But on the negative side, we’re coming in with angles of -π and π,
which give us the positive and negative 1/3π angles, the two Complex ones. That hints at there being
two more versions of the graph we could draw to complete the picture, just like we had an extra one to complete the square root sideways parabola. We can get these missing two by multiplying the
function with a unit turned 1/3 of the way around a circle, like this. And then one more time, like this.
So let’s draw them, one by one separately at first. This is the one we already saw, and then we
get this one, which completes the Real valued graph and gives us the continuation of the two negative
sides. But notice how the curves for Complex values just end up in a vacuum, not connecting to any of
the thicker lines. Their new home has been defined by the previous stage: here’s both of them together.
Finally, we get this one, that for some reason just contains the funny separated lines without
any part of the Real graph. These are in fact the solutions we’ve already seen for Complex values of f(x),
they just all appear in the opposite order. And here’s all three together. I know, this looks messy.
There’s too much going on in this picture. But there’s an amazing Elegance hiding behind it
First let’s just notice that we’ve again fully reconstructed the shape of the colored curves for x³
And now let’s just focus on the thicker line depicting results for the Real valued inputs.
What we’re actually looking at are three copies of the x³ shape pressed onto the Complex plane
in a manner similar to what flowers look like when they’re physically pressed onto a page of a book. To show you what I mean by that, we’ll briefly change perspective. Let me hide the graphs
and we’ll go back to the familiar coordinate system where the horizontal axis represents x,
and the vertical f(x). Now we slightly upgrade it: we’ll declare the vertical axis to be just the
Real component of f(x), and add another axis, going directly towards you, for the Imaginary component of f(x).
We can get away with using three dimensions here because we’re only considering
Real valued inputs for x, so it’s a mapping of one dimension to two. And now we draw our three graphs again.
This is what they look like in the space we’ve created. We can
look at the whole resulting shape as three cubic curves. Turning the coordinate system around to face straight down the x axis, we reveal the three
planes these curves occupy. We see that these planes are offset from each other by a third
of the way around a circle. Note that we were measuring the angles going from the positive
side of one curve to the positive side of another. Now when we look at our simplified model again,
we have a better idea what the shapes we’re seeing actually represent. We’ve just compressed these
rotations into a visual deformation, a sort of fake perspective in two dimensions. We’re not
seeing the true picture, but there’s an amazing amount of symmetry preserved. These three curves,
when inverted, represent graphs of three regular cubic functions, and all three of these functions,
when run through the same z expression we’ve used this whole time, yield the exact same shape,
with the quasi-focus points appearing in the same places. So we’ve made a long pit stop at
the third root, but it’s time to move on. We’re going to put multivaluedness behind us again to
Down to and Around Zero
have a simpler image to look at, and make our way towards the next interesting power: zero. Of course,
that’s just a straight line through 1. Let’s look at the journey between the power of 1 and
zero again: two simple straight lines separated by the thick woods of multivalued roots inbetween them.
And as we approach the power of zero, we can see a discontinuity form: the dreaded zero
the power of zero. Going even just the tiniest bit under zero power results in values approaching
the vertical axis exploding to Infinity. So we have this situation right over the power of zero,
and this right under it. There is something remarkable happening at the singularity when
both values are exactly zero, and to understand it, we’ll consider the limits as we approach it from
different directions. The most commonly considered ones are x^0, our function itself,
as x goes to zero from the positive and negative sides, which both work out unsurprisingly to 1.
The same happens if we simultaneously approach zero in both base and exponent. The negative side
limit only exists if we allow approaching from Complex numbers, but it’s there. If we fix the
The Big Bang
base to zero and approach the zero-th power from the positive side, we also unsurprisingly get zero.
But if we plug any negative number in the exponent, forget the limit, we get this peculiar
mathematical object called complex Infinity. Its formally used symbol is just the regular Infinity
we’re used to, but I find this confusing because it can easily be mistaken for positive Infinity.
Wolfram Alpha uses a notation with a tilde over it to make it more distinct and I will adopt that.
So what is this beast? To understand it, we need to consider how zero works in the Complex world.
In the Reals, zero is the only number that isn’t either positive or negative.
And in the Complex world, since there are infinitely many directions, this ambiguity encompasses all of them.
Zero is basically a circle of radius zero in the Complex plane, its direction is undetermined. Which
in some sense means we need to consider them all when operating on it. Raising this number
to any negative power turns it into division by zero, which is commonly left undefined, but within
a special set called the Extended Complex Numbers, that is, Complex numbers with this omnidirectional
Infinity added to it, this operation is actually allowed since it yields a single definitive answer.
That means that our Singularity goes through a kind of Big Bang; the infinitely small circle of
zero instantly becomes an infinitely large circle, a place where all Complex numbers, regardless of
the direction they were headed out in, will meet. Keep that in mind as we travel down to the -1 power.
At the end of this journey, we get a familiar function, x to the -1 power, better known as 1/x,
which has this property of looping back from Infinity around the x value of zero. Well now
you know where Pac-Man went and where he’s coming back from. Along the way, we’re passing multivalued
roots, just like we did on the positive side, and finally we form the two foci of the hyperbola
in a similar way like the single focus showed up for the parabola. There’s a lot more to say
about these phenomena, but I’m going to leave that for another time. This inverted version of
the behavior we’ve seen on the positive side continues on down to the lower powers, like this.
I suspect all this stopping and restarting we’ve done might have felt a bit frustrating
so what I want to do now is to just let the animation run from -4 where we currently stand, back up to 5, so you can see the whole story come together. And since we’re
done explaining, we’ll crank up the range and resolution to make it more pretty. Here we go:
I hope you liked this journey! If you’d like to play around with this model,
I’ve included links to Desmos graphs where you can fiddle around with the parameters to get a feeling for how this model behaves. Links in the description. As a bonus,
let me show you some other notable examples visualized this way. The exponential function;
x to its own power
and back over constant 1 to my favorite: Gaussian function, the normal distribution.
And let’s enhance it a bit. and then we go back over the exponential function, and we draw e^ix.
Which we remember to be cos x + i sin x, so if we just take out the i-sine, we get the cosine on its own.
Thank you so much for watching. I hope you enjoyed and learned something new.
I’m curious to know what you think of the approach taken here, what questions it inspires, so please leave a comment and let me know. Here are some other videos
of mine, and consider subscribing if you haven’t already. Have a lovely day. Bye!