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Intro
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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?
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And what’s the deal with the Pac-Man like behavior around division by zero? It made no sense to me
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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
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range of these functions to Complex numbers does give us a much deeper insight. But we no longer
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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
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cocooned caterpillar turning into a butterfly. I was never quite satisfied with that. So today’s video
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will be taking a somewhat unorthodox approach to indulge the intuition of my younger self.
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We’ll introduce complex arithmetic as it’s needed, and draw almost everything in glorious
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uncomplicated 2D. The goal is to watch the wings of the butterfly form and grow. I hope you enjoy!
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Let’s first just see what happens if we start with f(x) = x, the simplest possible
The Messy Powers
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linear graph. Make the first power of x explicit, and vary that power up to 2 in discrete steps.
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The negative side seems to be jumping up and down and disappearing in between. Let’s look at
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a few specific examples. For x to the power of 1.5, we only get a graph in the first quadrant, where
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both x and f(x) are positive. Both the domain and range of this function are Real numbers
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greater than or equal to zero. This is because x to the power of 1.5 is actually x to 3/2, which is
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a square root of x cubed. For negative values of x, x cubed will be negative and when
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that negative ends up under a square root, there won’t be any real valued answers. So right off the bat,
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half of our shape is missing. In this case we can still recover it, because it’s just the other
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square root. There are always two. We can get both by treating our expression as an equation
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instead of as a function. We’ll write y = √x³ and then square both sides. That gives us y² = x³
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and the graph looks like this. It’s just the piece we’ve already had, reflected about the s axis.
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Let’s just take that as it is and move on. What about x^1.4? Again, we make sense of this by
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turning the exponent into a fraction. In this case 14/10, that simplifies to 7/5, which makes f(x)
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the fifth root of x to the 7th power. The power under the root is odd, so it can yield a negative
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but that’s okay since the degree of root is also odd and we just get negative results out. We also
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don’t have to deal with multiple roots since there is only one odd degree root of a number in the Reals.
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This shape does look like a slightly bent line, an intermediary step on the way to a parabola.
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But the left side doesn’t look like it has any intention of bending upwards. Let’s keep exploring.
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What happens at x^1.6? That’s 16/10 or 8/5, so same story with the odd degree root; negatives
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are allowed. But this time it doesn’t really matter, since under the root we are raising all numbers
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to an even power, 8, making them always positive. That gives us a shape similar to a parabola, but it
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came out of the blue without following any pattern from the shapes we’ve seen before. So there isn’t
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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
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irrational power. That’s a whole other can of worms we won’t be opening today. Why is this so messy?
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And why is all the mess happening when x takes on negative values? It turns out that negativeness
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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
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the top right corner, and it’s also available in the description. Raising negative numbers to
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non-integer powers can have the effect of “breaking down” negativeness and taking numbers away from the
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Real line altogether. Let’s do a quick rundown of what these so-called Complex numbers are and how
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they work. All the Real numbers can be arranged in an order on a line relative to zero. We get
About Complex Numbers
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Complex numbers when we define another direction for numbers orthogonal to the Reals. This forms a
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two-dimensional plane, and specific complex numbers are points on this plane. All the Reals are
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of course included, but there’s also all these other numbers above and below the Real line. Numbers that
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lie specifically on this new vertical line are called Imaginary, and they are the same as Reals in
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every way except for the direction in which they are spanning, just like negatives are equivalent
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to the positive, except for the direction. And just like a unit of movement for positive Reals is 1
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and for negatives -1, imaginary numbers also have their own version of the number 1, called i,
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that happens to point at a right angle away from the Real line. So any number on the Complex plane
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can be expressed as a combination of Real and Imaginary, like x and y coordinates of a point
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on a regular coordinate system. And we express this combination by adding them together. An example of
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a complex number would be 2 + 3i, and you can think of it as walking away from zero, first by
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2 in the positive direction, and then by 3 more in the Imaginary direction. This also works
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as you would expect in the other quadrants. What can we do with this? Within the regular Reals, we
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have this concept of absolute value, which gives us the magnitude of a number independent from
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its direction. The same thing exists for Complex numbers, but this time we have a whole circle of
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infinitely many complex numbers that are the same distance away from zero, and therefore all
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share the same absolute value or magnitude. And this is what we we’ll need for our journey today.
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Instead of considering just positive and negative numbers, we’ll be plugging in the entire range of
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directions for each magnitude and tracking what happens to them as we vary the power we raise x to
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in our expression. By doing that, we are extending the domain and range of our functions to Complex
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numbers. Two coordinates go in, two coordinates come out. This is not as simple as drawing a line.
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Basically all the points on the plane map to some other points, also on on a plane, and we can’t plot
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them against each other because we can’t draw in four dimensions. This is where the unorthodox part
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comes in. We’re going to compress the behavior of the Complex valued function onto a single Complex plane.
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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
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into the Complex plane. We do this by redefining it as all Complex number z of the form z = x + f(x)i,
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so actually x + x²i, and for now we’ll just use Real inputs for x. This keeps the shape exactly
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the same. Only now we have the advantage that we can also plug a Complex number in for x and see
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where the value of z we get ends up. Let’s first see what this looks like, and then we can talk
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about what’s going on behind the scene. First we’ll take all Complex numbers of magnitude 1. We’ll
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start taking them from the Real value of -1 here, going around through the negative Imaginary
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half of the circle to positive 1, and then back to -1 around through the positive Imaginary half.
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All right, now we draw the z values in the same order. Watch the bigger Complex plane on the
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right. The curve we get starts at the regular point of the parabola
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for x = -1, travels to the regular point for positive 1,
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and comes back to x = -1 another way. Let’s do the same for x of magnitude 1/2.
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And now we zoom in a bit, and fill in magnitudes between 0 and 1.
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You’ll notice that for magnitudes of 1 and 1/2, which were drawn twice, the curves appear a
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bit brighter than the rest. In this visualization, curves drawn over each other enhance brightness.
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Let me remove them and dim the axes so we can observe where else this is happening. We have
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a bright spot right here, and it appears exactly where the curve for magnitude 1/2 formed a kink.
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This is the focus of the parabola. We’ll mark it with a white dot. Now remember, we’re getting this
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by plugging the function into our expression for z shown on the left and graphing the values of
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z on the complex plane for chosen ranges of x. The main thicker shape that coincides with the
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graph of the parabola we got from Real values of x, and the colored curves from plugging in a
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series of Complex values of the same magnitude, ranging from 0 to 1. With this established, we
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can play around with the function. I’m going to extend it to a general second degree polynomial.
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By changing the B and C terms, we can move the parabola, so let’s vary them just slightly to see
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what happens. Don’t worry about specific values we’re using for B and C, this doesn’t really matter.
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All right, the shape formed by the colored curves is starting to turn, but the bright spot
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we found still follows the focus. Flipping the parabola by changing the sign of the A
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parameter also doesn’t disturb that property. This gives us an opportunity to vary the A parameter
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gradually back from -1 up to 1 and watch the parabola transform between downward and upward
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facing. In between it will hit zero, making the function linear, so we’ll get our first chance
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to observe a parabola transforming into a line and vice versa. Let’s take the whole screen for
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that. Okay, the focus gets launched out, the curves re-form into concentric circles, and then the focus
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reappears from the other side, reassembling into a mirrored image of the whole setup.
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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
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the range, like this, they will follow along longer. If we were to extend them to Infinity, they would
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keep following along. The focus travels along this straight line, aligned with this feature
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that looks like a ray, formed where the curves intersect themselves. We’re seeing a hidden layer
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of the behavior of the parabola pop out. And now that we’re a bit more familiar with this
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way of graphing and how it behaves, we’re ready to go back to the experiment we started at the
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beginning of the video: varying the power we raise a single instance of x to. We’ll start with a simple
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f(x) = x, and for this regular line, the curves formed from Complex values of x would just be
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concentric circles, like we’ve seen before. Let’s remove them and take the power up a notch to 1.1
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Now we have two lines on the negative side. What’s going on here? Let’s draw in just one curve
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for x of magnitude 1, again starting at -1, and going all the way back around to -1.
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We started with -1 and ended with -1, but somehow our curve didn’t connect back to itself.
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How is this possible? I’m sure you’ve heard of the famous Euler’s Identity, the one that states
Overshooting with Euler
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that e^iπ = -1. This is actually a special case of representing a Complex number
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in terms of magnitude and an angle. The full form would look like this. So a Complex number z given
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by this expression would be M distance away from the origin, at a Theta angle in respect to the Real
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axis, which means its Real and Imaginary parts could be obtained trigonometrically like this.
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The famous special form of Euler’s Identity is just a Complex number of magnitude 1 and an angle of
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π radians: halfway around around a circle. And here’s a catch: we can hit that same spot where
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the number -1 lies with an angle of π, but also with an angle of 3π, or any odd number of
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increments of π. And we can go the other way around too. Remember, I’ve shown you that when
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we pick the Complex values of x we put into our function, we start at -1 and travel this
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way around to get back to -1 again. The way these values are generated behind the scene
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is that we’re ranging the angle from -π to π. The round bracket on the left just means that
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we’re only including values strictly larger than -π, because this ensures that every
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point is represented only once. Why this range in particular? It follows certain conventions in
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Complex arithmetic, but long story short, you could say it’s the range that gives us the nearest of
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the infinitely many representations of a Complex number we can get in this form. We reach all angles
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with an overall minimum amount of travel from the angle of zero. Now let’s look at raising a Complex
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number in this form to a power. The exponent is going to distribute. It’s going to affect the
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magnitude normally, like it would any Real number, and it’s going to multiply the angle. Now we can
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finally examine the case of our function. We were raising -1 to the 1.1 power. We’ll do it first
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as the angle of -π. Okay, the magnitude we can get rid of, because 1 to the power of anything is just 1,
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and we bring the 1.1 next to π so the angle is easier to comprehend. Now we do the same for
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the angle of positive π. These two expressions end up in different places on the Complex plane
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and escape our initial constrained range. One overshoots a bit into the positive Imaginary
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side, and one into the negative. Neither of those two numbers are Real, they are both detached from
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the Real line. If we were drawing the graph of the function in an ordinary way, these values, and
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generally all the values for negative inputs of x, would simply not be shown. But we’re not
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drawing the graph of the function. We are instead, remember, showing numbers of the form z = x + f(x)i
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on the Complex plane. And these numbers exist for any input of x we can think of,
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at least for functions we’ve considered so far. In the specific specific case of our function, this
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results in lines on the negative side slightly deflected from each other. All right, enough
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background. Let’s fill in some more curves and slowly crank the power the rest of the way up to 2.
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Okay let’s look at that again, rolling back to 1… and again to 2. We can see the results
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for negative x values rejoin each other as we reach the next integer power,
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and in doing so, wrapping the span between the negative and positive sides over itself to
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form the focal point. Let’s push this model a bit further. How does x² become x³?
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Looks like the focus that already existed survives, and another one forms through the same wrapping back
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onto itself motion. But there’s a problem: there is no such thing as a focus for the graph of x³,
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let alone two such points. This is where the geometric definition of a focus and the behavior
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of our system go their separate ways. But never mind, it looks interesting, so why don’t we crank
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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
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less sense to think of these points as foci. We get them in areas where the graph of the function
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bends, but also elsewhere seemingly unrelated to what the graph is doing. But if we add lower power
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terms to the polynomial, so that we express all the possible extrema, the points will travel to align
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with them. I increased the range and density of the curves so you can better see these features.
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Pause here if you’d like to take a closer look. This phenomenon of curves splitting between integer
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powers and joining at them has to do with roots. Any non-integer power is really a combination
Roots
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of a root and raising to a power, like we’ve seen earlier. Let’s write Real numbers a bit more
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explicitly using Euler’s identity. Positive Real numbers can be represented by any angle that is
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an even multiple of π, because all of those are on the Real line on the positive side of zero.
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2π is one full revolution. Likewise, negative Real numbers can have any angle that is an
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odd multiple of π. Multiplying those angles by a specific integer always yields one definitive
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location on the circle, no matter what choice of K we make. If we multiply an odd number by 3 [i.e.],
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it will always be odd, no matter which odd number we chose. And if we multiply an even number by any
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integer at all, it will stay even, reflecting the fact that positive numbers stay positive under any
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integer power. We have multiple representations of the same numbers, but we get the same result
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under exponentiation with integers. However if we multiply those different angle representations
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by a fraction, we get different results, the most familiar of which is the case of 1/2, the square root.
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The 1/2 will make the period cycle just one π, which means it will sometimes be even and
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positive and sometimes odd and negative. We get the two roots of 1, and the same happens for
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any positive number. There’s quite a lot of nuance here, and we’ll explore it by driving our graphing
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method further down, below the first power. Let’s again first just go down a notch to 0.9. Here we
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see a similar separation happening like before, but this time, the lines are not crossing over
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each other. They are instead opening up a gap between them. Let’s follow this down to 0.5. 1/2.
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The square root, that gives us half of a parabola turned on its side. The lines for negative values
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of x got compressed onto the negative side of the Real axis, with a bit sticking out on the positive side.
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And that bit is exactly touching the focus of this half drawn sideways parabola. The proper focus
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is back. How about we add the negated version of this function in as well, and complete the picture?
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Right the parabola is now fully reconstructed, along with the same shape of curves we got for
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x squared, and we also got this extra bit that looks like the handle of a pitchfork. Of course, this is
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not the actual graph of the function you see on the screen. We’re doing this z-value shenanigans
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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
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get even more interesting at our next stop: 1/3 or the cube rout. Just doing what we’ve done before,
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we get to this picture. But there’s something off here: this function does actually have Real outputs
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for negative values of x. If we drew it the regular way, it would look like this: a perfect and complete
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inverse of x cubed. That’s because within the Reals, there is only one cube root of a number, and it’s
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always the same sign as the number itself. So why do we get this other picture instead? We have to
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look at Complex exponentiation again to find that out. To keep it simple, we’ll again just consider
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the general positive and negative cases for the number 1, expressed using Euler’s Identity.
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It works the same for other magnitudes. First we have the positive case e the power of 2kπ. We raise that
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to the 1/3 power, which gives us 2/3 K π. As we iterate K, we get the following angles; and then
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they repeat. Only one of them, this one, is a Real number, and it’s positive. For the negative case,
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we’ll get 2/3 K + 1/3, all times π. As we iterate K, that will give us these three angles, of which
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again, only one is Real, this time negative. The way we draw our graph, the positive side gets
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represented by an angle of zero, that stays zero, so we get positive outputs for positive inputs and
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everything just looks normal. But on the negative side, we’re coming in with angles of -π and π,
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which give us the positive and negative 1/3π angles, the two Complex ones. That hints at there being
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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
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function with a unit turned 1/3 of the way around a circle, like this. And then one more time, like this.
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So let’s draw them, one by one separately at first. This is the one we already saw, and then we
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get this one, which completes the Real valued graph and gives us the continuation of the two negative
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sides. But notice how the curves for Complex values just end up in a vacuum, not connecting to any of
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the thicker lines. Their new home has been defined by the previous stage: here’s both of them together.
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Finally, we get this one, that for some reason just contains the funny separated lines without
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any part of the Real graph. These are in fact the solutions we’ve already seen for Complex values of f(x),
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they just all appear in the opposite order. And here’s all three together. I know, this looks messy.
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There’s too much going on in this picture. But there’s an amazing Elegance hiding behind it
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First let’s just notice that we’ve again fully reconstructed the shape of the colored curves for x³
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And now let’s just focus on the thicker line depicting results for the Real valued inputs.
Flower Pressing
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What we’re actually looking at are three copies of the x³ shape pressed onto the Complex plane
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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
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and we’ll go back to the familiar coordinate system where the horizontal axis represents x,
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and the vertical f(x). Now we slightly upgrade it: we’ll declare the vertical axis to be just the
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Real component of f(x), and add another axis, going directly towards you, for the Imaginary component of f(x).
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We can get away with using three dimensions here because we’re only considering
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Real valued inputs for x, so it’s a mapping of one dimension to two. And now we draw our three graphs again.
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This is what they look like in the space we’ve created. We can
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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
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planes these curves occupy. We see that these planes are offset from each other by a third
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of the way around a circle. Note that we were measuring the angles going from the positive
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side of one curve to the positive side of another. Now when we look at our simplified model again,
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we have a better idea what the shapes we’re seeing actually represent. We’ve just compressed these
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rotations into a visual deformation, a sort of fake perspective in two dimensions. We’re not
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seeing the true picture, but there’s an amazing amount of symmetry preserved. These three curves,
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when inverted, represent graphs of three regular cubic functions, and all three of these functions,
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when run through the same z expression we’ve used this whole time, yield the exact same shape,
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with the quasi-focus points appearing in the same places. So we’ve made a long pit stop at
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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
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have a simpler image to look at, and make our way towards the next interesting power: zero. Of course,
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that’s just a straight line through 1. Let’s look at the journey between the power of 1 and
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zero again: two simple straight lines separated by the thick woods of multivalued roots inbetween them.
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And as we approach the power of zero, we can see a discontinuity form: the dreaded zero
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the power of zero. Going even just the tiniest bit under zero power results in values approaching
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the vertical axis exploding to Infinity. So we have this situation right over the power of zero,
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and this right under it. There is something remarkable happening at the singularity when
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both values are exactly zero, and to understand it, we’ll consider the limits as we approach it from
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different directions. The most commonly considered ones are x^0, our function itself,
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as x goes to zero from the positive and negative sides, which both work out unsurprisingly to 1.
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The same happens if we simultaneously approach zero in both base and exponent. The negative side
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limit only exists if we allow approaching from Complex numbers, but it’s there. If we fix the
The Big Bang
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base to zero and approach the zero-th power from the positive side, we also unsurprisingly get zero.
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But if we plug any negative number in the exponent, forget the limit, we get this peculiar
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mathematical object called complex Infinity. Its formally used symbol is just the regular Infinity
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we’re used to, but I find this confusing because it can easily be mistaken for positive Infinity.
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Wolfram Alpha uses a notation with a tilde over it to make it more distinct and I will adopt that.
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So what is this beast? To understand it, we need to consider how zero works in the Complex world.
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In the Reals, zero is the only number that isn’t either positive or negative.
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And in the Complex world, since there are infinitely many directions, this ambiguity encompasses all of them.
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Zero is basically a circle of radius zero in the Complex plane, its direction is undetermined. Which
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in some sense means we need to consider them all when operating on it. Raising this number
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to any negative power turns it into division by zero, which is commonly left undefined, but within
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a special set called the Extended Complex Numbers, that is, Complex numbers with this omnidirectional
28:21
Infinity added to it, this operation is actually allowed since it yields a single definitive answer.
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That means that our Singularity goes through a kind of Big Bang; the infinitely small circle of
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zero instantly becomes an infinitely large circle, a place where all Complex numbers, regardless of
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the direction they were headed out in, will meet. Keep that in mind as we travel down to the -1 power.
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At the end of this journey, we get a familiar function, x to the -1 power, better known as 1/x,
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which has this property of looping back from Infinity around the x value of zero. Well now
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you know where Pac-Man went and where he’s coming back from. Along the way, we’re passing multivalued
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roots, just like we did on the positive side, and finally we form the two foci of the hyperbola
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in a similar way like the single focus showed up for the parabola. There’s a lot more to say
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about these phenomena, but I’m going to leave that for another time. This inverted version of
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the behavior we’ve seen on the positive side continues on down to the lower powers, like this.
Enjoy!
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I suspect all this stopping and restarting we’ve done might have felt a bit frustrating
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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
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done explaining, we’ll crank up the range and resolution to make it more pretty. Here we go:
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I hope you liked this journey! If you’d like to play around with this model,
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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,
Bonus Functions
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let me show you some other notable examples visualized this way. The exponential function;
31:05
x to its own power
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and back over constant 1 to my favorite: Gaussian function, the normal distribution.
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And let’s enhance it a bit. and then we go back over the exponential function, and we draw e^ix.
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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.
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