So the story goes: the tortoise challenges Achilles to a footrace.

"If you give me a head start," it says, "any start at all, you can never win.".
Achilles laughs and decides to be a good sport: he'll only run twice as fast as the tortoise.

The tortoise explains: "If you want to pass me, first you have to move to where I am. By the time you get there, I'll have walked ahead a little bit."

"While you cross the next distance, I will move yet again. No matter how many times you try to catch up, I'll always be some small distance ahead. Therefor, you cannot beat me."

Achilles realizes that talking tortoises are not a sign of positive mental health, so he decides to find a wall to run into instead. It will either confirm the theory, or end the pain.

See, the race is actually unnecessary, because the problem remains the same.
In order to reach the wall, Achilles first has to cross half the way there.

Then he has to go half that distance again, and again. No matter how many times he repeats this, there will always be some distance left. So if Achilles can't cross this distance in a finite amount of steps, why is he wearing that stupid helmet?

The distance travelled forms a never ending sequence of expanding sums.
We have to examine the entire sequence, rather than individual numbers in it.

By definition, the distance travelled and distance to the wall always add up to $ 1 $. So one simple way to resolve this conundrum is to say: Well yes, it's going to take you infinitely long to glue all those pieces together, but only because you already spent an infinite amount of time chopping them up!
But that's not a very mathematically satisfying answer. Let's try something else.

The distance to the wall is always equal to the last step taken. We know that each step is half as long as the previous one, starting with $ \frac{1}{2} $. Therefor, the distance to the wall must decrease exponentially: $ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, … $, getting closer to zero with every step.

But why can we say that this gap effectively closes to zero after 'infinity steps'? The number that we're building up is $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + … \,$

We know our sum will never exceed $ 1 $, as there is only $ 1 $ unit of distance being divided. This means $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + … \leq 1 $, which eliminates every number past the surface of the wall—but not the surface itself.

Suppose we presume $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + … < 1 $ and hence that this number lies some tiny distance in front of the wall.

Well in that case, all we need to do is zoom in far enough, and we'll see our sequence jump past it after a certain finite number of steps.

If we try to move it closer to the wall, the same thing happens. This number simply cannot be less than $ 1 $. Therefor $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + … \geq 1 $

The only place $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + … \, $ can be is exactly $ 0 $ units away from $ 1 $. If two numbers have zero distance between them, then they are equal.

What we've actually done here is applied the principle of limits: we've defined a procedure of steps that lets us narrow down the interval where the infinite sum might be. The lower bound is the sequence of sums itself: it only increases towards $ 1 $, never decreases. For the upper bound, we established no sum could exceed $ 1 $. Therefor the interval must shrink to nothing, and the sequence converges.

The purpose of a limit is then to act as a supercharged fast-forward button. It lets us avoid the infinite amount of work required to complete sums like $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + … $ and simply skip to the end. To do so, we have to step back, spot the pattern, and pin down where it ends. So limits allow us to literally reach the unreachable. But in fact, you already knew that.

As soon as you learned to divide, you found $ 2 \div 3 = 0.666… = 0.6 + 0.06 + 0.006 + …\hspace{2pt} $
Even in primary school the opportunity to examine infinity is there. Rather than tackle it head on, it's simply noted and filed. Eight years later it's regurgitated in the form of cryptic epsilon-delta definitions.

But then there's those pesky consequences again. By allowing the idea of infinity, we can invent an entire zoo of paradoxical things. For example, imagine a lamp that's switched on ($1$) and off ($0$) at intervals that decrease by a factor of two: on for $ \frac{1}{2} $ second, off for $ \frac{1}{4} s $, on for $ \frac{1}{8} s $, off for $ \frac{1}{16} s $, …
After $ 1\,s $, when the switch has been flipped an infinite amount of times, is the lamp on or off?

Another way to put this is that the lamp's state at $ 1\,s $ is the result of the infinite sum $ 1 - 1 + 1 - 1 + … $
Intuitively we might say each pair of $ +1 $ and $ -1 $ should cancel out and make the entire sum equal to $ 0 $.
But we can pair them the other way, leading to $ 1 $ instead. It can't be both.

If we zoom in, it's obvious that no matter how close we get to $ 1\,s $, the lamp's state keeps switching. Therefor it's meaningless to attempt to 'fast forward' to the end, and the limit does not exist. At $ 1\,s $ the lamp is neither on nor off: it's undefined. This infinite sum does not converge.

But actually, we overcomplicated things. Thanks to the power of limits, we can ask a simpler, equivalent question. Given a lamp that switches on and off every second, what is its state at infinity? The answer's the same: it never settles.

Let's imagine the rational numbers.

Actually, hold on. Is this really a line? The integers certainly weren't connected.

Rather than assume anything, we're going to attempt to visualize all the rational numbers. We'll start with the numbers between $ 0 $ and $ 1 $.

Between any two numbers, we can find a new number in between: their average. This leads to $ \frac{1}{2} $.

By repeatedly taking averages, we keep finding new numbers, filling up the interval.

If we separate out every step, we get a binary tree.

You can think of this as a map of all the fractions of $ 2^n $. Given any such fraction, say $ \frac{13}{32} = \frac{13}{2^5} $, there is a unique path of lefts and rights that leads directly to it. At least, as long as it lies between $ 0 $ and $ 1 $.

Note that the graph resembles a fractal and that the distance to the top edge is divided in half with every step. But we only ever explore a finite amount of steps. Therefor, we are not taking a limit and we'll never actually touch the edge.

But we can take thirds as well, leading to fractions with a power of $ 3^n $ in their denominator.

As some numbers can be reached in multiple ways, we can eliminate some lines, and end up with this graph, where every number sprouts into a three-way, ternary tree. Again, we have a map that gives us a unique path to any fraction of $ 3^n $ in this range, like $ \frac{11}{27} = \frac{11}{3^3} $.

Because we can do this for any denominator, we can define a way to get to any rational number in a finite amount of steps. Take for example $ \frac{21}{60} $. We decompose its denominator into prime numbers and begin with $ 0 $ and $ 1 $ again.

There is a division of $ 2^2 $, so we do two binary splits. This time, I'm repeating the previously found numbers so you can see the regular divisions more clearly. We get quarters.

The next factor is $ 3 $ so we divide into thirds once. We now have twelfths.

For the last division we chop into fifths and get sixtieths.

$ \frac{21}{60} $ is now the 21st number from the left.

But this means we've found a clear way to visualize all the rational numbers between $ 0 $ and $ 1 $: it's all the numbers we can reach by applying a finite number of binary (2), ternary (3), quinary (5) etc. divisions, for any denominator. So there's always a finite gap between any two rational numbers, even though there are infinitely many of them.

The rational numbers are not continuous. Therefor, it is more accurate to picture them as a set of tick marks than a connected number line.

To find continuity then, we need to revisit one of our earlier trees. We'll pick the binary one.
While every fork goes two ways, we actually have a third choice at every step: we can choose to stop. That's how we get a finite path to a whole fraction of $ 2^n $.

But what if we never stop? We have to apply a limit: we try to spot a pattern and try to fast-forward it. Note that by halving each step vertically on the graph, we've actually linearized each approach into a straight line which ends. Now we can take limits visually just by intersecting lines with the top edge.

Right away we can spot two convergent limits: by always choosing either the left or the right branch, we end up at respectively $ 0 $ and $ 1 $.

These two sequences both converge to $ \frac{1}{2} $. It seems that 'at infinity steps', the graph meets up with itself in the middle.

But the graph is now a true fractal. So the same convergence can be found here. In fact, the graph meets up with itself anywhere there is a multiple of $ \frac{1}{2^n} $.

That's pretty neat: now we can eliminate the option of stopping altogether. Instead of ending at $ \frac{5}{16} $, we can simply take one additional step in either direction, followed by infinitely many opposite steps. Now we're only considering paths that are infinitely long.

But if this graph only leads to fractions of $ 2^n $, then there must be gaps between them. In the limit, the distance between any two adjacent numbers in the graph shrinks down to exactly $ 0 $, which suggests there are no gaps. This infinite version of the binary tree must lead to a lot more numbers than we might think.
Suppose we take a path of alternating left and right steps, and extend it forever. Where do we end up?

We can apply the same principle of an upper and lower bound, but now we're approaching from both sides at once. Thanks to our linearization trick, the entire sequence fits snugly inside a triangle.

If we zoom into the convergence at infinity, we actually end up at $ \class{orangered}{\frac{2}{3}} $.
Somehow we've managed to coax a fraction of $ 3 $ out of a perfectly regular binary tree.

If we alternate two lefts with one right, we can end up at $ \class{orangered}{\frac{4}{7}} $. This is remarkable: when we tried to visualize all the rational numbers by combining all kinds of divisions, we were overthinking it. We only needed to take binary divisions and repeat them infinitely with a limit.

Every single rational number can then be found by taking a finite amount of steps to get to a certain point, and then settling into a repeating pattern of lefts and/or rights all the way to infinity.

If we can find numbers between $ 0 $ and $ 1 $ this way, we can apply the exact same principle to the range $ 1 $ to $ 2 $. So we can connect two of these graphs into a single graph with its tip at $ 1 $.

But we can repeat it as much as we like. The full graph is not just infinitely divided, but infinitely big, in that no finite box can contain it. That means it leads to every single positive rational number. We can start anywhere we like. Is your mind blown yet?

No? Ok. But if this works for positives, we can build a similar graph for the negatives just by mirroring it. So we now have a map of the entire rational number set. All we need to do is take infinite paths that settle into a repeating pattern from either a positive or a negative starting point. When we do, we find every such path leads to a rational number.
So any rational number can be found by taking an infinite stroll on one of two infinite binary trees.

Wait, did I say two infinite trees? Sorry, I meant one infinitely big tree.
See, if we repeatedly scale up a fractal binary tree and apply a limit to that, we end up with almost exactly the same thing. Only this time, the two downward diagonals always eventually fold back towards $ 0 $. This creates a path of infinity + 1 steps downward. While that might not be very practical, it suggests you can ride out to the restaurant at the end of the universe, have dinner, and take a single step to get back home.

Is it math, or visual poetry? It's time to bring this fellatio of the mind to its inevitable climax.

You may wonder, if this map is so amazing, how did we ever do without?
Let's label our branches. If we go left, we call it $ 0 $. If we go right, we call it $ 1 $.

We can then identify any number by writing out the infinite path that leads there as a sequence of ones and zeroes—bits.

But you already knew that.

See we've just rediscovered the binary number system. We're so used to numbers in decimal, base 10, we didn't notice. Yet we all learned that rational numbers consist of digits that settle into a repeating sequence, a repeating pattern of turns. Disallowing finite paths works the same, even in decimal: the number $ 0.95 $ can be written as $\, 0.94999…\, $, i.e. take one final step in one direction, followed by infinitely many steps the other way.

When we write down a number digit by digit, we're really following the path to it in a graph like this, dialing the number's … er … number. The rationals aren't shaped like a binary tree, rather, they look like a binary tree when viewed through the lens of binary division. Every infinite binary, ternary, quinary, etc. tree is then a different but complete perspective of the same underlying thing. We don't have the map, we have one of infinitely many maps.

Which means we can show this graph is actually an interdimensional number portal.
See, we already know where the missing numbers are. Irrational numbers like $ π $ form a never-repeating sequence of digits. If we want to reach $ π $, we find it's at the end of an infinite path whose turns do not repeat. By allowing such paths, our map leads us straight to them. Even though it's made out of only one kind of rational number: division by two.

So now we've invented real numbers. How do we visualize this invention? And where does continuity come in? What we need is a procedure that generates such a non-repeating path when taken to the limit. Then we can figure out where the behavior at infinity comes from.

Because the path never settles into a pattern, we can't pin it down with a single neat triangle like before. We try something else. At every step, we can see that the smallest number we can still reach is found by always going left. Similarly, the largest available number is found by always going right. Wherever we go from here, it will be somewhere in this range.

We can set up shrinking intervals by placing such triangles along the path, forming a nested sequence.

What we've actually done is rounded up and down at every step, to find an upper and lower bound with a certain amount of digits. This works in any number base.

Let's examine these intervals by themselves. We can see that due to the binary nature, each interval covers either the left or right side of its ancestor. Because our graph goes on forever, there are infinitely many nested intervals. This tower of $ π $ never ends and never repeats itself, we just squeezed it into a finite space so we could see it better.

If we instead approach a rational number like $ \frac{10}{3} = 3.333…\, $ then the tower starts repeating itself at some point. Note that the intervals don't slide smoothly. Each can only be in one of two places relative to its ancestor.

In order to reach a different rational number, like $ 3.999… = 4 $, we have to establish a different repeating pattern. So we have to rearrange infinitely many levels of the tower all at once, from one configuration to another. This reinforces the notion that rational numbers are not continuous.

If the tower converges to a number, then the top must be infinitely thin, i.e. $ 0 $ units wide. That would suggest it's meaningless to say what the interval at infinity looks like, because it stops existing. Let's try it anyway.

There is only one question to answer: does the interval cover the left side, or the right?

Oddly enough, in this specific case of $ 3.999…\, $ there is an answer. The tower leans to the right. Therefor, the state of the interval is the same all the way up. If we take the limit, it converges and the final interval goes right.

But we can immediately see that we can build a second tower that leans left, which converges on the same number. We could distinguish between the two by writing it as $ 4.000…\, $ In this case the final interval goes left.

If we approach $ 10/3 $, we take a path of alternating left and right steps. The state of the interval at infinity becomes like our paradoxical lamp from before: it has to be both left and right, and therefor it is neither, it's simply undefined.

The same applies to irrational numbers like $ π $. Because the sequence of turns never repeats itself, the interval flips arbitrarily between left and right forever, therefor it is in an undefined state at the end.

But there's another way to look at this.
If the interval converges to the number $ π $, then the two sequences of respectively lower and upper bounds also converge to $ π $ individually.

Remember how we derived our bounds: we rounded down by always taking lefts and rounded up by always taking rights. The shape of the tower depends on the specific path you're taking, not just the number you reach at the end.

That means we're approaching the lower bounds so they all end in $ 0000… \, $ Their towers always lean left.

If we then take the limit of their final intervals as we approach $ π $, that goes left too. Note that this is a double limit: first we find the limit of the intervals of each tower individually, then we take the limit over all the towers as we approach $ π $.

For the same reason, we can think of all the upper bounds as ending in $ 1111 …\, $ Their towers always lean right. When we take the limit of their final intervals and approach $ π $, we find it points right.

But, we could actually just reverse the rounding for the upper and lower bounds, and end up with the exact opposite situation. Therefor it doesn't mean that we've invented a red $ π $ to the left and green $ π $ to the right which are somehow different. $ π $ is $ π $. This only says something about our procedure of building towers. It matters because the towers is how we're trying to reach a real number in the first place.

See, our tower still represents a binary number of infinitely many bits. Every interval can still only be in one of two places. To run along the real number line, we'd have to rearrange infinitely many levels of the tower all at once to create motion. That still does not seem continuous.

We can resolve this if we picture the final interval of each tower as a bit at infinity. If we flip the bit at infinity, we swap between two equivalent ways of reaching a number, so this has no effect on the resulting number.

In doing so, we're actually imagining that every real number is a rational number whose non-repeating head has grown infinitely big. Its repeating tail has been pushed out all the way past infinity. That means we can flip the repeating part of our tower between different configurations without creating any changes in the number it leads to.

That helps a little bit with the intuition: if the tower keeps working all the way up there, it must be continuous at its actual tip, wherever that really is. A continuum is then what happens when the smallest possible step you can take isn't just as small as you want. It's so small that it no longer makes any noticeable difference. While that's not a very mathematical definition, I find it very helpful in trying to imagine how this might work.

Finally, we might wonder how many of each type of number there are.
The natural numbers are countably infinite: there is a procedure of steps which, in the limit, counts all of them.

We can find a similar sequence for the even natural numbers by multiplying each number by two. We can also alternate between a positive and negative sequence to count the integers. The three sequences are all countably infinite, which means they're all equally long.
There are as many even positives as positives. Which is exactly as many as all the integers combined. As counter-intuitive as it is, it is the only consistent answer.

But we can take it one step further: we can find such a sequence for the rational numbers too, by laying out all the fractions on a grid. We can follow diagonals up and down and pass through every single one. If we eliminate duplicates like $ 1 = 2/2 = 3/3 $ and alternate positives and negatives, we can 'count them all'. So there are as many fractions as there are natural numbers. "Deal with it", says Infinity, donning its sunglasses.

The real numbers on the other hand are uncountably infinite: no process can list them all in the limit. The proof is short: suppose we did have a sequence of all the real numbers between $ 0 $ and $ 1 $. We could then build a new number by taking all the bits on the diagonal, and flipping zeroes and ones.
That means this number is different from every listed number in at least one digit, so it's not on the list. But it's also between $ 0 $ and $ 1 $, so it should be. Therefor, the list can't exist.

This even matches our intuitive explanation from earlier. There are so many real numbers, that we had to invent a bit at infinity to count them, and find something that would tick at least once for every real number. Even then we couldn't say whether it was $ 0 $ or $ 1 $ anywhere in particular, because it literally depends on how you approach it.

We're going to go for a drive.

We'll graph speed versus time. We have kilometers per hour vertically, and hours horizontally. We've also got a speedometer—how fast—and an odometer—how far.

Suppose we drive for half an hour at 50 km/h.

We end up driving for 25 km. This is the area of spanned by the two lengths: $ 50 \cdot \frac{1}{2} $, a rectangle.

Now we hit the highway and maintain 120 km/h for the rest of the hour. We go an additional 60 km, the area of the second rectangle, $ 120 \cdot \frac{1}{2} $.
Whenever we multiply two units like speed and time, we can always visualize the result as an area.

Because we crossed 85 km in one hour, this is equivalent to driving at a constant speed of 85 km/h for the duration. The total area is the same.

If this were a race between two different cars, we would see a photo finish. The distance travelled in kilometers is identical at the 1 hour mark. Where they differ is in their speed along the way, with the red car falling behind and then catching up.

The difference is visible in the slope of both paths. The faster the car, the more quickly it accumulates kilometers. If it drove 25 km in half an hour, then its speed was 50 km/h, $ \frac{25}{0.5} $. This is the distance travelled divided by the time it took, vertical divided by horizontal.

Slope is a relative thing. If we shrink the considered time, the distance shrinks along with it, and the resulting speed is the same. What we're really doing is formalizing the concept of a rate of change, of distance over time.

Constant speed means a constant increase in distance. We can directly relate the area being swept out left to right with the accumulated distance by each car. This is clue number 1.

Now suppose the red car starts ahead by 10 km and drives the same speeds.
It will also end up 10 km ahead after 1 hour, its path has simply been shifted by 10 units. The slope is unchanged: it doesn't matter where you are and where you've been, only how fast you're going right now. It's what's called an instantaneous quantity, it describes a situation only in the moment. This is clue number 2.

In order to get ahead, the red car had to drive there. So we can imagine it started earlier, $ \frac{1}{5} $ of an hour, driving for 10 km at the same speed. Again, the equality holds: area swept out equals accumulated distance, we add another $ 50 \cdot \frac{1}{5} $. Constant slope still equals constant speed.

One curve describes how the other changes in the moment, therefor the two quantities are linked somehow. We add up area to go from speed to distance; we find slope to go from distance to speed. We're going to examine this two-way relationship more.

Real cars don't start or stop on a dime, they accelerate and decelerate. So we're going to try more realistic behavior.

Suppose the speed follows a curve. In one hour, the car starts from 0 km/h, accelerates to over 100 km/h and then smoothly decelerates back to standstill. The distance travelled also curves smoothly, from 0 to 60 km, so we've driven 60 km in total.

We can immediately see that at the point where the car was going fastest, the distance was increasing the most. Its slope is steepest at that point. The relationship between the two curves holds.

But actually measuring it is a problem. First, there are no more straight sections to measure the slope on. If we take two points on a curve, the line that connects them doesn't touch the curve, it crosses it at an angle.

Second, we can no longer measure the area by dividing it into rectangles, or any other simple geometric shape. There will always be gaps. We can solve both of these problems with a dash of infinity.

We'll start with area. We have to find an upper and a lower bound again.
We're going to divide the curve into 4 sections.

First, the upper bound. We find the highest value in each section and make a rectangle of that height. This approach is too greedy and overestimates.

The lower bound is similar. We find the smallest value in each interval and make rectangles of that height.
This underestimates and leaves areas uncovered.

If we do 7 divisions instead. We can see that the upper bound has decreased: there is less excess area. The lower bound has increased: the gaps are smaller and more area is covered.

With 10 divisions, it's even better. It seems the upper and lower bounds are approaching each other.

And the same at 13 divisions. If we keep doing this, our slices will get thinner and thinner, and we'll be adding more of them together. If we take a limit, each slice becomes infinitely thin, and there are infinitely many of them. Let's step back and see what that means.

Take for example the sequence of lower bounds.

Because every slice is equally wide, we can glue them together into a single rectangle per step.
Its width $ w $ is the thickness of a single slice, and its height $ h $ is the sum of the heights of the slices.

In the limit, this rectangle becomes both infinitely thin and infinitely tall. This is a tug of war between Zero and Infinity where at first sight, they both seem to win. That's a problem. Luckily, we're not interested in the rectangle itself, but rather its area.

We can change a rectangle's sides without changing its area. We multiply its width by one factor (e.g. $ 2 $), and divide the height by the same amount. The area $ 2w \cdot \frac{h}{2} $ is unchanged. Hence, we can normalize our rectangles to all have the same width, for example $ 1 $.

We can do the same for the upper bounds. We can see that both areas are converging on the same value. This is the true area under the curve, which is neither zero nor infinite. In this tug of war, both parties are equally matched.

Now our sequence looks very different: it's approaching a definite area, sandwiched between red and blue.

If we take the limit, we get the area under our curve.

This way we can find the area under any smooth curve. This process is called integration. The symbol for integration is $ \int_a^b $ where $ a $ and $ b $ are the start and end points you're integrating between. The S-shape stands for our sum, adding up infinitely many pieces.

We can then integrate one curve to make another, by sweeping out area horizontally from a fixed starting point. We move the end point to a time $ T $ and plot the accumulated value along the way. Using limits, we can do this continuously. This takes us from speed to distance travelled. The quantity $ \,\mathrm{d}t\, $ is the infinitesimal width of each slice, an infinitely small amount of time.

Now we just need to figure out the reverse and find slopes. We'll go back to our failed attempt from earlier.

If we shrink the distance we're considering, our slope estimate gets closer to the true value. But if we try to take a limit, we end up dividing $ 0 $ by $ 0 $.

Instead we need to normalize our sequence again so it doesn't vanish.

We only care about slope: the ratio of the two right sides. Which means, if we scale up each triangle, the ratio is unchanged. That just comes down to multiplying both sides by the same number. Again we can scale them all to the exact same width.

Now we've created a limit that does converge to something rather than nothing.

This finite value is the slope at the point we were homing in on. Because we can apply this process at any point on the curve, we can find the exact slope anywhere. This is called finding the derivative or differentiation.

We can also apply this process over an entire curve to generate a new one. So now we know how to go the other way: distance to speed. Mathematically, we are dividing an infinitesimal piece of the distance, the curve $ \,\mathrm{d} \class{slate}{f(t)}\, $, by an infinitesimal slice of time $ \,\mathrm{d} t\, $. Working with infinitesimal formulas is tricky however. There's always an implied limit being taken in order to reach them in the first place.

We can note that if we shift the distance curve up or down, the speed is unchanged. When you take a derivative, any constant value you've added to your function simply disappears. This shows again that speed is always in the moment, it only describes what's going on in an infinitely short piece of curve.

Differentiation is then like x-ray specs for curves and quantities, and it's turtles all the way down. For example, if we differentiate speed, we get acceleration. This is another rate of change, of speed over time. We see the car's acceleration is initially positive, speeding up, and then goes negative, to slow down, i.e. accelerate in the opposite direction.
Note: The acceleration has been divided by 4 to fit.

If we integrate acceleration to get speed, we have to count the second part as negative area: it is causing the speed to decrease.

We can see that the point of maximum speed is the point where the acceleration passes through $ 0 $. One of the most useful applications of derivatives is indeed to find a maximum or minimum of a curve more easily. No matter where it is, the slope at such a point must always be horizontal.

Let's end this with a more exciting example. What's tall, fast and makes kids scream?

A roller coaster! We'll construct a little track by welding together pieces of circles and lines.

Alas, we shouldn't be too proud of our creation. Even though it looks smooth, there's something very wrong. This is how you build roller coasters when you don't want people to have fun. To see the problem, we need to use our x-ray specs.

We differentiate the height into its slope. It has sharp corners all over the place. Even though the track itself looks smooth, it doesn't change smoothly. The slope is constant in the straight sections and changes rapidly in the curved sections.

If we take the derivative of the slope, i.e. find the slope's slope, we get a measure of curvature. It's positive inside valleys, negative on top of crests. This graph is even worse: there are sharp peaks and cliffs. Note that in the formula, we are now dividing by the square of the infinitesimal distance $ \mathrm{d}x $. This is like going two levels down on the hyperreal number line and back up again.

We can see better if we replace the second derivative with the 2D curvature.
This is the radius of the circle that touches the curve at a given point. As this radius gets infinitely big on straight sections, we use its inverse, $ \class{teal}{κ} $. Because of how we built the track, $ κ $ switches between $ 0 $ and a constant positive or negative value.
At every switch, there will be a corresponding change in force, a jerk.

Let's simulate a ride. As riders go through our curved sections, their inertia will push them to the outside of the curve. From their point of view, this is a centripetal force up or down. We'll plot the (subjective) vertical G force including gravity. It starts at a comfy 1 G, but then swings wildly between 0.5 G and 1.25 G.

Even though the track seems smooth, we can see that the vertical G's are not. Every time we enter a curve, we experience a sudden jerk up or down. This is due to the jumps in the curvature. The G's are themselves curved, because the rider's sense of gravity decreases as the cart goes vertical. The sharp dips below 0.5 G are not simulation errors: this is actually what it would feel like.

To really highlight the problem, we need to x-ray the G's and derive again. G forces are a form of acceleration. The derivative of acceleration is a change in force, called jerk. Whenever it's non-zero, you feel jerked in a particular direction.

To fix this, we need to alter the curve of the track and smooth it out at all the different levels of differentiation. Here I've applied a relaxation procedure. It's like a blur filter in photoshop: we replace every point on the track with the average of its neighbours. We get a subtly different curve. Its height hasn't changed much at all, it's just a little bit less tense.

But this minor change has a huge effect on both slope and radius of curvature. They are completely smoothed out, with all corners and jumps removed.

If we do another simulation, the G force graph looks completely different. There are no more jumps.

But the real difference is in jerk. There are no more actual jerks, only smooth oscillations. Instead of bruises, riders will get butterflies. Thanks to calculus, we avoided that painful lesson without ever having to ride it ourselves.

Please check your pockets for loose items. Lost property will not be returned.

Let's start with the original, unrelaxed track. Thanks to calculus, we can simulate head-bobbing so you can get a feel for how jerky this is. Even virtually, this isn't very pleasant.

This is the improved track. Notice the smooth transitions in and out of curves.

And that's how you make sweet roller coasters: by building them out of infinitely small, smooth pieces, so you don't get jerked around too much.

We'll start with the classic real number line, marked at the integer positions, and poke around.
We imagine the line continues to the left and right indefinitely.

But there's a problem with this visualization: by picturing numbers as points,
it's not clear how they act upon each other.
For example, the two adjacent numbers $ \class{blue}{2} + \class{green}{3} $ sum to $ \class{red}{5} $ …

… but the similarly adjacent pair $ \class{blue}{-2} + \class{green}{-1} = \class{red}{-3} $.
We can't easily spot where the red point is going to be based on the blue and green.

A better solution is to represent our numbers using arrows instead, or vectors.
Each arrow represents a number through its length, pointing right/left for positive/negative.

The nice thing about arrows is that you can move them around without changing them.
To add two arrows, just lay them end to end. You can easily spot why $ \class{blue}{-2} + \class{green}{-1} = \class{red}{-3} $ …

… and why $ \class{blue}{2} + \class{green}{3} = \class{red}{5} $, similarly.
As long as we apply positives and negatives correctly, everything still works.

Now let's examine multiplication. We're going to start with $ \class{blue}{1} $ and then we'll multiply it by $ \class{green}{1.5} $ repeatedly.

With every multiplication, the vector gets longer by 50 percent.
These vectors represent the numbers $ \class{red}{1}, \class{red}{1.5}, \class{red}{2.25}, \class{red}{3.375} $, $ \class{red}{5.0625} $, a nice exponential sequence.

Now we're going to do the same, but multiplying by the negative, $ \class{green}{-1.5} $, repeatedly.

The vectors still grow by 50%, but they also flip around, alternating between positive and negative.
These vectors represent the sequence $ \class{red}{1}, \class{red}{-1.5}, \class{red}{2.25}, \class{red}{-3.375}, \class{red}{5.0625} $.

But there's another way of looking at this. What if instead of flipping from positive to negative, passing through zero, we went around instead, by rotating the vector as we're growing it?

We'd get the same numbers, but we've discovered something remarkable: a way to enter and pass through the netherworld around the number line. The question is, is this mathematically sound, or plain non-sense?

The challenge is to come up with a consistent rule for applying these rotations. We start with normal arithmetic. Multiplying by a positive didn't flip the sign, so we say we rotated by $ 0^\circ $. Multiplying by a negative flips the sign, so we rotated by $ \class{green}{180^\circ} $. The lengths are multiplied normally in both cases.

Now suppose we pick one of the in-between nether-numbers, say the vector of length $ 1.5 $, at a $ 90^\circ $ angle. What does that mean? That's what we're trying to find out! We'll write that as $ \class{green}{1.5 \angle 90^\circ} $ (1.5 at 90 degrees). It could make sense to say that multiplying by this number should rotate by $ \class{green}{90^\circ} $ while again growing the length by 50%.

This creates the spiral of points: $ \class{red}{1 \angle 0^\circ} $, $ \class{red}{1.5 \angle 90^\circ} $, $ \class{red}{2.25 \angle 180^\circ} $, $ \class{red}{3.375 \angle 270^\circ} $, $ \class{red}{5.0625 \angle 360^\circ} $. Three of those are normal numbers: $ +1 $, $ -2.25 $ and $ +5.0625 $, lying neatly on the real number line. The other two are new numbers conjured up from the void.

Let's examine this rotation more. We can pick $ 1 $ at a $ \class{green}{45^\circ} $ angle. Multiplying by a $ 1 $ probably shouldn't change a vector's length, which means we'd get a pure rotation effect.

By multiplying by $ \class{green}{1 \angle 45^\circ} $, we can rotate in increments of $ 45^\circ $.
It takes 4 multiplications to go from $ +1 $, around the circle of ones, and back to the real number $ -1 $.

And that's actually a remarkable thing, because it means our invented rule has created a square root of $ -1 $.
It's the number $ \class{green}{1 \angle 90^\circ} $.

If we multiply it by itself, we end up at angle $ \class{green}{90} + \class{green}{90} = \class{blue}{180^\circ} $, which is $ \class{blue}{-1} $ on the real line.

But actually, the same goes for $ \class{green}{1 \angle 270^\circ} $.

When we multiply it by itself, we end up at angle $ \class{green}{270} + \class{green}{270} = \class{blue}{540^\circ} $. But because we went around the circle once, that's the same as rotating by $ \class{blue}{180^\circ} $. So that's also equal to $ \class{blue}{-1} $.

Or we could think of $ +270^\circ $ as $ -90^\circ $, and rotate the other way. It works out just the same. This is quite remarkable: our rule is consistent no matter how many times we've looped around the circle.

Either way, $ \class{blue}{-1} $ has two square roots, separated by $ 180^\circ $, namely $ \class{green}{1 \angle 90^\circ} $ and $ \class{green}{1 \angle 270^\circ} $.
This is analogous to how both $ 2 $ and $ -2 $ are square roots of $ 4 $.

Complex multiplication can then be summarized as: angles add up, lengths multiply, taking care to preserve clockwise and counterwise angles. Above, we multiply two random complex numbers a and b to get c.

When we start changing the vectors, c turns along, being tugged by both a and b's angles. It wraps around the circle, while its length changes. Hence, complex numbers like to turn, and it's this rule that separates them from ordinary vectors.

We can then picture the complex plane as a grid of concentric circles. There's a circle of ones, a circle of twos, a circle of one-and-a-halfs, etc. Each number comes in many different versions or flavors, one positive, one negative, and infinitely many others in between, at arbitrary angles on both sides of the circle.

Which brings us to our reluctant and elusive friend, $ \class{blue}{i} $. This is the proper name for $ \class{blue}{1 \angle 90^\circ} $, and the way complex numbers are normally introduced: $ i^2 = -1 $. The magic is that we can put a complex number anywhere a real number goes, and the math still works out, oddly enough. We get complex answers about complex inputs.

Complex numbers are then usually written as the sum of their (real) X coordinate, and their (imaginary) Y coordinate, much like ordinary 2D vectors. But this is misleading: the ugly number $ \class{red}{\frac{\sqrt{3}}{2} + \frac{1}{2}i } $ is actually just $ \class{green}{1 \angle 30^\circ} $ in disguise, and it acts more like a $ 1 $ than a $ \frac{1}{2} $ or $ \frac{\sqrt{3}}{2} $. While knowing how to convert between the two is required for any real calculations, you can cheat by doing it visually.

But looking at individual vectors only gets us so far. We study functions of real numbers by looking at a graph that shows us every output for every input. To do the same for complex numbers, we need to understand how these numbers-that-like-to-turn, this field of vectors, change as a whole.
Note: from now on, I'll put $ +1 $, i.e. $ 0^\circ $ at the 12 o'clock position for simplicity.

When we apply a square root, each vector shifts. But really, it's the entire fabric of the complex plane that's warping. Each circle has been squeezed into a half-circle, because all the angles have been halved—the opposite of squaring, i.e. doubling the angle. The lengths have had a normal square root applied to them, compressing the grid at the edges and bulging it in the middle.

But remember how every number had two opposite square roots? This comes from the circular nature of complex math. If we take a vector and rotate it $ 360 ^\circ $, we end up in the same place, and the two vectors are equal. But after dividing the angles in half, those two vectors are now separated by only $ 180 ^\circ $ and lie on opposite ends of the circle. In complex math, they can both emerge.

Complex operations are then like folding or unfolding a piece of paper, only it's weird and stretchy and circular. This can be hard to grasp, but is easier to see in motion. To help see what's going on, I've cut the disc and separated the positive from the negative angles in 3D.

When we square our numbers to undo the square root, the angles double, folding the plane in on itself. The lengths are also squared, restoring the grid spacing to normal.

After squaring, each square root has now ended up on top of its identical twin, and we can merge everything back down to a flat plane. Everything matches up perfectly.

Thus the square root actually looks like this. New numbers flow in from the 'far side' as we try and shear the disc apart. The complex plane is stubborn and wants to stay connected, and will fold and unfold to ensure this is always the case. This is one of its most remarkable properties.

There's no limit to this folding or unfolding. If we take every number to the fourth power, angles are multiplied by four, while lengths are taken to the fourth power. This results in 4 copies of the plane being folded into one.

However, things are not always so neat. What happens if we were to take everything to an irrational power, say $ \frac{1}{\sqrt{2}} $? Angles get multiplied by $ 0.707106... $, which means a rotation of $ 360^\circ $ now becomes $ \sim 254.56^\circ $.

Because no multiple of $ 360 $ is divisible by $ \frac{1}{\sqrt{2}} $, the circular grid never matches up with itself again no matter how far we extend it. Hence, this operation splits a single unique complex number into an infinite amount of distinct copies.

For any irrational power $ p $, there are an infinite number of solutions to $ z^p = c $, all lying on a circle. For a hint as to why this is so, we can look at Taylor series: an arbitrary function $ f(z) $ can be written as an infinite sum $ a + bz + cz^2 + dz^3 + ... \,$ When z is complex, such a sum doesn't just represent a finite amount of folds, but a mindboggling infinite origami of complex space.

Our region of interest is the disc of complex numbers less than $ 2 $ in length. I've marked the circle of ones as a reference.

We take an arbitrary set of numbers, like this grid, and start applying the formula $ f(z) = z^2 + c $ to each. Rather than use vectors, I'll just draw points, to avoid cluttering the diagram.

First we square each number. That is, their lengths are squared, their angles are doubled.
The squaring has a dual effect: numbers larger than $ 1 $ grow bigger and are pushed outwards, numbers less than $ 1 $ grow smaller and are pulled inwards.

Next, we reset the grid back to neutral, keeping the numbers in their new place.
We also pick a random value for the constant $ \class{green}{c} $, e.g. $ \class{green}{0.57 \angle -59^\circ} $.

Now we add $ \class{green}{c} $ to each point, completing one round of Julia iteration, $ f(z) = z^2 + c $. As a result, some numbers have ended up closer towards the origin (i.e. $ 0 $), others further away from it. The combination of folding + shifting has had a non-obvious effect on the numbers.

We begin the second iteration and square each number again. Any number not inside the critical circle of $ 1 $ in the middle will get pushed out again. The other numbers continue to linger in the middle.

If we zoom out, we can see the larger numbers are spiralling outwards and are permanently lost. The minor nudge by $ \class{green}{c} $ won't be enough to bring them back.

Others remain in the middle, being drawn in, but are also at risk of being pushed out of the circle by $ \class{green}{c} $.

Resetting the grid again, we add the same value $ \class{green}{c} $ to our vectors again to finish. At this point, our original grid of numbers has been completely jumbled up.

If we continued this process would any numbers remain in the middle? Or would they eventually all get flung out? Unfortunately it's very hard to see what's going on while iterating forwards, because we lose track of where each point came from.

So we're going to go backwards instead. We'll establish a safe-zone of all numbers less than $ 2 $, forming a solid disc of all those which aren't irretrievably lost. We want to know where all these numbers can possibly come from. To help track these points, I've coloured one area in a different shade.

First we have to shift the numbers again, this time in the opposite direction to subtract $ c $.

Now we apply the square root to find $ z_{n-1} = \pm \sqrt{z_n - c} $, which is a Julia iteration in reverse.

After one backwards iteration, the disc has been squished down into an oval at an angle.
These are all the points that will definitely stay in the middle after one iteration.

When we apply the second iteration, a pattern starts to develop. Because of the repeated unfolding, we create two bulges wherever there was previously only one.

At the same time, the square root alters the length of each number as well. As a result, we squeeze in the radial direction, scaling down earlier features as they combine with newly created ones.

After 4 iterations, we start to see the first hints of self-similarity. The shape's lobes are sprouting into spirals.

But all we've really done is narrow down our blue safe-zone to include only those points that 'survive' up to 5 Julia iterations.

Remarkably this seems to distort the fractal evenly: our highlighted circles don't stretch into ovals. This is not a coincidence. Complex operations are indeed stubborn, in that they all preserve right angles everywhere. To do so, the mapping must act like a pure scaling and rotation at every point, without shearing off in any particular direction. This is what allows the fractal to look like itself at different scales.

Skipping ahead to iteration 12, we've definitely abandoned the realm of neat, traditional geometry.
Despite curving wildly, the total mapping $ z_{12} $ still has this property of evenness, which is properly referred to as a conformal mapping.

After 128 iterations, we end up with this intricate dragon-like shape, approximating the safe zone for the true fractal map $ z_\infty $. The numbers that make up the blue area are the hardiest points that will survive the next 128 attempts on their life. All the others will definitely get flung out.

Yet this complicated shape is merely the result of folding over and over again, adding a simple constant in between. If we perform a forwards Julia iteration, i.e. squaring and shifting, we see this shape matches up with itself, and looks identical before and after.

For different values of $ \class{green}{c} $, the fractal morphs into other shapes. There's literally an infinite variety to discover. Some sets are made up of disconnected parts. In this case, $ |c| $ is large enough to push the solid disc away from the center in a single iteration, but not so far that some points can't fold back in. If $ |c| $ gets much larger, the set vanishes.

For a smaller $ c $, Julia sets are solid. Even a small shift in the value of $ c $ can accumulate into a large difference. Here we zone in on some fluffy clouds right outside the 'solid zone'. Oddly enough, it seems when $ c $ is not inside of its own Julia set, the set is not solid.

This area of fractal space is dubbed Seahorse Valley, for rather obvious reasons.

Nearby, we find these jewel-like spirals.

Buried deep inside, there are remarkable combinations of shapes, like this pearl necklace covered in something resembling palm trees.

And we can even make snowflakes. The dramatic changes due to $ c $ reveal the chaotic nature of fractals. Mathematically, chaos occurs when even the tiniest change can accumulate and blow up to an arbitrarily large effect.

If we change our iteration formula, for example to a fourth power $ f(z) = z^4 + c $, the entire shape changes. Because each iteration now turns one bulge into four, the resulting shape has four-fold rotational symmetry.

Again, different values of $ \class{green}{c} $ make different shapes, precipitating dramatic changes.

To understand the effect of $ c $ we need to make a Mandelbrot set. This is similar to a Julia set, but the formula is applied differently. We'll use $ z^2 + c $ again. Instead of different starting values $ z_1 $, we choose different values of $ c $ and start with $ z_1 = 0 $ every time. Because $ c $ is no longer constant, the mapping stops being a simple folding operation. Each iteration is now unique and not so easy to visualize.

Because the Mandelbrot set traverses all possible values of $ c $ across its surface, it has a part of every associated Julia set in it. Around any number $ \class{green}{c} $ it looks like the Julia set which has that value as its constant. Here, we move towards the three-way cross at the bottom of the Mandelbrot set. The Julia set develops similar features.

Where the Mandelbrot set is round and bulbous, the Julia set is too.

The spirals and seahorses from earlier are located here. You can literally see the shapes on both sides of the valley evolving towards horseheads and spirals respectively. But the Mandelbrot set acts like a map to Julia sets in a much more direct way: anywhere the Mandelbrot set is filled in (blue), the corresponding Julia set is solid too. The white areas are values of $ c $ which create disconnected Julia sets.

That the Mandelbrot set is a 'pixel-perfect' map of Julia sets is a big clue. It reflects that they're actually both slices of a single higher dimensional object. By viewing these slices as we travel through, we can get a vague idea of its shape and complexity. In this object, every point in the Mandelbrot set is connected to the center of the corresponding Julia set. Actually picturing this 4D object is a challenge.

But like any fractal, the Mandelbrot set also contains copies of itself, buried inside its edge. This is just one of the many varied copies. As a result, deep Mandelbrot zooms can reach astonishing levels of beauty in complexity. This is best done with specialized software that can calculate with hundreds of digits of precision.

Take for example a sine wave $ w(x) $.

For the wave to propagate across a distance, its values have to ripple up and down over time.
The rate of change over time is drawn on top. This is the vertical velocity at every point. Both the wave and its rates of change undergo a complicated numerical dance.

But to properly describe this motion, we have to go one level deeper. We have to examine the rate of change of the vertical velocity of the wave. This is its vertical acceleration. We see that green vectors tug on blue vectors as blue vectors tug on the wave.

It's easier to see what's going on if we center the vectors vertically. The acceleration appears to be equal but opposite to the wave itself.

But that's just a lucky coincidence. If we shift the wave up by one unit, its opposite shifts down by a unit. Yet its velocity and acceleration are unaltered. So acceleration is not simply the opposite of the wave.

What's actually going is that the green vectors match the curvature of the wave, positive inside valleys, negative on top of crests. Intuitively, this can be explained by saying that waves tend to bounce towards an average level: this is going to pull the value up out of valleys and down from peaks.

But curvature is the rate of change of the slope, and slope is the rate of change over a distance. So to describe real waves, we need to relate 'second level' change over time and change over distance, each deriving twice. This is Complicated with a capital C.

Let's try this with complex numbers instead. Until now, we had a 2D graph, showing the real value of the wave over real distance. We're going to make the wave's value complex. Mapping a 1D number (distance) to a 2D number (the wave function), means we need a 3D diagram.

The complex plane is mapped into the old Y direction (real) and the new Z direction (imaginary).

To make a complex wave, we do the thing complex numbers are best at: we make them turn, and make a helix. In this case, our wave function is simply the variable number $ 1 \angle x $ , a constant length with a smoothly changing rotation over distance.

To make the wave move, we can simply twist it in-place. Which we now know is the same as multiplying by $ 1 \angle t $. If we plot the complex velocity of each point, at first sight this might not look any simpler than the real wave. But in fact, these vectors are not changing in length at all, unlike the real version. Just like the wave they pull on, they undergo a pure rotation.

At all times, the velocity is offset by $ 90^\circ $ from the wave itself. And that means that described in complex numbers, wave equations are super easy. Instead of involving two derivatives, i.e. the rate of rate of change, we only need one. There is a direct relationship between a value and its rate of change. The necessary rotation by $ 90^\circ $ can then be written simply as multiplying by $ i $.

To recover a real wave from a complex wave, we can simply flatten it back to 2D, discarding the imaginary part. By using complex numbers to describe waves, we give them the power to rotate in place without changing their amplitude, which turns out to be much simpler.

In fact, flattening the wave has a perfectly reasonable complex interpretation: it's what happens when we average out a counter-clockwise wave (positive frequency) with a clockwise wave (negative frequency). By twisting each in opposite directions, the combined wave travels along, locked to the real number line.

But if we add up two arbitrary complex frequencies, their sum immediately turns into a spirograph pattern that manages to evolve and propagate, even as it just rotates in place. Though the original waves both had a constant amplitude of $ 1 $, the relative differences in angles (i.e. the phase) allows them to cancel out in surprising ways.

Neither curve is actually moving forward: they're just spinning in place, creating motion anyway. This is actually what quantum superposition looks like, where two or more complex probability waves combine and interfere. Where the combined wave cancels out to zero, that's where two separate possible states are simultaneously cancelling out each other. The fact that the underlying numbers are complex doesn't prevent them from describing real physics, indeed, it seems that's how nature actually works.