Picture the number line.
Start at zero and walk to the right. 1, 2, 3, 4. You can keep going, and there’s never a last number. Someone can always say “plus one” and you’ve got a new one.
Now go back through zero and walk the other way. −1, −2, −3, −4. Same thing. No last number on that side either.
So that’s two infinities already, one in the positive direction and one in the negative. Most people stop there. But there’s a third place to look, and it isn’t further along the line at all. It’s in the gaps.
Before going on, one thing about the word “infinity.”
It’s easy to picture infinity as a giant number sitting at the far end of the line, like a finish line nobody can reach. Mathematicians don’t quite see it that way. The whole numbers are unbounded above, which means there’s no largest one. Infinity isn’t the biggest number. It’s the reason there is no biggest number.
Mathematicians do use the symbols +∞ and −∞, and in some systems they’re added to the number line as special end markers. But they aren’t ordinary numbers. They’re closer to a way of saying “this keeps going.”
Now stop travelling outward. Pick two numbers that are right next to each other, like 0 and 1, and look at the space between them.
There are numbers in there: 0.5, 0.25, 0.9, 0.001. Make the space smaller. Look only between 0 and 0.1. There are still numbers in there. Smaller again, between 0 and 0.01, then 0 and 0.001, then 0 and 0.000001. It doesn’t matter how small you make the gap. There are still infinitely many numbers inside it.
Mathematicians call this density. Between any two different real numbers, however close, there’s always another one, and then another between that one and the next.
So infinity doesn’t only show up when you go farther. It also shows up when you look closer. The number line is endless going out, and it’s endless going in.
This is what I think the “decimal place” part of the original idea is really pointing at. You can keep adding digits, and each new digit lets you zoom in a little more. The space never runs out.
There’s a second thing in the decimal places, and it’s a different thing from the first.
Take π. It starts 3.14159 and carries on forever without ending and without settling into a repeating pattern. The golden ratio, φ, does the same. It starts 1.618 and keeps going.
It’s tempting to file these two under “infinity,” because their digits never stop. But look at the actual sizes. π is a bit more than 3. φ is a bit more than 1.6. They’re small, ordinary, finite numbers. You could mark both on a ruler.
Their digit strings are endless. Their values are not.
This is also where it gets easy to mix up two ideas. Not every endless decimal is a strange number. Even 1/3 gives you 0.3333… forever, and 1/3 is as plain a fraction as there is. What makes π and φ special is that their decimal expansions don’t eventually repeat a pattern. These are called irrational numbers, because they can’t be written as one whole number divided by another.
So there are actually two different things happening in the decimal place. One is that you can zoom into the number line without end. The other is that some single numbers need endless digits to write down. They sound alike, but they aren’t the same.
π is the number that connects a circle’s distance around to its distance across. It was being approximated long before the ancient Greeks. Early civilizations had working estimates of it. And it wasn’t Pythagoras who came up with it, even though his name often gets attached to anything geometric.
The first surviving systematic calculation people usually point to belongs to Archimedes, and it’s a nice one to picture.
Draw a circle. Draw a polygon inside it, with its corners touching the circle. Draw another polygon outside it, with its sides touching the circle. The circle is trapped between the two. The inside shape is slightly too small and the outside shape is slightly too big, so the circle’s edge must be somewhere in between.
Now give both polygons more sides. The shapes hug the circle more tightly. More sides again, and tighter again.
There’s no polygon where this has to stop. You can always add another side. And yet the number you’re closing in on, π, is perfectly finite. It’s an endless process aimed at a fixed target.
Now take a regular pentagon, the five-sided shape with all sides equal.
Draw a line from one corner to the next corner but one, across the inside. That’s a diagonal. Now compare the length of that diagonal to the length of one side. The ratio is φ, the golden ratio, which works out to (1 + √5) ÷ 2.
Draw all the diagonals and you get a five-pointed star, a pentagram. Inside it there’s a smaller pentagon, and the same ratio turns up again between the pieces of the star.
The Pythagorean tradition is where the Pythagoras part of the original idea fits best. Ancient Greek mathematicians ran into lengths that couldn’t be written as a simple fraction of each other, and the stories connect this to Pythagoras’s followers, with the name Hippasus often mentioned. Some historians have even suggested that the pentagon and the golden ratio were part of how this was first noticed. The honest version is that nobody knows exactly who found what or when, and the evidence we have was written down much later. So “the Pythagorean tradition” is about as far as it’s safe to go.
What matters here is that φ is finite, but it keeps coming back out of the shape. A single ratio shows up again and again, inside the same figure.
That “again and again” leads into the last piece.
There’s a shape called the pentaflake. Start with a pentagon. Replace it with six smaller pentagons: five arranged around one in the middle. Then take each of those six and do exactly the same thing. Then do it again to every new one.
The result is a fractal, with the same pattern showing up at smaller and smaller scales. And φ is built into it. The numbers that describe how much smaller each new pentagon is, and where it sits, involve the golden ratio.
I should be careful here, because it’s easy to overstate. A pentagon isn’t a fractal, and the golden ratio isn’t a fractal. It only becomes one when you take a pentagon and keep applying the same rule to it. The recursive self-similarity is what makes it a fractal. The pentagon just supplies the shape.
But it does give a third way of meeting infinity. Not endlessly farther, and not endlessly in between, but endlessly again.
There’s one more twist, and it’s the one that makes “infinity is infinite” feel truer than it first sounds.
The whole numbers are infinite. The fractions are infinite too, and there are infinitely many of them between any two points. You might expect the real numbers, which include π and φ and everything else on the line, to simply be infinite in the same way.
They aren’t. A mathematician named Georg Cantor showed that the real numbers are a bigger infinity than the whole numbers. You can make a list of the whole numbers, or even of all the fractions, and in a particular mathematical sense every one gets its turn. You can’t do that with the real numbers. Whatever list you try, some real numbers are always left out.
So there isn’t just one infinity. There are different sizes of it.
This is a separate idea from positive and negative infinity. Plus infinity and minus infinity aren’t two sizes. They’re two directions. Cantor’s sizes are about how many things are in a collection, not which way you’re heading.
So go back to the number line one more time.
Look outward and there’s no end. Look inward and there’s no last number to land on. Write out π and there’s no final digit. Keep adding pentagons to a pentaflake and there’s no final round.
I don’t think infinity is a single place at the end of anything. It looks more like what happens whenever something has no point where it’s finished. You can meet it going out, going in, writing something down, or repeating a pattern.
Maybe that’s why it’s so hard to point at. It isn’t one thing that never ends. It’s lots of things that never end, in different ways.