In my younger and more vulnerable years, I hated Math. By hate I mean loathe. That’s quite a statement to read from someone currently employed as a statistician, at the same time taking his master’s in statistics. How could fifth grade me - who once shamed in front of his classmates for not having memorized the multiplication tables for numbers 7, 8, and 9 – ever imagine that in the dawn of his twenties he would be eating and drinking mathematics, modelling regression equations at work, and analyzing Markov chains in class?
I put the blame on my high school Calculus classes, where I first encountered and understood the concept of infinity. Back then, I walked the short distance between school and home. While walking, I’d count the streetlights, the red cars that sped along the highway, the children that walked by holding balloons or toy cars tied to the end of recycled shoe strings. I was on such a walk home one day when the realization hit me: there are more numbers than there are things to count them with. You could write a number on every item, every person, every animal, every plant, and every being that is on Earth and still not run out numbers. You could write a number on every planet, on every star, on every asteroid, on every comet – and you would still have numbers left (an infinity of them, in fact).
From then on, this became my favorite game. In my free time, I explored the idea of asymptotes. I read about nonterminating numbers. I investigated on why the math teachers from my high school would always answer me that five divided by zero equals infinity. I started chasing this elusive number: the number to span all numbers.
Infinities within infinities
There was a joke from The Big Bang Theory in which a character was revealed to have been using a certain dating app to meet with women, and was therefore asked by his friends how many he had met with since. “Two,” he said, “I don’t have to remind you that there’s an infinite set of numbers between that and zero.” That little bit of throwaway math gag was quite the hit for me, but I wondered how many other viewers caught up with it.
When talking about infinities, mathematicians often define two categories: those that are countable, and those that are uncountable. When counting things, what we are really doing is associating them to what we call the Natural Numbers. These are the numbers 1, 2, 3,4, 5, and so on. We know that we have three books on a coffee table if we can say that one book is book number one, another is book number two, and the last is book number three. Countable sets are sets whose members we can line up and state which member gets assigned to which number. The last number that gets assigned is then the "count" of the members belonging to that set (e.g. the last number assigned to the books was three, so we say we have three books).
A countable infinity, therefore, is a set that has an infinite number of elements, but these are distinct elements that we can – hypothetically speaking – count. We probably won't stop counting – but that’s beside the point. Imagine the number of people to ever be born on this planet starting from this year. Taking the assumption that the planet and the human race will live forever, then we expect an infinite number of people to be born henceforth. But we can count people. For example, we can consolidate census listings for every country starting from this year. If we were to count that manually we can pick a starting point, then say which person is person 1, which is person 2, which is person 3, ad infinitum.
But what about an uncountable infinity? An uncountable infinity is a set of infinite elements whose members cannot be lined up like with countable sets. We cannot count their elements because we can't tell which element should be element 1, which is element 2, which is element 3. Think about all the fractional numbers between 0 and 1. What comes after 0? Is it 0.1? Is it 0.01? Or is it 0.0000000001? This is where the idea of uncountability arises - we can't associate these elements to our counting numbers because we can't even tell what these elements are. Between any two elements, we can always find another new, distinct element. Between 0 and 1, there is 0.5. Between 0 and 0.5, there's 0.25. Between 0 and 0.25, there's 0.125. You see the point.
Remember that line from The Fault In Our Stars that some infinities are bigger than other infinities? So we see that there's an infinity between 0 and 1 - what more between 0 and 2? Between 0 and 3? If between 0 and 1, between 1 and 2, between 2 and 3, and between every whole number there is an infinite set of other numbers, then what does that say for the entire number line? It's an infinity containing an infinity of other infinities.
Zeno’s paradox
The Greek philosopher Zeno of Elea introduced to us this paradox: suppose that Achilles and a turtle went on a race. Let's assume that the turtle moves at a rate of 1 unit of distance per minute, while Achilles moves at 5 units per minute. The turtle is given a head start of ten minutes - that means the turtle has already covered a distance of ten units before Achilles even begins running. But Achilles runs faster, and in three minutes he will have covered fifteen units of distance, while the turtle has only done three plus his ten-unit headstart, a total of thirteen units. We can see that in three minutes Achilles should have already outrun his opponent.
How do we measure distance anyway? Geometry has an answer for this in what is called the Ruler Postulate. How do you measure the length of, say, your notebook? You take a ruler, then you position the edges between two numbers on the ruler. The length of the notebook corresponds to how many numbers – marked by ticks on the ruler – fall between the edges. Likewise, we can put a number line as a ruler between Achilles and the turtle to measure the distance between them. Consider the time when the turtle has just finished its ten-minute head start and Achilles is about to begin running. With Achilles positioned on zero, the turtle should be on ten.
But how many numbers fall between 0 and 10? We have shown previously that an infinite set of numbers exists on the number line between any two whole numbers. Therefore, there should numerically be an infinite distance between Achilles and the turtle. Note as well that the turtle is continuously moving as Achilles attempts to cross this infinity, opening up even more infinities between them. Beyond reason and real-world logic, Achilles could run forever and still never catch up to his slower opponent.
This brings me to the most painful infinity I ever encountered.
I was eighteen then, in college, and in love with a soft-spoken muse who had a fondness for Taylor Swift and young adult novels. I knew her from high school. But soon the time came when we both moved to college, and I went to a campus in Quezon City while she absconded all the way to Laguna.
Even before, things were never entirely constant between us. Whether the magnetism that pulled at my core towards her was shared between us was a dichotomy that switched poles every day: some days we were two asteroids on a collision course with each other, in others I was a comet making a flyby to the orbit of a planet that would be miles away by the time I arrived. Some days she rained on me with all sorts of stories about her day, but on others, she would be quiet and would refuse to respond to whatever I said or did.
When college happened, the polarities stopped fluctuating. Suddenly the eventual moments of clarity, when things between us were good, stopped happening entirely. Communication between us took a standstill, and every day I died waiting for a phone call, or a single text. The silence lasted for a month, then two months. Before I knew it, a semester had passed, and the last time I heard from her was the day she said goodbye before leaving for Laguna.
But I kept running, like Zeno’s Achilles, doomed to span an infinity that kept ever expanding before me, crossed what appeared to be a short distance between me and the girl of my dreams. But despite how fast I ran, we could not get any closer.
Ever wondered what the quotient should be when a number is divided by zero? Everyone knows that such a division is impossible. We have a term for that operation; we say it is indeterminate, or undefined. But why stop there? We can find an approximate answer to division by zero by observing the behavior of the quotient when a number is divided by a divisor that is getting nearer and nearer to zero. For example, 5 divided by 1 is 5. How about we make the divisor smaller, change it into something nearer to zero. How about 5 divided by 0.5: that’s 10. Let’s change 0.5 again into something even nearer to zero: 5 divided by 0.25 is 20.
5 divided by 0.125 is 40.
5 divided by 0.001 is 5,000.
5 divided by 0.00001 is 500,000.
5 divided by 0.000001 is 5,000,000.
Notice that when 5 is divided by numbers getting nearer and nearer to zero, the quotient is getting larger and larger: from 5, to 10, to 40, to 5,000, and eventually reaching 5,000,000. How about 5 divided by 0.000000001? That's 5,000,000,000. So we know that division by zero is impossible, but division by a number that is getting smaller and smaller (i.e., approaching zero) yields a value that is getting larger and larger. Zero, in this case, is something referred to in Calculus as an asymptote. It is a value that we’ll approach across infinity, but will never actually reach.
And that was how I, eighteen and in love, saw her: as my asymptote. And I was trying to bridge a gap that would not diminish. I ran and ran, but there was this infinity between us, and always I came up short. This is the most painful infinity: being so close, yet so infinitely far.
I would be stuck in that infinity for another year until at last she called.
She said: she should have talked to me sooner.
She said: she just was never really sure of her feelings for me.
It was my turn to be quiet. I hardly open my mouth, afraid that if I did, she’d catch a faint sound of a heart shattering into tiny little pieces. I wanted to tell her: I’ve been running infinities for you. I wanted to tell her: I’m willing to run infinities more, for you.
But that night, over the phone, she was asking me to stop.
D.D.
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