Thinkin’ about numbers

If you don’t like numbers, please don’t read this post. PLEASE. I’m begging you. Get out while you can.

I think about numbers a lot, because they’re fascinating. A lot of times I think about them in the context of sports, or science, or finance. Numbers are just cool. They are an exclusively human creation, and yet they appear all the time throughout nature. This world we live in has inherent mathematical properties, and we are lucky to be conscious and intelligent and aware to discover them for ourselves.

And each time I try to learn something new, I inevitably travel down a rabbit hole that forces me to write blog posts like this.

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You’ve probably heard of the Fibonacci sequence. You start with two consecutive 1’s, and then each number after that is the sum of the two numbers that precede it.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

The Fibonacci sequence appears in many biological settings: branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern, the arrangement of a pine cone, and the family tree of honeybees.

In fact, the spiral of hurricanes match the Fibonacci sequence. It’s pretty extraordinary.

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Here’s an odd quirk of math. Take the number 1. Then add 2. Then add 3. Then repeat this forever:

1 + 2 + 3 + 4 + 5 + 6 + … + ∞ = ?

What’s the result? Infinity, right? No! The answer is not infinity. It is -1/12. That doesn’t look right. How can the sum of all positive integers equal a negative number? And why the heck is it -1/12? I cannot answer that. But I can link you to the absolutely ridiculous proof. I advise not to read through for fear of a massive migraine.

The YouTube channel Numberphile also did a video on this seemingly impossible answer. They break it down into (somewhat) understandable chunks.

(EDIT: As my friend Dan points out, -1/12 may be incorrect. It is hotly debated.)

Have you ever heard of Graham’s number? It is a number so big, so frikkin’ big, that it holds the record for the largest number ever used in a serious mathematical proof.

Literally, no one has ever used a larger number.

It’s difficult to visualize how big this number is, but here is an attempt. Let’s start with the smallest measurable unit – the Planck volume. It is 1.616199×10−35 meters, which is about 10−20 times smaller the diameter of a proton. A proton. So, yes, the Planck volume is small. You could fit a lot of those in the observable universe, about 10185 of them. That’s how many of the smallest thing (Planck units) you can fit in the biggest thing (universe).

Graham’s number is WAY bigger than that.

Calculating Graham’s number requires tetration and hyperoperations and it’s pretty much the most complicated thing ever. I recommend this post and then I recommend taking a break from the internet.

From Wikipedia:

Using Knuth’s up-arrow notation, Graham’s number G is:

<br /> \left.<br />  \begin{matrix}<br />   G &=&3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots\cdots \uparrow}3 \\<br />     & &3\underbrace{\uparrow \uparrow \cdots\cdots\cdots\cdots \uparrow}3 \\<br />     & &\underbrace{\qquad\;\; \vdots \qquad\;\;} \\<br />     & &3\underbrace{\uparrow \uparrow \cdots\cdot\cdot \uparrow}3 \\<br />     & &3\uparrow \uparrow \uparrow \uparrow3<br />  \end{matrix}<br /> \right \} \text{64 layers}<br />

where the number of arrows in each layer, starting at the top layer, is specified by the value of the next layer below it; that is:

G = g_{64},\text{ where }g_1=3\uparrow\uparrow\uparrow\uparrow 3,\  g_n = 3\uparrow^{g_{n-1}}3,

and where a superscript on an up-arrow indicates how many arrows there are. In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G =g64 with g63 up-arrows between 3s.

Equivalently,

G = f^{64}(4),\text{ where }f(n) = 3 \uparrow^n 3,

And there you have it. Graham’s number.

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One of the hardest concepts I learned in finance was the Black-Scholes model*, a mathematical model of a financial market containing certain derivative investment instruments. The model was first published by Fischer Black and Myron Scholes in their 1973 paper, The Pricing of Options and Corporate Liabilities.

*Business Week recently named the Black-Scholes model the 60th most disruptive idea in the last 85 years, right between “the sharing economy” and cable news. It was ranked higher than: YouTube, OPEC, and the bar code. And yet most people have no idea what it is (for good reason, probably).

Let us take a look at the formula:

66c077aa44cab1df32603df78b81057e

 

I spent many nights trying to uncover the mystery of this Black-Scholes formula. It didn’t make much sense when I learned it in college, and it doesn’t make much sense now, but there’s no doubt that it was a breakthrough:

Now the formula is used everywhere. If you have a mortgage, your right to prepay is an option. Your right to default and turn over the house to the lender if it’s underwater is an option. Every simple mortgage has these two options embedded in it. Seven hundred trillion dollars of this stuff is sloshing around the earth.

The key idea behind the model is this: you hedge an option by buying and selling the underlying asset in just the right way and you can eliminate risk. This type of hedging is called delta hedging and is the basis of more complicated hedging strategies that our friends in investment banks and hedge funds use.

I’m not sure if I explained that well enough. But, uh, let’s move on.

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It is always in your favor to order a larger pizza. Even if you are not hungry. Again, this is because of math.

Let’s take a small 10 inch pizza and a large 16 inch pizza. And let’s assume that the price of the small is $10 and the price of the large is $16. It looks like you’re paying the same price for the same amount of pizza, but, well, you’re not.

We all know the formula for the area of a circle: πr2. Right? This is a thing we still know? So, let’s take a look at the area of each pizza:

10 inch pizza = π * (5)= 78 square inches
16 inch pizza = π * (8)2 = 201 square inches

So, in this scenario, you are paying 60% more money but you’re actually getting almost three times as much pizza. And so you are paying much less per square inch.

10 inch pizza = $10 = 78 square inches = 12.8 cents per square inch of pizza
16 inch pizza = $16 = 201 square inches = 8.0 cents per square inch of pizza

And this is one of the many excuses  for why I always order the large.

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I once had a theory that everyone in the world could get rich.

You start with person A. Let’s call him Edgar. Edgar has 1 dollar. He sells that dollar to Sally for 2 dollars. Sally then sells that dollar to Johan for 3 dollars. You repeat this forever. Each time, a different person is getting $1 richer. Go around the world one million times, and everyone makes one million dollars, right? Right? Did I just find a loophole?

No, what I inadvertently discovered was a ponzi scheme.

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The difference between 1 million and 1 billion is just insanely large. We look at $1 million and we think: Wow, that’s a lot of money. Then we look at $1 billion and think: Wow, that’s really a lot of money! But the two numbers are nowhere close to each other.

If you were to count to 1 million, it would take you about 12 days. This is assuming you didn’t sleep, but I urge you to sleep, because I don’t want another Russian Sleep Experiment fiasco on my hands.

If you were to count to 1 billion, it would take you THIRTY ONE YEARS. This is, again, assuming you didn’t sleep. Put another way: the difference between 1 and 1 million is the difference between now and two weeks ago (can’t wait for Thanksgiving!). The difference between 1 and 1 billion is the difference between now and 1983, when the top song was Every Breath You Take and swatches were a thing.

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In conclusion, I like numbers.

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