The Math You Weren’t Allowed to Do

You probably learned a bunch of things in school math about what you can and can’t do. When you were a first grader, perhaps you learned that that you can’t subtract 5 from 2, but later on, you learned about negatives: 2 – 5 = -3.

You also might have been told that you can’t divide 2 by 6, but then you learn about fractions. And by now, you are no doubt an expert at splitting 2 pizzas among 6 people.

Even so, there is much more that you may not have known…

1. You Can’t Count to Infinity

Actually, you can. It can be done via ordinal numbers.

You start out counting by

1, 2, 3, 4, 5, 6, 7,… 700,… 30,000,000, etc.

When you played a “what’s the highest number?” game with someone, every time you said a number, they countered by saying your number plus one, that is, unless you said infinity. Because infinity plus one is still infinity, right?

Here is where ordinals come into play. The ordinal number \omega (ω, omega) is defined as the first number after ALL of the positive integers. No matter what normal number they might say, whether it’s ten billion or a googol, the ordinal number \omega is far, far larger. It is practically infinity.

But then you can add one to it, and it becomes an even bigger number. Add two, and it becomes even bigger.

\omega, \omega+1, \omega+2,..., 2\omega,...,3\omega,...,n\omega,....,\omega^2,...

What the heck is going on? If you count an infinite number of numbers after omega, you get two omega? Is this two times infinity? And then three omega? And then omega squared?

It turns out to keep on going. Eventually you will get \omega^\omega, and then \omega^{\omega^\omega}, etc. And then you reach \Omega (big omega), which is larger than all things that can be written in terms of little omegas. And then you can make bigger things than that, with no end.

So the next time someone claims infinity is the largest number, you can confidently reply, “infinity plus one.”

2. You Can’t Divide by Zero

Actually, under certain conditions, you can.

The field of complex analysis is largely based around taking contour integrals around poles. Another word for pole is singularity. And another word for singularity is something you get when you divide by zero.

Consider the function y = 1/x. When x is 1, y is 1, and when x is 5, y is 1/5. But what if x is 0? What happens? Well, 1/0 is undefined. However, if you look at a graph, you see that the function spikes up to infinity at x = 0.

What you do in complex analysis is integrate in a circle around that place where it spikes to infinity. The result in this case, if done properly, is 2\pi i. It’s quite bizarre.

3. You Can Only Understand Smooth Things

Actually, there is much theory on crazy, “pathological” functions, some of which are discontinuous at every point!

The image above is kind of misleading, as it is a graph of the Cantor function, which is actually continuous everywhere (!), but nonetheless manages to rise despite having zero derivative almost everywhere.

There is another function with the following properties: it is 1 whenever is x is rational and 0 whenever x is irrational. Yet this function is well understood and is even integrable. (The integral is 0.)

Then you have things that are truly crazy:

The boundary of that thing is nowhere smooth, and is one of the most amazing things that have ever been discovered. Yet it is generated by the extraordinarily simple function z^2 + c, which most people have seen and even studied in school.

4. You Must “Do the Math” and Not Draw Pictures

Actually, math people use pictures all the time. The Mandelbrot set (the previous picture) was not well understood until computer images were generated. There is no such thing as doing the math in a “correct” way. Some fields are quite based on pictures and visualizations.

How else would anyone have thought, for example, that the Mandelbrot set would be so complex? Without seeing that in pictures, how would we have realized the fundamental structure behind the self-similarity of nature?

Yeah, that’s a picture of broccoli. Not a mathematical function. Broccoli.

5. If It Doesn’t Make Sense, It’s Not True

Actually, many absurd things in math can be perfectly reasonable.

What’s the next number after 7?

8, you say. But why 8? What’s wrong with saying the next number after 7 is 0? In fact, I can define a “number” to only include 0, 1, 2, 3, 4, 5, 6, and 7. Basic operations such as addition and multiplication can be well defined. For example, addition is just counting forward that many numbers. So 6 + 3 = 1, because if you start at 6 and go forward 3, you loop back around and end up at 1.

Even weirder is the Banach-Tarski Paradox, which states a solid sphere can be broken up into a finite number of pieces, and the pieces can be reassembled to form TWO spheres of the exact same size as the original!

I hope this was understandable for everyone. May the reader live for ω+1 years!

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