math

# Musings on Zero

Studying math, I was often taught that 0/0 was undefined, but yet somehow alot of equations seemed to rely on it. For example, x=x could be rewritten as x/x=1, and this was true along the entire graph of x/x, except at 0. Reality, while it can be described by math, has always had an independence that makes math look silly, but always in the case where we try to describe something as real when it isn’t. Zero is going to be the joke of this article. Ready to break math?

1^1 = 1, naturally, and 2^1 = 2. Exponents – at least in their original formulation – are meant to represent how many times a number is multiplied by itself. Of course, some clever quack (or quacks) figured he didn’t need to restrict exponents to integers or even numbers greater than zero. Raising a number to the negative power results in division of 1 by that number:

1^-1 = 1/1

It allows for:

1^-1 * 1^1 = 1

which allows for combining exponents such that, if they add to zero, it simply results in 1.

1^-1 * 1^1 = 1^(1-1) = 1^0 = 1

The peculiar part is when we start trying to apply this to zero itself. Consider either of the following functions:

f(x) = x^0
f(x) = x^x

Looking at the graph, the first function obviously stays at 1 until it reaches zero, at which point, it implies:

0^0 = 1

i.e. zero raised to itself, or, zero not raised at all is 1.

The second function is similar in that all the points on the graph lead up to same conclusion.

That all seems fine and dandy until we get to raising to a negative number. Let’s consider the following function:

f(x) = 0^x

We are given two rules for interpreting this:

x < 0 : f(x) = 1/0^(-x)
x > 0 : f(x) = 0^x

Notice that when x < 0, we flip its sign and make it the divisor. This is quite nice for the following:

[ f(x) | x < 0 ] * [ f(x) | x > 0 ] = 1

And note, that’s the same rule as given above. To put it in a more abstract way:

a^x * a^-x = 1

So, notably:

0^x * 0^-x = 1

Right? 😀 Here’s the silly part. Recall the following:

0^1 = 0
0^2 = 0^3 = 0

It can then be shown that:

0^-1 = 1/0^1 = 1/0
0^-2 = 1/0^2 = 1/0

Since equality allows for substitution (duh), it is true that:

0^1 = 0^2 = 0^3 …

Likewise:

0^-1 = 0^-2 = 0^-3 …

Recall now, the rule that allows us to add exponents, and let us apply this to the case of 0.

a^x * a^-x = 1
0^x * 0^-x = 1

Now let’s suppose that x = 1.

0^x * 0^-x = 1
0^1 * 0^-1 =1
0^1 * 1/0^1 = 1
0 * 1/0 = 1
0/0 = 1

And that’s the answer we expect. But here’s something funny we can do. Let’s step back and substitute 0^1 with, equivalently, 0^2.

0^1 * 0^-1 = 1
(0^2) * 0^-1 = 1

Next, let’s operate on this equation with the rule of adding exponents:

a^b * a^c = a^(b+c)

0^2 * 0^-1 = 1
0^(2-1) = 1
0^1 = 1
0 = 1

An voilà, math is broken.

Conversely, we can do the same with 0^-1, as in the following:

0^1 * 0^-1 = 1
0^1 * (0^-2) = 1
0^(1-2) = 1
0^-1 = 1
1/0 = 1

If it’s not obvious yet, it should be: When working with something that doesn’t exist (i.e. zero), you can make anything happen. The rule for zero is that it isn’t there.

The way reality works tends to mock theories that are too abstract. Recently, I was reminded of Zeno’s paradoxes, which are basically statements utilizing infinite division to demonstrate impossibilities. Diogenes the Cynic said nothing but simply walked across a room to demonstrate that Zeno was wrong. In short, no matter what you theorize, if it doesn’t align itself with reality, it’s clearly wrong. But poor theories aren’t totally useless; they can make for some humorous brain teasers… and bad jokes. You know that even if Zeno’s paradox was only half as bad as it was yesterday, it’d still never go away.