People through history never really paid much attention to the number zero. They did understand the concept of nothing but they thought that zero had no special relevance in mathematics. As mathematics developed, so did its notation. People started seeing the relevance of zero only in notation; it was a number they used to easily see the difference between 1, 10, 100… Zero began being regarded as a real number in 9th century in India.

Zero we now know is probably the most complicated number we know of.

Why is this number giving us so many problems?

There are few things you absolutely cannot do; that are dividing with zero and have zero to the power zero. You probably heard this many times but have you ever wondered why is that true?

What is the logic that people use when they say that something divided by zero is infinity?

Well, when you divide two real numbers, let’s say 16 divided by 4, you’d continuously subtract 4 from 16 until you get to zero.

1. $16 – 4 = 12$
2. $12 – 4 = 8$
3. $8 – 4 = 4$
4. $4 – 4 = 0$

We got to zero in 4 steps which means that $16 : 4 = 4$.

What if you want to divide 16 by zero? This means that you should repeatedly subtract zero from 16.

1. $16 – 0 = 16$
2. $16 – 0 = 16$
3. $16 – 0 = 16$
4. ….

This means that no matter how many steps we take, we’d never finish our division.

This leads us to thinking that $20 : 0 = \infty$. Why can’t we simply conclude this? If we observe dividing any other number with 0 we should, by the same process, conclude that the coefficient of any real number and zero is equal to $\infty$.

Now we have: $\frac{1}{0} = \infty = \frac{2}{0}$. From here 1 is equal to two which is obviously not true.

Let’s take a look at a graph of a function $f(x) = \frac{1}{x}$.

Now let’s get to $0^0$.

You already know that any number to the power of zero is equal to one.

$a^0 = 1$ for every real number $a$, except zero

And that zero to the power of any real number is equal to zero.

$0^a = 0$ for every real number $a$, except zero

What happens if we observe these two statements together?

Let’s take a look at a graph of a function $f(x) = x^x$. If we observe limits that go from the right and from the left to the zero, we get the same value – 1.

$\lim_{x \to 0} + x^x = 1 \lim_{x \to 0} – x^x = 1$

Does this mean that we can conclude that $x^x = 1$? This is true only on the surface. There is a lot more than a simple real number line. But you also have complex numbers, so you have to include imaginary axis. Now there are a lot of different ways of approaching the origin. This is where the statement fails, because there are many different limits in complex plane. A generalization of this theorem is that any two bodies in R3 that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other.

Zeno of Elea was a Greek philosopher and mathematician, whom Aristotle called the inventor of dialect. He is especially known for his paradoxes that contributed to the development of logic.

Image credit: Colm Kelleher on TED-Ed

The first paradox we’ll explain goes like this:

A runner wants to run a certain distance, for example 500 meters. But if he wants to reach that distance he must first reach the first 250 meters, and to reach that he must first run 125 meters. Since space is infinitely divisible, we can repeat that forever. Thus, the runner has to reach an infinite number of midpoints in a finite time which is impossible. This leads us to thinking that anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is an illusion.

One assumption in this paradox is that space is infinitely divisible. Another is that it is impossible to reach infinite number of points in a finite time. But why time couldn’t be infinitely divided to?

$Distance = speed * time$

$Time = distance / speed$

If you divide an infinitesimal distance by a finite speed, you get an infinitesimal time.

This means that time can be infinitely divided; that’s why infinitesimal distance can be travelled in an infinitesimal time, and the times add up to a finite time in the same way that the infinite number of distances adds up to a finite distance. This is where this paradox fails.

## Achilles and the tortoise

Image credit: Affordable Housing Institute

Let’s say that, for some reason, Achilles and the tortoise are racing. The tortoise, if of course a lot slower that the Achilles and for that reason she has a 100 meters head start.

Achilles sprints 100 meters and catches up to a tortoise. By that time tortoise also moved, but only 10 meters.
Now Achilles sprints another 10 meters to catch up to a tortoise. But now, tortoise moved another meter. And this keeps happening. The paradox is that the Achilles can never catch the tortoise which is nonsense because he obviously has to.

Image credit: IB Maths Resources

So how can an infinite process end?

If we have the distance of 2 meters and we’re constantly dividing it by two we get:

$S = 1 + (\frac{1}{2}) + (\frac{1}{4}) + (\frac{1}{8}) + …$

If we divide whole sum by 2:

$\frac{S}{2} = (\frac{1}{2}) + (\frac{1}{4}) + (\frac{1}{8}) + (\frac{1}{16}) + …$
Now let’s subtract those two sums:

$\frac{S}{2} = 1$
Which means that $S = 2$.

This is a well behaved sum. At infinity it is equal to a real number exactly. But when we’re talking about infinity, we know that it doesn’t have the last step. So how can something without a last step be completed? That is the paradox. But even though this is an infinite process, it can be completed. This is the same problem as drawing a length of an irrational number. How can you draw accurately segment whose length has infinitely many decimals? But for example, if you have a right triangle with length of both of its legs 1, its hypotenuse will be $\sqrt{2}$, which is an irrational number.

Back in the 1924, two mathematicians, Stefan Banach and Alfred Tarski stated that it is actually possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original.

Later on, number of pieces was reduced to five by Julia Robinson, which is also the minimal number of pieces required for this theorem.

You may thing we’re talking nonsense and this can’t be true, but according to Banach Tarski theorem it is completely possible.

What if you broke a glass ball? It is clear that you couldn’t just rearrange broken pieces and get two new balls identical to the original one.

Banach Tarski theorem works because it is not a physical sphere we’re talking about, but rather a mathematical sphere- an infinite collection of points.

Now let’s take a look at a set of natural numbers from one to infinity.

$N = {1, 2, 3, 4…}$
This set is obviously infinitely large. Now let’s take a look at a set which contains only even numbers from the set of natural numbers.

$E = {2, 4, 6, 8…}$
Also, the set of odd numbers:

$O = {1, 3, 5, 7…}$

At the first look you might say that there are half as many numbers in sets that contain only even or odd numbers as there are in the original set of natural numbers since we only took every other number, but every even number is made by multiplying every number of N by two, and every odd number is made by multiplying every number of N by two and subtracting 1. Therefore, sets of even and odd numbers are both infinitely large.

Let’s thing about what we just did. We created two infinitely large sets from one. This is why it is possible to break one sphere into two spheres identical to the original.

A generalization of this theorem is that any two bodies in R3 that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other.