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This section describes the very special and often overlooked virtue of the numbers Zero and One.

Perhaps the most common and useful techniques in all levels of mathematics can be boiled down to “add zero or multiply by 1; cleverly”. This is because zero and one are incredibly special numbers in mathematics. Here we will give a very brief explanation of what we mean by being ‘clever’ and a few things about what makes these numbers special.

#### Multiplying by One... Cleverly

The number ‘one’ has a special role in mathematics, which is one you almost certainly learned more than a decade ago as the simple rule that “anything times one is itself”. It turns out that this seemingly simple rule can be extremely useful when we combine it with the other rule you probably learned long ago; that ‘anything divided by itself is one’.

This allows us to multiply a term by $1$ ”cleverly” in the sense that we choose something useful to multiply and divide by simultaneously.

You’ve actually already used this technique when finding common denominators for fractions, but it turns out that this is a fact we will abuse over and over to help us manipulate difficult functions and expressions. Thus we will often multiply and divide by something so that we can simplify a term.

1 : What is so clever about multiplying by one?
We multiply cleverly to make it seem like we aren’t wasteing our time. We multiply cleverly because it helps us solve something somehow. We multiply cleverly in order to introduce a factor that will cancel or otherwise help evaluate or simplify the expression we are working on. We multiply cleverly because we were told to.

Zero has two different primary roles that we will discuss here. The first is adding zero in a way that can help with simplifying a problem we have. The second is it’s role in multiplication.

First off we will consider the “adding zero cleverly”. The key aspect of zero is that “anything plus zero is itself.” This is a little harder to see currently when it will be useful, but it will crop up a lot later on and becomes a more and more useful tool over time. Consider the following example of factoring, which will be covered extensively in our exploration of polynomials.

As mentioned, further examples of “adding zero cleverly” will be seen as we explore future topics (and will become more and more prevalent if you move into higher level math courses, like calculus).

2 : We add zero cleverly so that...
We can cancel it back out and not change anything. Because we are told to. We don’t, it just generates extra work. We can introduce a key needed term to factor or otherwise simplify an expression into a more workable/usable form without changing the expression.

#### Zero: The Annihilator of Reality!

The other major exploit we use with zero centers around its role in multiplication. We observe that zero is incredibly special with multiplication; specifically that any (finite) number times zero is zero (math people have a special name for this too, zero is called the “annihilator of the real numbers”).

The key thing here though, is that zero is the only number that does this. So what we will actually exploit is the following: if we know $a \cdot b = 0$ then either $a$ or $b$ must be zero.

3 : We call zero the annihilator of real numbers because...
It’s basically a terminator. It lets us factor things. It doesn’t change a value if you add it to something. It is the only number such that; if you multiply by it, it annihilates the value. In essence, if the product of two numbers is zero, then one of those numbers must be zero.