**1 :**What is so clever about multiplying by one?

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

*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 ”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.

#### Adding Zero... Cleverly

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.

Thus adding and subtracting (aka “adding zero cleverly”) ends up making the factoring much easier to see and compute.

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...

#### 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 then either or **must** be
zero.

Unfortunately we can quickly see that we don’t. If you want to try and claim that
one of either or *must* be a specific number, say (again, feel free to use any
number you want here), we could easily come up with a pair of numbers where
neither nor are . In this case we could choose and , and neither of those are
.

This is because we could let or be any number we want, and force the
other to make the computation correct because of how the real numbers
work. If we fix the value as *any* number we want (other than zero), then
making we have a valid pair of numbers so that . This means that we can’t
really figure out anything about or without knowing at least one of the
two.

But in that statement of we can see why zero is a special case. If is zero then that fraction fails to exist. If is zero, then that fraction can’t work for any value of (meaning that any value of will still not result in ). So, in fact, a product of numbers is zero means one of those numbers is zero as well as the fact that any (finite) number times zero is zero.

**3 :**We call zero the annihilator of real numbers because...

*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.