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