An equals sign is magic; like wishes or curses.

Two things being equal in mathematics is an incredibly strong statement. Again it’s important to remember that math is all about precision. Thus when we say something like “$x = 4$”, this is meant to be a precise statement. This might seem trivial or silly to point out, but the human brain is impressive at building equivalence, and in mathematics there is a very important difference between equality and equivalence.

An equivalence is where you can convert one thing to another through some outside mechanism. For example, we could say that the letters of the alphabet and the numbers one through twenty six are equivalent by assigning each letter the corresponding number in sequence (eg A = 1, B = 2, C = 3, ..., Z = 26). But we would not say the letters of the alphabet and the numbers one through twenty six are equal.

In mathematics equality means they are absolutely, and in every way, the exact same thing. This is sort of like the difference between ‘congruence’ and ‘similarity’ in geometry. Things can be similar (equivalent) because they are somehow “basically the same thing”, but being congruent means that two shapes are exactly the same, which is what equality requires. This means that in general it is very difficult to claim two things are equal, but if you already know that two things are equal, it allows you to do a lot of useful things with that knowledge.

^[1] (Generally speaking it’s very difficult to show things are equal, but math teachers often skip that part by merely providing the equality to the students. Although understandable, this often hides the incredibly rigorous requirements required to show that two things are equal, and thereby hides how powerful the equality sign really is from the students.) In this class we will rarely deduce two things are equal (but it will happen), but we will often have circumstances where we know things are equal, or we define something to be equal to something else (this is how variable substitution works).

One of the chief reasons to draw this distinction is due to the prevalence of a technique in mathematics which is used so frequently that students tend to develop a reflexive urge to do it whenever they can. This is the “setting an expression equal to zero” technique. This is a very power technique, and one that we will use in this course, but it’s also important to know why and when it applies. After all, as we have been saying in this section, claiming two things are equal is a big deal in mathematics!

It’s important to realize that by introducing an equal sign yourself you are making an incredibly powerful statement. This means that you should never set something equal to zero without being able to explain why doing so is allowed and useful. More specifically ‘because that’s how I get the solution’ is not a valid response when asked why you are setting something equal to zero. You should be able to explain why the equation or expression being zero represents the solution. Consider the following example and explanation to see what we mean.