**1 :**What is so special about equal signs?

This section describes the very special and often overlooked virtues of the ‘equals sign’. It also includes when and why you should “set something equal to zero” which is often overused or used incorrectly.

#### 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 “”, 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.

*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.

When we think about this question we might consider what it means to ‘hit the
ground’. Specifically, what would the height of the ball be when it hit the ground?
Since is telling us how high off the ground the ball is, then when it hits the ground
would be zero. Thus we would want to calculate the answer to the equation; Notice
that we *appear* to be “setting to zero”, but that’s not what we *actually did*,
or at least, there was a line of thinking that led up to getting . The zero
didn’t come from the nature of the function we are looking at, it came from
the context of the model; we knew that the ball hits the ground when the
height is zero, so we knew the height of the ball at the time of interest should
be zero and so we ‘plugged it in’. Meaning that the value *happened* to be
zero, and that’s why we ‘set it equal to zero’, not because of the equation
itself.