## Math As Language: Understanding the Equals Sign

But we can do better. In 1557, Robert Recorde invented the equals sign, written with two parallel lines (=), because “noe 2 thynges, can be moare equalle”.

“2 + 3 = 5″ is much easier to read. Unfortuantely, the meaning of “equals” changes with the context — just ask programmers who have to distinguish =, == and ===.

A “equals” B is a generic conclusion: what

*specific relationship*are we trying to convey?

**Simplification**

I see “2 + 3 = 5″ as “2 + 3 can be simplified to 5″. The equals sign transitions a complex form on the left to an equivalent, simpler form on the right.

**Temporary Assignment**

Statements like “speed = 50″ mean “the speed is 50, for this scenario”. It indicates that we

*decided*this equivalence. We could have picked any value, but chose one useful for the problem at hand.

**Fundamental Connection**

Consider a mathematical truth like $\displaystyle a^2+b^2=c^2$, were a, b, and c are the sides of a right triangle.

I read this equals sign as “must always be equal to” or “can be seen as” because it states a permanent relationship, not a coincidence. The arithmetic of $\displaystyle 3^2+4^2=5^2$ is a simplification; the geometry of $\displaystyle a^2+b^2=c^2$ is a deep mathematical truth.

The formula to add 1 to 100 “can be seen as” geometric rearrangement, combinatorics, averaging, or even list-making.

**Factual Definition**

Statements like

**Constraints**

Here’s a tricky one. We might write

x + y = 5

x – y = 3

which indicates conditions we

*want*to be true. I read this as “x + y should be 5, if possible” and “x – y should be 3, if possible”. If we satisfy the constraints (x=4, y=1), great!

If we can’t meet both goals (x + y = 5; 2x + 2y = 9) then the “equations” could be true individually but not together.

**Example: Demystifying Euler’s Formula**

Untangling the equals sign helped me decode Euler’s formula:

A pedant might say it’s just a simplification and break out the calulus to show it. This isn’t enlightening: there’s a fundamental relationship to discover.

$\displaystyle e^{i\pi}$ refers to the same destination as -1. Two fingers pointing at the same moon.

They are both ways to describe “the other side of the unit circle, 180 degrees away”. -1 walks there, trodding straight through the grass, while $\displaystyle e^{i\pi}$ takes the scenic route and rotates through the imaginary dimension. This works for any point on the circle: rotate there, or move in straight lines.

*that’s*what their equality means. Move beyond a generic equals and find the deeper, specific connection (“simplifies to”, “has been chosen to be”, “refers to the same concept as”).

Happy math.