The Equal Sign

I've long wanted to write about the equal sign but I've been putting it off. Now I have an excuse to write about it. It is reported that students don't understand what the equal sign means. I don't blame them because the symbol '=' is overloaded. It is given four meanings in grade-school curriculum.

The first meaning is reduction of computation, as in arithmetics. An example is 2 × 5 + 4 × 3 = 10 + 12 = 22. In this case, the reduction from a formula 2 × 5 + 4 × 3 to a value requires us to compute 2 × 5 = 10 and 4 × 3 = 12 first because of operator precedence, and then compute 10 + 12 = 22. When arithmetics is taught, the equal sign is taken to mean reduction. We kind of gloss over the fact that there is a specific order of reduction, but the order of reduction was never formally taught.

The second meaning of the equal sign, by the time you learn algebra, is constraint solving. You are asked to solve for the value of the variables given a system of constraints, for example, 2x - y = 7 and x + y = 5. The answer is x = 4 and y = 1. Here the equal sign represents a goal, and you are asked to give solution for the goal.

The third meaning of the equal sign, if you haven't flunk out of math class yet, is definition. You would learn in pre-calculus about how to define a function, e.g. f(x) = x² - 3x + 7. The formula x² - 3x + 7 has been assigned a name f(x), and there is some implicit notion of variable scoping and substitution. For example, if you were to compute f(4) - 5, you would first reduce the function f(4) to its definition x² - 3x + 7, and then substitute x for 4, which gives you 4² - 3 × 4 + 7 = 11. Then you compute 11 - 5, which gives you 6. The equal sign here allows you to give a long formula a short name.

Finally, we encounter the fourth meaning in a very subtle way when we talked about constraint solving and function definition. That is, we also use the equal sign to denote a substitution. For example, you can verify that the system of constraints 2x - y = 7 and x + y = 5 yields the solution x = 4 and y = 1 if you substitute x for 4 and y for 1, so that 2 × 4 - 1 = 8 - 1 = 7, and 4 + 1 = 5. And to compute f(4), you will compute x² - 3x + 7 in the context of x = 4 because of the function definition.

All four notions of the equal sign are different, but they all use the same symbol. That's why students get confused. And it takes a computer scientist to tell a mathematician about that!