Why does algebra work?

When I was in middle school, my teachers taught me the mechanics of algebra — try to isolate to variable by adding, subtracting, multiplying, and dividing both sides. And then in later years, my teachers expanded that toolbox to include stuff like logarithms and whatnot.

But I think that it’s easy to get lost in the details and overlook one crucial piece of logic. It’s the basis of all algebra, but it’s a kind of hidden fact that becomes important in more complicated problems.

Let’s say we have the equation $2x + 1 = 5$, and we’re trying to find all possible values of $x$.

By instinct, one might jump directly into subtracting $1$ from both sides and dividing both sides by $2$:

$$\begin{gather*} 2x + 1 = 5 \\ 2x + 1 – 1 = 5 – 1 \\ 2x = 4 \\ \frac{2x}{2} = \frac{4}{2} \\ x = 2 \\ \end{gather*}$$

But here’s why the algebra works: these clauses are linked by “if and only if”s.

$2x + 1$ equals $5$ if and only if $2x$ equals $4$, which is true if and only if $x = 2$. Therefore, we can conclude that $2x + 1 = 5$ if and only if $x = 2$.

One way to look at it is that we’re replacing the original equation with equivalent forms. $2x + 1 = 5$ says exactly the same information as $x = 2$, just written slightly differently. That’s really the whole idea of algebra (and logic) — replacing one statement with another exactly equivalent until it’s obvious what your result is.

That’s why, in general, it’s not a productive idea to multiply both sides by zero: $2x + 1 = 5$ is not exactly equivalent to $0 \cdot (2x + 1) = 0 \cdot 5$ because the latter is true no matter what $x$ is. Therefore, we’ve lost some information in multiplying by zero.

“If and only if” is abbreviated $\Leftrightarrow$, and I wish that I was taught algebra with that notation. It’d be easier to remember that $\sqrt{x^2} = 1$ is not exactly equivalent to $x = 1$ (it can be $-1$ too!) if I had that symbol to remind me.

January 31, 2013, 11:15am by Casey