Casey Chu

- Why does algebra work?
- In middle school, our teachers taught us the mechanics of algebra: isolate the variable by adding, subtracting, multiplying, and dividing on both sides. And then in later years, our toolboxes expanded to include logarithms, exponentials and other, fancier operations.
- However, I think it’s easy to get lost in the mechanics and not realize
*why*it all works out. It’s one crucial piece of logic, the basis of all algebra, but it’s easily overlooked, despite becoming 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 $x$. We solve it by subtracting $1$ from both sides and then dividing both sides by $2$:
- $2x + 1 = 5,$
- $2x + 1 - 1 = 5 - 1,$
- $2x = 4,$
- $\tfrac{2x}{2} = \tfrac{4}{2},$
- $x=2.$
- The key to this working? 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 $x = 2$. - That is, one way to look at it is that we’re replacing the original equation with equivalent forms. $2x + 1 = 5$ is true exactly in all the cases that $x = 2$ is true. 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.
- Another tricky case where remembering this concept helps is when squaring numbers. Remember that $x^2 = a$ is not exactly equivalent to $x = \sqrt{a}$, since $x$ can be $-\sqrt{a}$ too.
- “If and only if” is abbreviated $\Leftrightarrow$, and I think using it generously can help keep algebra tidy and correct.