Algebra appreciation

We take algebra for granted, at least those who still remember it. But it’s not as obvious as we think (as those who don’t remember it can attest.) Consider Boolean algebra: it’s very similar to ordinary algebra yet we find it highly surprising (Boolean algebra is basically just algebra for logic). Boring old algebra is similarly surprising when you think about it.

To be precise, I’m using algebra here to refer to elementary algebra. It’s the one we’re all familiar with: solving an equation for a variable (x).

Algebra lets us solve problems that would be extremely difficult without. Easy equations can be solved trivially in your head: for example, say you have $8 and you give $3 to your son and the rest to your daughter. Obviously, you gave her $5, but you could also write out the equation and solve it: x+3=8Subtracting 3 from both sides we get x=5

But when it gets more complicated, it’s much more convenient to use the rules of algebra to manipulate the equation to isolate x: for example, say you’re at an amusement park and the pricing is as follows. $12 to get in and $3 per ride. You have $30 and a 50% off coupon. How many rides can you afford? You could count it out, or write out the equation and solve it using algebra: \frac{12+3x}{2}=30This is tedious to solve without algebra. But it’s very easy to solve with algebra: multiply through by 2, then subtract 12, and divide by 3, yielding x=16.

As you can imagine, when the equations get more complicated it becomes practically impossible to solve them without using algebra. Algebra provides a set of tools to systematically solve equations. We can even program computers with these rules so that they can solve equations for us algorithmically.

Manipulating equations to solve them is extremely useful, and not intuitively obvious. Without algebra, we would be greatly inconvenienced.