Why Arithmetic Comes Before Algebra

You already know how to add, subtract, multiply, and divide. Honestly, you could probably solve a fair number of the problems in this course on sight. So why would we open an entire course with a chapter on arithmetic?

Problem

You said you already know this stuff, so here is a quick one, no pencil: what is $4 \times 19 \times 25$? Look for a friendly pair before you multiply.

Show a hint

Which two of $4$, $19$, $25$ multiply to a round number?

Show the full solution

You may multiply in any order, so reach for the pair that makes life easy: $4 \times 25 = 100$, and then $100 \times 19 = \boxed{1900}$. You never had to face $19 \times 25$ head-on. That is the whole game: spotting the structure that turns a hard-looking problem into an easy one.

Start with the name on the door, Prealgebra. People don't fully agree on what the word means. We think of it as the bridge between arithmetic and algebra.

Arithmetic is the world of adding, subtracting, multiplying, and dividing, plus a few bolder moves like squaring a number or taking a square root. You've met most of it already. The hardest part is usually a word problem. “If Maya has 6 stickers and Theo has 9, how many do they have together?” As you get older the numbers grow larger, but problems like that never really get harder.

Arithmetic is perfect for counting stickers. But the moment a question gets richer (forecasting a satellite's orbit, modeling how a rumor spreads through a school, or counting the routes a single message can take across the internet), we need a bigger toolbox.

That toolbox is algebra. Algebra is the language of all higher mathematics. It takes the ideas you already trust from arithmetic and makes them general, so they work far beyond the one problem where you first met them.

Here's a small taste. With arithmetic, you can check a single case.

Algebra hands you the far more powerful statement.

Problem

Now use that same splitting idea. A fast way to compute $7 \times 99$ is to write it as $7 \times (100 - 1)$. What do you get?

Show a hint

$7 \times 100$ is easy. Now take away $7 \times 1$.

Show the full solution

Distribute across the subtraction: $7 \times (100 - 1) = 7 \times 100 - 7 \times 1 = 700 - 7 = \boxed{693}$. The same law that looked abstract with letters just saved you from multiplying by $99$.

And in more advanced mathematics, a, b, and c might not even be ordinary numbers, and “+” and “×” might not be the addition and multiplication you know today. But we're getting a little ahead of ourselves.

So our first goal is simple. We lay down the rules of arithmetic carefully, and show you why each one is true.

That word, why, is the heart of this course. You're ready now to ask not just how a calculation works, but why it works. That's the difference that lets you bend a familiar technique to crack an unfamiliar problem. So throughout, we'll rarely just tell you how something works; we'll show you why.

You'll be able to explain (really explain, not just compute) each of these.

You'll know these not because you memorized a rule or tapped buttons on a calculator, but because you understand the mathematics underneath them.

Why Begin with Arithmetic?