MiloQuanticaPrealgebraOpen in the course

Prealgebra · Lesson 1.1

Why Begin with Arithmetic?

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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 4×19×25? 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: 4×25=100, and then 100×19=1900. You never had to face 19×25 head-on. That is the whole game: spotting the structure that turns a hard-looking problem into an easy one.

Start with the word Prealgebra. People don't fully agree on what it means. We use it to mean the material that comes between arithmetic and algebra.

Arithmetic is adding, subtracting, multiplying, and dividing, plus a few more operations like squaring a number or taking a square root. You already know most of it. 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 harder questions (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) need more than arithmetic.

They need algebra. Algebra is arithmetic written with letters, where a letter stands for a number. That letter can be any number, so one algebraic statement covers every case instead of just one.

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 7×99 is to write it as 7×(100−1). What do you get?
Show a hint
  • 7×100 is easy. Now take away 7×1.
Show the full solution
Distribute across the subtraction: 7×(100−1)=7×100−7×1=700−7=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.

You're ready now to ask not just how a calculation works, but why it works. If you know why a method works, you can use it on problems that look nothing like the ones where you learned it. So throughout the course, we will 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.