There are only two operations

You learned four operations as four separate things, each with its own table to memorize. Two of them were a convenient lie.

Start with an honest one. Addition is motion. $a + b$ means “stand at $a$, take $b$ steps.” Positive $b$, step right. Negative $b$, step left.

Walk the queued steps. You start at $-7$; the walker takes a $+3$ step, then a $-4$ step.

Here’s the first collapse. There is no separate “subtraction” machine. To subtract, you add the opposite:

$-7 - (-3)$ looks like two operations fighting. It isn’t. Rewrite it: $-7 + 3$. Subtracting a negative is adding a positive: two left-facing arrows cancelling into one right-facing one. The number line above already showed you this: a $-4$ step and “subtract $4$” are the identical motion.

The other operation that’s secretly pure motion: multiplying by $-1$ is a reflection across zero. It resizes nothing. It just flips the side.

Press $\times(-1)$ a few times and watch the counter. After an even number of flips you’re back on the positive side; after an odd number you’re negative. The sign of a product is nothing more than this parity: count the negative factors, even means positive, odd means negative.

Second collapse, exact same shape as the first. There is no separate “division” machine. To divide, you multiply by the reciprocal:

Subtraction undoes addition. Division undoes multiplication. Each pair is one operation paired with its inverse, its undo button.

This finally explains why dividing by zero is banned. $a \div 0$ would mean $a \times \frac{1}{0}$, multiplying by the reciprocal of zero. But every number times zero is zero, so $\frac{1}{0}$ (a number you could multiply zero by to get $1$) simply does not exist.

It’s not “infinity.” It’s “no such number.” The operation has nothing to return, so it’s left undefined.

Here is everything you will ever do to a number, on one card:

• add ⇄ subtract (inverses)
• multiply ⇄ divide (inverses)

Four names, two operations, two undo buttons. Keep this. The next two lessons build expressions out of these operations; the lesson after that solves equations by pressing the undo buttons in the right order. “Apply the inverse” is about to become the most important verb in the course.