The Laws of Arithmetic

Addition and multiplication are just counting, twice

Remember the successor machine from What Is a Number? — every natural number is $0$ wearing some $S$'s, and "add $b$" means "apply the successor $b$ times." That's addition: counting on, with a head start. Same fucking move, one floor down.

Multiplication is the same trick stacked once more. $a \times b$ means add $a$ to itself $b$ times — repeated addition, the way addition was repeated succession. Three groups of four:

$3 \times 4 = 4 + 4 + 4 = 12.$

So we have two operations, both born from counting. Now here's the thing the Federation of Boring Textbook Authors never tells you: these operations come with guarantees — properties so reliable we build the entire skyscraper of algebra on them. And every one of those guarantees can be proven from the counting picture. Let me show you what they've been hiding.

Commutativity: the rectangle that ends the argument

The claim is $a + b = b + a$ and $a \times b = b \times a$. Order doesn't matter. Obvious? It is NOT obvious — it's a damn theorem, and here's the proof that makes it undeniable.

Lay out $a \times b$ as a rectangle of dots: $a$ rows, $b$ columns.

$\begin{matrix} \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \end{matrix}$

Count it by rows: $3$ rows of $4$, that's $3 \times 4 = 12$. Now turn your head ninety degrees and count the SAME dots by columns: $4$ columns of $3$, that's $4 \times 3 = 12$. Same dots, same count, two readings. That's the whole proof — $a \times b = b \times a$ because rotating a rectangle doesn't change how many dots it has. For addition, line up $a$ dots then $b$ dots, walk the line backwards: same total. Commutativity is a fact about geometry, not a vibe. I have this proof laminated on the lab wall next to a portrait of Peano looking smug.

Associativity: grouping is free

The claim is $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$. The parentheses say which addition to do first, and associativity says the answer doesn't care.

For addition this is just "a pile of dots is a pile of dots" — group the first two, then add the third, or group the last two first; you're counting the same heap. For multiplication, picture a box of dots: $a$ deep, $b$ wide, $c$ tall. Slice it into $c$ layers of $a \times b$ dots each — that's $(a \times b) \times c$. Or slice it into $a$ walls of $b \times c$ — that's $a \times (b \times c)$. Same box, same dots. The grouping is bookkeeping; the box doesn't move.

Distributivity: the bridge, the engine, the whole damn point

Here is the law that matters more than all the others combined:

$a \times (b + c) = a \times b + a \times c.$

This is the ONLY law that mentions both $+$ and $\times$ in the same breath — it's the bridge between them. And it has a picture so clean it should be straight-up illegal to teach algebra without it. Take a rectangle $a$ tall and $(b + c)$ wide, then chop it with a vertical line into a $b$-wide piece and a $c$-wide piece:

$\underbrace{a \times (b+c)}_{\text{whole slab}} = \underbrace{a \times b}_{\text{left piece}} + \underbrace{a \times c}_{\text{right piece}}.$

The whole area equals the sum of the two pieces. That's it. That's distributivity — it's just "the area of a rectangle splits when you split a side."

Why do I care so much? Because distributivity is the engine of all of algebra. FOIL? Distributivity twice. Factoring? Distributivity run backwards. Combining like terms, expanding $(x+3)(x-5)$, multiplying matrices forty lessons from now — all distributivity. Every single damn one of them. File this away: it detonates everywhere.

Subtraction and division are not new operations

Now I'm going to take two things you thought were operations and demote them to questions.

What is $7 - 3$? It is the answer to: "what do I add to $3$ to get $7$?" Subtraction is addition asked backwards. $7 - 3 = 4$ because $3 + 4 = 7$. There's no new machinery — it's the inverse question for $+$.

What is $12 \div 3$? It is the answer to: "what do I multiply $3$ by to get $12$?" Division is multiplication asked backwards. $12 \div 3 = 4$ because $3 \times 4 = 12$.

This "inverse question" idea is one of the deepest in the course, and it's about to pay off violently. The negatives (next lesson) exist only to make subtraction always answerable. Fractions exist only to make division always answerable. Hold onto this — it's the master move, and I will beat it into you like contraband chalk on a blackboard until it sticks.

Order of operations: a convention, not a law

Last thing, and I need you to actually hear me: order of operations is not a law of nature. When you see $2 + 3 \times 4$, the answer is $14$, not $20$ — but only because we all agreed that $\times$ binds tighter than $+$. It's grammar, not arithmetic. The Babylonians could've agreed otherwise and the universe wouldn't have blinked.

The convention — multiplication before addition, parentheses override everything — exists so we can write $2 + 3 \times 4$ without drowning in brackets. It saves ink. That's the whole reason. Don't confuse a punctuation rule (convention) with the commutative law (a theorem). One is a fact about reality; the other is a fact about us. Mixing them up is the kind of bullshit that gets theorems wrong in the middle of a proof.

You now know WHY arithmetic works, not just that it does. Go run the gauntlet.