Unit 23: Making Up Rules

Any rule will do

An operation takes two numbers and gives you one back. Adding is one. Multiplying is another.

But nothing says those are the only ones. Define a brand new one:

a ★ b = 2a + b

Then 3 ★ 4 = 6 + 4 = 10. It is a perfectly good operation, and it was invented one sentence ago.

Commutative: does order matter?

An operation is commutative if swapping the two numbers changes nothing.

Adding is: 3 + 5 and 5 + 3 both give 8. Multiplying is too.

Subtracting is not: 3 - 5 is not 5 - 3. Neither is dividing. And neither was ★, as you just found out.

Associative: does grouping matter?

An operation is associative if the brackets make no difference.

(2 + 3) + 4 and 2 + (3 + 4) both give 9. Adding is associative, and so is multiplying.

Subtracting is not: (10 - 5) - 2 is 3, but 10 - (5 - 2) is 7. Which is exactly why brackets exist.

The identity leaves things alone

Every well-behaved operation has an identity — the number that changes nothing.

For adding it is 0, since n + 0 = n. For multiplying it is 1.

And the inverse of a number is whatever takes you back to the identity: -5 undoes 5 under adding, and 1/5 undoes it under multiplying.

Distributive: the one that links them

Multiplying distributes over adding:

3 × (4 + 5) = 3 × 4 + 3 × 5

That is the rule behind every bracket you opened in unit 6, and behind mental arithmetic tricks like 7 × 99 = 7 × 100 - 7.

Read the definition, then obey it

Every made-up operation comes with its own rule, and the only skill is following it exactly.

a ♠ b = a^2 - b. So 5 ♠ 3 is 25 - 3 = 22.

Do not let the strange symbol hurry you. Write out the definition with the numbers substituted in, then work it out.

Solving with a strange operation

You can be given the answer and asked for the input.

With a ♠ b = a^2 - b, suppose 4 ♠ x = 10. Then 16 - x = 10, so x = 6.

Substitute, then it is an ordinary unit 3 equation. The symbol was never the difficult part.

The whole operation on one grid

For an operation on only a few values, you can write down every answer in a grid — row for the first input, column for the second.

Then the whole operation is visible at once, and its properties can be seen rather than worked out.

If the grid is symmetric about its diagonal, the operation is commutative. That is a very satisfying thing to be able to check by looking.

Adding on a clock

Take the numbers 0, 1, 2, 3 and "add, then take the remainder after dividing by 4".

2 + 3 = 5, and 5 leaves remainder 1. So 2 + 3 = 1 in this system.

It is a complete, well-behaved operation on just four values — and it is exactly how a clock face works, which is why it feels familiar rather than strange.

Finding the identity in a table

The identity is the value whose row is an exact copy of the headings — it changes nothing.

In clock-4 addition that is 0, because 0 + n = n every time.

And each value has an inverse: the partner that takes you back to the identity. 1 pairs with 3, since 1 + 3 = 0 on a clock of four.

Closed, or not

An operation is closed on a set if the answer always stays inside it.

Adding two whole numbers gives a whole number — closed. Subtracting them does not: 3 - 5 escapes to the negatives.

That escape is exactly what forced the invention of negative numbers in unit 8. Closure is not a technicality; it is why new numbers get invented.

A checklist for any new operation

When you meet an operation you have never seen, run through four questions.

Is it closed? Is it commutative — does order matter? Is it associative — do brackets matter? Is there an identity?

Each answer is a quick test with small numbers, and together they tell you almost everything about how the operation behaves.

Three properties a relation might have

A relation can be reflexive — everything relates to itself, like a = a.

Symmetric — if a relates to b then b relates to a, like "is a sibling of".

Transitive — if a relates to b and b to c, then a to c, like <.

Equality has all three. < has only the last one, since 3 < 3 is false and 3 < 5 says nothing about 5 < 3.

Divisibility is a relation too

"Divides into" is a relation on the whole numbers, and it has an interesting mix of properties.

Every number divides itself, so it is reflexive. If a divides b and b divides c, then a divides c — transitive.

But it is not symmetric: 3 divides 12, and 12 certainly does not divide 3.

The order of operations

When several operations meet, there is an agreed order: brackets, then powers, then times and divide, then plus and minus.

So 2 + 3 × 4 is 14, not 20.

This is not a law of nature. It is a convention everyone agreed on so that one written expression means one thing.

A relation compares

A relation is not an operation. It does not give you a number — it gives you a *yes or no*.

< is a relation: 3 < 5 is true. So is =, and "is a factor of".

Some relations are symmetric — if a = b then b = a. < is not: 3 < 5 says nothing good about 5 < 3.