Taking Away

Subtraction tells two kinds of story. A take-away story: 8 grapes, 3 eaten, 5 remain. A comparing story: 8 grapes and 5 crackers, and the grapes win by 3. Different pictures, one sign — 8 − 3 = 5 and 8 − 5 = 3 both wear the minus. The answer to a subtraction is called the difference.

For a comparing story, nothing needs to go away. Count up instead: start at 5 and climb to 8 — six, seven, eight — three climbs. Counting up finds the gap, and the gap is the difference.

Both jobs are one question in disguise: 8 − 5 asks what number added to 5 gives 8? Taking away removes 5 and sees what remains; comparing measures the gap from 5 up to 8. Either way, the answer completes 5 + ? = 8. Subtraction is addition asked backward.

That is why counting up works, and why it is often the fastest tool. For 13 − 9, climbing from 9 to 13 takes four steps; taking 9 away one at a time takes nine.

On the number line, a − b is the point b steps left of a — and also the distance from b up to a. The two meanings meet in one definition: a − b is the unique number x with b + x = a. Notice what addition never had: order now matters. 8 − 5 and 5 − 8 are different questions, and only one has an answer among the counting numbers.

So subtraction is neither commutative nor associative — (10 − 4) − 3 and 10 − (4 − 3) disagree — and it is not always possible: within the counting numbers, a − b exists only when b is at most a. That only when is a genuine boundary of the number system you own so far.

Eight grapes. Three go away. Count what is left: five. We write 8 − 3 = 5. The minus sign means some went away.

The minus sign has a second job. Eight grapes, five crackers. Match them up, one to one. Three grapes have no partner. 8 − 5 = 3 more grapes.

(ℕ, +, 0) is a commutative monoid with cancellation, but not a group: nothing except 0 has an inverse, so subtraction is a partial operation, defined exactly on pairs with b ≤ a. Computer arithmetic keeps a total stand-in, truncated subtraction or monus: a ∸ b = max(a − b, 0) — primitive recursive and total, at the price of discarding information below zero.

The honest repair is to build ℤ: take pairs (a, b) of naturals — read a minus b — and identify (a, b) with (c, d) whenever a + d = b + c. The quotient is a group, subtraction becomes total, and ℕ embeds via a ↦ (a, 0). At last 6 − 9 has a value: −3.

The pair construction is the Grothendieck group of a commutative monoid, with its universal property: any monoid homomorphism into a group factors uniquely through it, and K(ℕ) ≅ ℤ. Group completion is a functor, left adjoint to the forgetful functor from abelian groups to commutative monoids — subtraction is precisely what the adjunction freely adds.

Monus has its own theory: it makes ℕ a commutative ordered semiring in which a ∸ b is the residual of addition — the largest x with b + x ≤ a — an adjoint characterization again. In recursion theory a ∸ b is a standard primitive recursive function; in resource semantics its refusal to go below zero is the point, not a defect.