Adding & Subtracting Integers

Think of positives as money you have and negatives as money you owe.

Same signs: add the amounts and keep the sign. −6 + −4 means you owe 6 then owe 4 more, so you owe 10: −10.

Different signs: subtract the smaller size from the bigger one and keep the sign of whichever was bigger. −9 + 4 means you owe 9 and pay back 4, so you still owe 5: −5.

Subtraction is the shortcut worth memorising. Every subtraction can be rewritten as addition: 5 − 8 becomes 5 + (−8) = −3. Once every problem is addition, you only need the two rules above.

Subtraction is defined as addition of the additive inverse: a − b := a + (−b). This is a definition rather than a separate operation, which is why "keep, change, change" works and why it always works.

Addition of signed numbers has a clean geometric statement. Adding a positive translates right on the number line, adding a negative translates left. Two same-signed values translate the same direction, so magnitudes add and the sign is preserved: |a| + |b|. Opposite signs translate against each other, so magnitudes partially cancel and the result carries the sign of the larger magnitude: |a| − |b| in absolute size.

The structure being used here is that ℤ forms an abelian group under addition: every element has an inverse, addition is commutative and associative. That is exactly what licenses rearranging −7 + 12 − 5 into any order you like, which is a habit worth forming before it is needed for combining like terms.