Rules for Addition

Addition needs structure, not memorisation. The four rules below — commutativity, associativity, identity, inverse — are the entire grammar of integer addition. Every "trick" you ever learned for adding signed numbers is just one of these four rules in disguise, applied step by step.

The four laws, said plainly

Commutative. Order does not matter: $a + b = b + a$. Walking 3 steps east then 5 east lands you in the same place as 5 east then 3 east.

Associative. Grouping does not matter: $(a + b) + c = a + (b + c)$. You can pair off whichever two are easiest first.

Identity. Adding zero changes nothing: $a + 0 = a$. Zero is the "do-nothing" identity element of addition.

Inverse. Every integer has a partner that cancels it out: $a + (-a) = 0$. This is what makes subtraction possible in $\mathbb{Z}$.

The number line is the proof picture

Think of addition as walking on the number line. A positive integer is a step to the right; a negative integer is a step to the left. The marker in the widget below lets you see this directly — place $a$ on the line and read off where $a + 5$ or $a + (-3)$ lands.

(Set a = 3 in the widget, then mentally add $-7$. You should land at $-4$. Now use the line to find $-5 + 8$.)

Same sign vs different sign — one rule, two cases

For integers with the same sign, add the absolute values and keep the common sign: $(-4) + (-7) = -(4 + 7) = -11$. Geometrically, you are walking the same direction twice, so distances add.

For integers with different signs, the two walks partially cancel. Subtract the smaller absolute value from the larger and keep the sign of whichever was bigger: $(-8) + 5 = -(8 - 5) = -3$.

Subtraction is addition in disguise. Because every integer has an additive inverse, $a - b$ is just $a + (-b)$. There is really only one operation here.

Try it

• Place $a = -6$ in the widget. Predict where $a + 9$ lands before checking.
• Use the associative law to compute $(-17) + 8 + 17 + (-8)$ in your head. Pair inverses first.
• Rewrite $12 - (-5)$ as an addition, then evaluate.
• Is there an integer $x$ with $x + 7 = 0$? Why?

Pause: which of the four laws would you need if you were proving that $a + b + c = c + a + b$ for all integers? List them before scrolling on.

Try it in code

A trap to watch for

Many beginners read the symbol $-a$ as "negative a" and assume it is automatically a negative number. It is not. The symbol $-a$ means "the additive inverse of $a$." If $a = -5$, then $-a = -(-5) = 5$, which is positive. The minus sign in $-a$ is an operator ("flip the sign of $a$"), not a label that says "this is negative." The fix: read $-a$ out loud as "the opposite of $a$" until the reflex sticks.

What you now know

You can carry out any integer addition by combining the four laws — commute to reorder, associate to regroup, use the identity to ignore zeros, use the inverse to convert subtraction. The same four laws will reappear in the next section for multiplication, plus one new partner: the distributive law that ties multiplication and addition together.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 1, §2 — the canonical commutativity/associativity/identity/inverse exposition for integer addition.
• Stewart, I. and Tall, D. (2015). The Foundations of Mathematics, 2nd ed. Oxford. Chapter 4 — the four laws presented as the definition of an abelian group, with the integers as the prototype example.
• Niven, I., Zuckerman, H., and Montgomery, H. (1991). An Introduction to the Theory of Numbers, 5th ed. Wiley. Chapter 1 — integer addition treated rigorously from the Peano axioms.