Chapter 1.2 — Properties of Rational Numbers

Closed Under Three Operations, Not Four

A set is closed under an operation if combining any two members of the set by that operation always produces another member of the same set.

• Addition: 3/4 + (−5/6) = (9 − 10)/12 = −1/12 — still a rational number.
• Subtraction: 2/5 − (−4/10) = (4 + 4)/10 = 8/10 — still rational.
• Multiplication: 1/5 × 7/10 = 7/50 — still rational.
• Division fails: 5 ÷ 0 is undefined, so division is the one operation under which rational numbers are not closed.

Does Order Matter? Commutativity

An operation is commutative when swapping the order of the two numbers leaves the answer unchanged: a ★ b = b ★ a.

• Addition holds: 3/5 + (−5/9) = 2/45, and (−5/9) + 3/5 = 2/45 — identical.
• Multiplication holds: 5/6 × 7/8 = 35/48, and 7/8 × 5/6 = 35/48 — identical.
• Subtraction fails: 7/6 − 5/9 = 11/18, but 5/9 − 7/6 = −11/18 — opposite signs, so order matters.
• Division fails: 5/8 ÷ 11/13 = 65/88, but 11/13 ÷ 5/8 = 88/65 — the two results are reciprocals of each other, not equal.

Does Grouping Matter? Associativity

An operation is associative when regrouping three numbers with brackets doesn't change the result: (a ★ b) ★ c = a ★ (b ★ c). Testing this with 3/5, 4/7 and 8/9 gives a clean pattern:

• Addition holds: both groupings of 3/5, 4/7 and 8/9 simplify to 649/315.
• Multiplication holds: both groupings simplify to 96/315.
• Subtraction fails: (3/5 − 4/7) − 8/9 = −271/315, while 3/5 − (4/7 − 8/9) = 289/315 — different values.
• Division fails: the two possible groupings give 168/180 and 189/160 — again different.

The pattern to remember: addition and multiplication are always well-behaved (closed, commutative, associative); subtraction and division are the two operations that repeatedly break these rules.

The Numbers That Change Nothing

An identity element leaves a number completely unchanged when the operation is applied to it.

For a = 4/9: 4/9 + 0 = 4/9 confirms 0 as the additive identity, and 4/9 × 1 = 4/9 confirms 1 as the multiplicative identity. These two numbers are the identities for every rational number, not just this example.

Undoing an Operation: Inverses

The additive inverse of a is −a, since a + (−a) = 0; for 4/9, the additive inverse is −4/9. Every rational number has exactly one additive inverse, no exceptions.

The multiplicative inverse (reciprocal) of a is 1/a, since a × 1/a = 1 — but this only works when a ≠ 0, because 1/0 is undefined. So every rational number except zero has a multiplicative inverse; zero is the single exception in the entire set.

Distributive Law

The distributive law links multiplication to addition and subtraction: a × (b + c) = a×b + a×c, and equally a × (b − c) = a×b − a×c.

Checking with a = 2/3, b = 1/4, c = 5/7: the left side, 2/3 × (1/4 + 5/7) = 2/3 × 27/28 = 54/84. The right side, (2/3 × 1/4) + (2/3 × 5/7) = 2/12 + 10/21 = 14/84 + 40/84 = 54/84. Both sides match, and the same equality holds when + is replaced by −.

Quick Reference — What Holds and What Doesn't

• Closure: yes for +, −, ×  |  no for ÷ (division by 0).
• Commutative: yes for +, ×  |  no for −, ÷.
• Associative: yes for +, ×  |  no for −, ÷.
• Identity: 0 for addition, 1 for multiplication.
• Inverse: every number has an additive inverse; every number except 0 has a multiplicative inverse.
• Distributive: multiplication distributes over both addition and subtraction.

Verifying a Property Directly

Naming a property is one skill; confirming it holds for specific numbers by working out both sides of an equation independently is another, and it's the more reliable way to be sure you actually understand it. For example: check that 5/4 + (−1/2 + −3/2) = (5/4 + −1/2) + −3/2.

Left side: −1/2 + −3/2 = −2, so 5/4 + (−2) = 5/4 − 8/4 = −3/4.
Right side: 5/4 + −1/2 = 5/4 − 2/4 = 3/4, so 3/4 + −3/2 = 3/4 − 6/4 = −3/4.
Both sides equal −3/4, confirming the associative property of addition. Computing each side separately and then comparing, rather than assuming the property applies, is the method to use whenever a question asks you to verify rather than simply state a property.

Why the Failing Cases Matter as Much as the Holding Ones

It's tempting to focus only on what works, but knowing exactly where a property breaks down is what separates real understanding from rote memorisation of the list. Being able to produce a quick counterexample — like 5/8 ÷ 11/13 ≠ 11/13 ÷ 5/8 — on demand shows the same understanding as restating the property in words, only more convincingly. Keep at least one concrete counterexample ready for each of subtraction and division, since those are the two operations where these properties consistently fail.

Try It Yourself

Before moving to the exercise, test whether you can apply — not just recite — these rules on a fresh set of numbers. Take a = 1/2, b = 2/3, c = −3/4 and check the distributive law over subtraction: a × (b − c) should equal a×b − a×c.

Left side: b − c = 2/3 − (−3/4) = 8/12 + 9/12 = 17/12, so a × (b − c) = 1/2 × 17/12 = 17/24.
Right side: a×b = 1/2 × 2/3 = 1/3 = 8/24, and a×c = 1/2 × (−3/4) = −3/8 = −9/24, so a×b − a×c = 8/24 − (−9/24) = 17/24.
Both sides agree, exactly as the property predicts — and working through one extra example like this in your own numbers (rather than the textbook's) is a reliable way to confirm you understand a property rather than having memorised one specific case of it. If you can repeat this check with numbers you pick yourself, in under a minute, you're genuinely ready for the exercise. Try the same check with a×(b+c) using three of your own fractions before moving on — it takes only a few lines of working and settles any lingering doubt far better than re-reading the rule a second time.

Beyond This Lesson

These six properties are exactly what Exercise 1.1 asks you to identify and apply in mixed, unlabelled questions — that is the real test of whether you understand them or just memorised the list. The same properties reappear when you start simplifying algebraic expressions and solving linear equations later this year, and again in polynomials in Class 9.