MathsICSE Class 8 · Rational Numbers

Properties of Rational Numbers: Closure, Commutative, Associative, Inverses (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

Rational numbers are closed under addition, subtraction and multiplication, and (excluding division by 0) division. Addition and multiplication are commutative and associative; $0$ is the additive identity, $1$ the multiplicative identity, $-a$ the additive inverse of $a$, $\frac{1}{a}$ the multiplicative inverse of $a \neq 0$, and multiplication distributes over addition.

Understand Properties of Rational Numbers

A rational number is a number $\frac{p}{q}$ with $p, q$ integers and $q \neq = 0$ . The CISCE Class VIII curriculum asks you to use the *general form* of an expression to describe these properties, so each one below is written with letters $a, b, c$ standing for any rational numbers.

Closure. Adding, subtracting or multiplying two rational numbers always gives a rational number. Division is closed only if you leave out division by $0$ , because $\frac{a}{0}$ means nothing.

Commutative. $a + b = b + a$ and $a \times b = b \times a$ . Subtraction and division are not commutative: $\frac{2}{3} - \frac{1}{2} = \frac{1}{6}$ but $\frac{1}{2} - \frac{2}{3} = - \frac{1}{6}$ .

Associative. $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$ . Subtraction and division fail this too.

Identities. $a + 0 = a$ (so $0$ is the additive identity) and $a \times 1 = a$ (so $1$ is the multiplicative identity).

Inverses. The additive inverse of $a$ is $- a$ because $a + (- a) = 0$ . The multiplicative inverse (reciprocal) of $a \neq = 0$ is $\frac{1}{a}$ because $a \times \frac{1}{a} = 1$ . The number $0$ has no multiplicative inverse. Only $1$ and $- 1$ are their own multiplicative inverses.

Distributive property. $a \times (b + c) = a \times b + a \times c$ . This is the tool for shortening long sums: factor out the repeated rational number.

A useful habit is to keep the denominator positive: $\frac{5}{- 8} = \frac{- 5}{8}$ . Many mistakes with inverses come from missing this sign.

Properties of Rational Numbers: formulas to remember

Commutative laws: $a + b = b + a, a \times b = b \times a$; Not true for subtraction or division.
Associative laws: $(a + b) + c = a + (b + c), (ab) c = a (b c)$; Lets you regroup to get friendly numbers.
Inverses: $a + (- a) = 0, a \times \frac{1}{a} = 1 (a \neq = 0)$; $0$ has no reciprocal.
Distributive law: $a (b + c) = ab + a c$; Read right to left to factor out a common rational number.

How to solve Properties of Rational Numbers questions

To verify or use a property, choose the right law first, then compute both sides separately.

1Write the property in general form and substitute the given rational numbers.
2Put every fraction over a positive denominator before adding.
3Compute the left side completely in its own line, then the right side in its own line.
4Reduce each result to lowest terms before comparing.
5Conclude with the name of the property used.

Properties of Rational Numbers in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Distributive law as shortcut

Suppose a shop sells notebooks at Rs 36 each and pens at Rs 14 each, and a school buys 25 of each.

Find the total cost using the distributive law.

Try it first, then show the solutionSolution

Total $= 25 \times 36 + 25 \times 14$ .
$= 25 \times (36 + 14) = 25 \times 50 = 1250$ .

Answer: Rs 1250.

Memory hook: Same multiplier on both sides? Pull it out front.

Tricks to remember Properties of Rational Numbers

Short memory aids. Say each one aloud once, then use it on a question the same day.

Two inverses, two answers
Additive inverse flips the sign; multiplicative inverse flips the fraction. Ask "which inverse?" before you write.
Use it when: Any inverse question.
Only subtraction and division misbehave
Addition and multiplication obey commutative and associative laws; subtraction and division break them, so test with numbers when asked.
Use it when: Counter-example questions.

Properties of Rational Numbers: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Verifying the associative law of addition · 3 marks · Short answerBoard level

Verify $a + (b + c) = (a + b) + c$ for $a = - \frac{2}{3}, b = \frac{3}{4}, c = - \frac{5}{6}$ .

Try it first, then show the solutionSolution

$b + c = \frac{9}{12} - \frac{10}{12} = - \frac{1}{12}$ .
$a + (b + c) = - \frac{8}{12} - \frac{1}{12} = - \frac{9}{12} = - \frac{3}{4}$ .
$a + b = - \frac{8}{12} + \frac{9}{12} = \frac{1}{12}$ .
$(a + b) + c = \frac{1}{12} - \frac{10}{12} = - \frac{9}{12} = - \frac{3}{4}$ .
Both sides equal, so the property is verified.

Answer: Both sides $= - \frac{3}{4}$ .

Example 2: Multiplicative inverse · 1 mark · MCQBoard level

The multiplicative inverse of $- \frac{5}{8}$ is:
(a) $\frac{5}{8}$ (b) $- \frac{8}{5}$ (c) $\frac{8}{5}$ (d) $- \frac{5}{8}$

Try it first, then show the solutionSolution

Product with the inverse must be $1$ : $- \frac{5}{8} \times (- \frac{8}{5}) = 1$ .

Answer: (b)

Example 3: Distributive law with three terms · 3 marks · Short answerHarder board level

Evaluate $\frac{3}{5} \times (- \frac{7}{8}) + \frac{3}{5} \times \frac{1}{8} + \frac{3}{5} \times (- \frac{5}{8})$ .

Try it first, then show the solutionSolution

Take $\frac{3}{5}$ common: $\frac{3}{5} \times (- \frac{7}{8} + \frac{1}{8} - \frac{5}{8})$ .
Bracket: $\frac{- 7 + 1 - 5}{8} = - \frac{11}{8}$ .
$\frac{3}{5} \times (- \frac{11}{8}) = - \frac{33}{40}$ .

Answer: $- \frac{33}{40}$

Example 4: Finding the multiplier · 3 marks · Short answerHarder board level

By what rational number should $- \frac{7}{12}$ be multiplied so that the product is the additive inverse of $\frac{5}{8}$ ?

Try it first, then show the solutionSolution

Additive inverse of $\frac{5}{8}$ is $- \frac{5}{8}$ .
Required $x$ : $- \frac{7}{12} \times x = - \frac{5}{8}$ .
$x = (- \frac{5}{8}) \div (- \frac{7}{12}) = \frac{5}{8} \times \frac{12}{7} = \frac{60}{56} = \frac{15}{14}$ .
Check: $- \frac{7}{12} \times \frac{15}{14} = - \frac{105}{168} = - \frac{5}{8}$ .

Answer: $\frac{15}{14}$

Example 5: Testing a claim · 4 marks · Case-basedHOTS / competency

Rina says "subtraction and division of rational numbers follow the same laws as addition and multiplication". Test her claim: (i) is subtraction commutative for $\frac{2}{3}$ and $\frac{1}{2}$ ? (ii) is division associative for $a = \frac{1}{2}, b = \frac{1}{4}, c = \frac{1}{8}$ ? (iii) For which rational numbers $a, b$ is $a - b = b - a$ ?

Try it first, then show the solutionSolution

(i) $\frac{2}{3} - \frac{1}{2} = \frac{1}{6}$ but $\frac{1}{2} - \frac{2}{3} = - \frac{1}{6}$ . Not equal.
(ii) $(a \div b) \div c = 2 \div \frac{1}{8} = 16$ . And $b \div c = 2$ , so $a \div (b \div c) = \frac{1}{2} \div 2 = \frac{1}{4}$ . Not equal.
(iii) $a - b = b - a \Rightarrow 2 a = 2 b \Rightarrow a = b$ .
So Rina is wrong; each failure is a counter-example.

Answer: Claim false. (i) $\frac{1}{6} \neq = - \frac{1}{6}$ ; (ii) $16 \neq = \frac{1}{4}$ ; (iii) only when $a = b$ .

Where students lose marks in Properties of Rational Numbers

The slips examiners see most often, and what to write instead.

Mistake: Giving 0 a reciprocal
No rational number multiplied by 0 gives 1.
Wrong working
"The multiplicative inverse of $0$ is $0$ ."
Right working
$0$ has no multiplicative inverse.
Do this: Remember: a reciprocal needs a non-zero number.
Usually costs: 1 mark in an inverse question.
Mistake: Mixing the two inverses
The additive inverse changes sign; the multiplicative inverse flips the fraction.
Wrong working
Multiplicative inverse of $- \frac{7}{9}$ written as $\frac{7}{9}$ .
Right working
Additive inverse $\frac{7}{9}$ ; multiplicative inverse $- \frac{9}{7}$ .
Do this: Test by multiplying (or adding) your answer back.
Usually costs: Method and answer marks.

Exam tips

In a "verify" question, show the left side and the right side as two separate full calculations, then write "LHS = RHS".
Name the property in words; examiners award a mark for it.
To disprove a property for subtraction or division, a single counter-example is enough.

Practice questions on Properties of Rational Numbers

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Which property is shown by $\frac{2}{3} \times (\frac{4}{5} + \frac{1}{5}) = \frac{2}{3} \times \frac{4}{5} + \frac{2}{3} \times \frac{1}{5}$ ?
(a) associative (b) commutative (c) distributive law (d) closure
Hint
Multiplication is spreading over a sum.
Answer
(c)
Q2 · 2 marks · Short answerBoard level
Verify that $a \times b = b \times a$ for $a = - \frac{5}{6}$ and $b = \frac{9}{10}$ .
Hint
Multiply numerators and denominators, then reduce.
Answer
$- \frac{5}{6} \times \frac{9}{10} = - \frac{45}{60} = - \frac{3}{4}$ and $\frac{9}{10} \times (- \frac{5}{6}) = - \frac{45}{60} = - \frac{3}{4}$ . Equal, verified.
Q3 · 3 marks · Short answerHarder board level
Evaluate by rearranging: $(- \frac{7}{5} \times \frac{3}{8}) \times \frac{- 5}{7}$ .
Hint
Pair the numbers whose product is easy.
Answer
$\frac{3}{8}$ . Since $(- \frac{7}{5}) \times (- \frac{5}{7}) = 1$ , the product is $1 \times \frac{3}{8}$ .
Q4 · 4 marks · Short answerHarder board level
Verify $a \times (b + c) = a \times b + a \times c$ for $a = - \frac{1}{2}, b = \frac{2}{3}, c = - \frac{3}{4}$ .
Hint
First $b + c = \frac{8}{12} - \frac{9}{12}$ .
Answer
$b + c = - \frac{1}{12}$ , so LHS $= (- \frac{1}{2}) (- \frac{1}{12}) = \frac{1}{24}$ . RHS $= - \frac{1}{3} + \frac{3}{8} = - \frac{8}{24} + \frac{9}{24} = \frac{1}{24}$ . Verified.
Q5 · 3 marks · Short answerHOTS / competency
Find the sum of the additive inverse of $\frac{5}{- 8}$ and the multiplicative inverse of $- \frac{3}{4}$ .
Hint
Write $\frac{5}{- 8}$ as $- \frac{5}{8}$ first.
Answer
Additive inverse of $- \frac{5}{8}$ is $\frac{5}{8}$ ; multiplicative inverse of $- \frac{3}{4}$ is $- \frac{4}{3}$ . Sum $= \frac{15}{24} - \frac{32}{24} = - \frac{17}{24}$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Which operations on rational numbers are commutative?Show answer
Addition and multiplication only.
2.What is the additive identity?Show answer
0.
3.What is the multiplicative inverse of -1?Show answer
-1.
4.Which rational number has no multiplicative inverse?Show answer
0.

Properties of Rational Numbers: frequently asked questions

Is division closed for rational numbers?

Only if division by 0 is excluded; with that exception the quotient is always rational.

What is the difference between an identity and an inverse?

An identity leaves a number unchanged (0 for addition, 1 for multiplication); an inverse brings it to the identity.