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Rational numbers · 2 marks

Can you explain why we need q ≠ 0 in the definition of a rational number?

Answer: Because pq means p ÷ q, and division by 0 has no meaning: no number times 0 gives a non-zero p, and for p = 0 every number would work. So p0 is not a number at all.

Step-by-step solution

Idea: Division is the reverse of multiplication: pq = r means r × q = p. Test this with q = 0.

  1. pq = r means q × r = p. (For example 123 = 4 because 3 × 4 = 12.)½ mark
  2. Take q = 0 and p ≠ 0, say 50. We would need 0 × r = 5, but 0 × r = 0 for every number r. No such r exists.½ mark
  3. Take q = 0 and p = 0: 00. Now 0 × r = 0 is true for every r, so there is no single answer.½ mark
  4. Either way p0 does not name one definite number, so it cannot be a rational number. That is why the definition insists on q ≠ 0.½ mark
p/q stands for p ÷ q, and dividing by zero gives either no answer (p ≠ 0) or no single answer (p = 0). So q must not be 0.

Answer to write in the exam

pq = r means q × r = p

If q = 0 and p ≠ 0: 0 × r = p has no solution (0 × r = 0)

If q = 0 and p = 0: 0 × r = 0 for every r, so no single value

∴ p0 is not defined, so q ≠ 0 is needed.

Common mistakes that cost marks

  • Saying 50 = 0 or 50 = 5. Neither is true, since 0 × 0 ≠ 5 and 0 × 5 ≠ 5.
  • Confusing 05 with 50. 05 = 0 is a perfectly good rational number; only a zero denominator is forbidden.
  • Saying ‘the answer is infinity’. Infinity is not a number on the number line, so it does not rescue the definition.

How this can come in the exam

MCQ (1 mark)

Which of the following is NOT a rational number?

  1. 07
  2. −71
  3. 70
  4. 7−1
Show answer

(C) 70
70 has a zero denominator, so it is not defined. 07 = 0, −71 = −7 and 7−1 = −7 are rational.

Assertion–Reason (1 mark)

Assertion (A): 0 is a rational number.
Reason (R): 0 can be written as 0q for any non-zero integer q.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
0 = 01 = 05, which has the form pq with q ≠ 0. R explains A.

Try one yourself

For which value of x is 3x − 4 not a rational number?

Show answer

When x − 4 = 0, that is x = 4: the denominator becomes 0.

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