Rational Numbers — Definition

• Define a rational number as a number that can be written in the form p/q, where p and q are integers and q ≠ 0.
• Distinguish rational numbers from whole numbers and integers.
• Write rational numbers in standard form (lowest terms, positive denominator).
• Recognise equivalent rational numbers obtained by multiplying or dividing numerator and denominator by the same non-zero integer.

Level 1 — Core idea

1. What is a rational number?
   Any number of the form p/q where p, q ∈ ℤ and q ≠ 0 is a rational number.
   Examples: 3/4, −5/2, 7/1 (= 7), 0/5 (= 0), −8/−3 (= 8/3).

2. Every integer is rational — Any integer n can be written as n/1.
   So −4 = −4/1 and 0 = 0/1 are rational numbers.

3. Numerator and denominator — In p/q, p is the numerator and q is the denominator. The denominator must never be zero.

Level 2 — Process and representation

1. Standard form — A rational number p/q is in standard form when:

   • p and q have no common factor other than 1 (coprime), and
   • q > 0 (denominator is positive).
     Example: −12/18 = −2/3 in standard form.

2. Equivalent rational numbers — Multiplying or dividing both p and q by the same non-zero integer gives an equivalent rational:
   p/q = (p×n)/(q×n) and p/q = (p÷d)/(q÷d) when d divides both.
   Example: 2/3 = 4/6 = 6/9.

3. Positive, negative, and zero rationals

   • If p and q have the same sign, the rational is positive.
   • If p and q have opposite signs, the rational is negative.
   • 0/q = 0 for any q ≠ 0.

Express −36/48 in standard form.

Step 1 — find HCF of 36 and 48: HCF = 12
Step 2 — divide numerator and denominator by 12:
         −36 ÷ 12 = −3;  48 ÷ 12 = 4
Step 3 — write with positive denominator: −3/4
Answer: −3/4
Check: −3 and 4 share no common factor except 1, and 4 > 0 ✓

Is 0.75 a rational number? Justify.

0.75 = 75/100 = 3/4
It can be written as p/q with q ≠ 0.
Yes, 0.75 is a rational number.

• Write −15/25 in standard form. (−3/5)
• Is −7 a rational number? How? (Yes; −7/1)
• Write two equivalent forms of 3/8. (e.g. 6/16 and 9/24)
• Which is not a rational number: 0/3, 4/0, −2/5? (4/0)

Open the Practice tab for graded questions on rational number definitions.