Rational Numbers

What you'll learn

• Define rational numbers and identify them in p/q form
• Find equivalent rational numbers and express them in standard form
• Compare and order rational numbers
• Add, subtract, multiply, and divide rational numbers
• Represent rational numbers on a number line

Key concepts

Definition — What is a Rational Number?

A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0.

Key insight: All integers and fractions are rational numbers.

Standard form of a rational number:

• Denominator is positive
• Numerator and denominator share no common factor other than 1 (HCF = 1)

Worked Example: Express −12/−18 in standard form. Both negative → −12/−18 = 12/18 HCF(12, 18) = 6 → 12/18 = 2/3

Worked Example: Express 15/−25 in standard form. Make denominator positive: −15/25 HCF(15, 25) = 5 → −3/5

Equivalent Rational Numbers

Two rational numbers are equivalent if one can be obtained from the other by multiplying or dividing numerator and denominator by the same non-zero integer.

p/q = (p×m)/(q×m) = (p÷m)/(q÷m) (where m ≠ 0)

Worked Example: Find three equivalent forms of −2/3. −2/3 = −4/6 = −6/9 = −8/12

To check equivalence: Cross-multiply.

• Are 3/−4 and −9/12 equivalent?
• 3 × 12 = 36 and (−4) × (−9) = 36 → Yes, equivalent

Comparing Rational Numbers

Step 1: Make all denominators positive. Step 2: Find the LCM of denominators. Step 3: Convert to equivalent fractions with the LCM as denominator. Step 4: Compare numerators.

Worked Example: Compare −3/4 and −5/6. LCM(4, 6) = 12 −3/4 = −9/12, −5/6 = −10/12 −9 > −10, so −3/4 > −5/6

Quick rule for negative rational numbers: Among negative rationals, the one with the larger absolute value is smaller.

• |−5/6| > |−3/4| so −5/6 < −3/4

Ordering example:

Arrange in ascending order: 1/2, −1/3, 3/4, −2/5 Convert to 60ths: 30/60, −20/60, 45/60, −24/60 Ascending: −24/60 < −20/60 < 30/60 < 45/60 That is: −2/5 < −1/3 < 1/2 < 3/4

Operations on Rational Numbers

Addition

Same denominator: Add numerators, keep denominator.

Different denominators: Find LCM, convert, then add.

Formula: a/b + c/d = (ad + bc) / bd

Worked Example: 2/5 + (−3/7) LCM(5, 7) = 35 = 14/35 + (−15/35) = (14 − 15)/35 = −1/35

Worked Example: −5/6 + 3/4 LCM(6, 4) = 12 = −10/12 + 9/12 = −1/12

Additive inverse of p/q is −p/q:

• Inverse of 3/7 is −3/7; their sum = 0

Subtraction

a/b − c/d = a/b + (−c/d)

Worked Example: 3/8 − (−5/6) = 3/8 + 5/6 LCM(8, 6) = 24 = 9/24 + 20/24 = 29/24 = 1⁵⁄₂₄

Multiplication

Formula: (p/q) × (r/s) = pr/qs

Sign rule: Same as for integers.

Worked Example: (−4/5) × (3/7) = (−4 × 3) / (5 × 7) = −12/35

Worked Example: (−2/3) × (−9/4) = 18/12 = 3/2 = 1½

Multiplicative inverse (reciprocal) of p/q = q/p:

• Reciprocal of −3/7 is −7/3
• (p/q) × (q/p) = 1

Division

Dividing = multiplying by the reciprocal:

(a/b) ÷ (c/d) = (a/b) × (d/c)

Worked Example: (−5/8) ÷ (3/4) = (−5/8) × (4/3) = −20/24 = −5/6

Worked Example: (−7/3) ÷ (−14/9) = (−7/3) × (9/(−14)) = (−7 × 9) / (3 × (−14)) = −63/(−42) = 3/2

Rational Numbers on the Number Line

Placing positive rational numbers:

To plot 3/5: divide the segment from 0 to 1 into 5 equal parts; mark the 3rd point.

Placing negative rational numbers:

To plot −2/3: divide the segment from −1 to 0 into 3 equal parts; mark the 2nd point from 0 (moving left).

Benchmark positions:

← −2  −3/2  −1  −1/2  0  1/2  1  3/2  2 →

Finding rational numbers between two given numbers:

Between 1/3 and 1/2:

• Method 1: Find mean → (1/3 + 1/2)/2 = (5/6)/2 = 5/12
• Method 2: Convert to same denominator → 2/6 and 3/6 → 5/12 lies between them

Infinitely many rational numbers exist between any two rational numbers (density property).

Worked Example: Find 3 rational numbers between −1/2 and 1/3. Convert: −3/6 and 2/6 Numbers between: −2/6, −1/6, 0/6, 1/6 Simplify: −1/3, −1/6, 0, 1/6

Quick check

1. Express −24/36 in standard form.
2. Which is greater: −5/8 or −7/12? Show working.
3. Calculate: (−3/4) + (5/6) − (1/3)
4. Evaluate: (−2/7) × (−14/6) ÷ (1/3)
5. Find two rational numbers between −3/5 and −1/2.

Open the Practice tab for graded questions on Rational Numbers.