Rational Numbers
Definition & Standard Form
A rational number can be written as p/q where p and q are integers and q ≠ 0. In standard form the denominator is positive and the fraction is in its simplest form: 6/−8 = −3/4.
Example 1: Write 6/−8 in standard form.
Make the denominator positive and simplify: −6/8 = −3/4.
Example 2: Is 0 a rational number?
Yes, since 0 = 0/1.
Quick recap
- Rational number = p/q with q ≠ 0.
- Standard form: positive denominator, simplest form.
✓ Quick check
Write 4/8 in standard form.
4/8 = ½ in simplest form.
Write −6/9 in standard form.
Divide both by 3: −6/9 = −2/3.
Comparison & the Number Line
Rational numbers can be placed on a number line, with negatives to the left of 0. To compare, a number further to the right is greater, so −2/3 is greater than −3/4. Between any two rational numbers there are many more.
Example 1: Which is greater, −3/4 or −2/3?
−2/3 is closer to 0, so it is greater.
Example 2: Name a rational number between 0 and 1.
½ (any fraction less than 1 works).
Quick recap
- Further right on the number line means greater.
- There are infinitely many rationals between any two.
✓ Quick check
Which is greater, ⅗ or ⅔?
As 9/15 and 10/15, ⅔ is greater.
Which is smaller, −½ or ½?
−½ is to the left of ½, so it is smaller.
Operations & Properties
To add or subtract rationals, use a common denominator. To multiply, multiply across; to divide, multiply by the reciprocal. Rationals follow the same properties as integers, and the additive inverse of a/b is −a/b.
Example 1: Add ⅓ + (−⅙).
2/6 − 1/6 = 1/6.
Example 2: Multiply (−⅔) × ¾.
(−2 × 3)/(3 × 4) = −6/12 = −½.
Quick recap
- Add/subtract with a common denominator; divide by multiplying by the reciprocal.
- Additive inverse of a/b is −a/b.
✓ Quick check
What is ½ + (−¼)?
2/4 − 1/4 = ¼.
What is (−⅖) ÷ ⅕?
(−⅖) × 5/1 = −10/5 = −2.
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