(i) is a prime number. It is irrational because does not have a complete root.
(ii) is not a prime number. 15 is root of so it is rational.
(iii) 0.3796 is terminating decimal expansion. Hence, it is rational.
(iv) 7.478478... = , it is non-terminating but repeating decimal expansion. Hence, it is rational.
(v) 1.101001000100001... is non-terminating and non-repeating decimal expansion. Hence, it is irrational.
Number Systems — Interactive Study Guide
Master the real number system, irrational numbers, surds, rationalisation, and exponent laws.
The Number Hierarchy
Think of numbers as nested boxes: Natural numbers are inside Whole numbers, which are inside Integers, which are inside Rational numbers, which are inside Real numbers.
Identifying Rational vs Irrational
| Number | Type | Reason |
|---|---|---|
| √4 | Rational | √4 = 2 (perfect square) |
| √7 | Irrational | 7 is not a perfect square |
| 0.333... | Rational | Recurring decimal = 1/3 |
| 0.10100100010... | Irrational | Non-terminating, non-recurring |
| π | Irrational | Non-terminating, non-recurring |
| 22/7 | Rational | It is p/q form (just an approximation of π) |
Rationalisation — Quick Method
To rationalise a denominator with surds, multiply top and bottom by the conjugate:
- Conjugate of (a + √b) is (a − √b)
- Conjugate of (√a − √b) is (√a + √b)
The denominator becomes rational because (a+b)(a−b) = a² − b².
Quick Self-Check
- Is √(16/9) rational or irrational? (Rational: = 4/3)
- Simplify: √50 + √18 (= 5√2 + 3√2 = 8√2)
- Rationalise: 1/(√3 + 1) (= (√3 − 1)/2)
- Find: 81/3 (= 2)