Steps:
- Draw a line and take AB = 9.3 units on it.
- From B, measure a distance of 1 unit and mark C on the number line. Mark the midpoint of AC as O.
- With 'O' as center and OC as radius, draw a semicircle.
- At B, draw a perpendicular to cut the semicircle at D.
- With B as center and BD as radius draw an arc to cut the number line at E. Thus, taking B as origin the distance BE =
Hence, point E represents on the number line.
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CBSE Class IX | Academic Year 2026-2027
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Mathematics | Number SystemsWeb Content • Interactive Notes
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.
Key Insight: Between any two rational numbers, there are infinitely many irrational numbers, and vice versa. The number line is “dense” with both types!
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)