This chapter sorts every number you have met into neat families, and then introduces a new one: irrational numbers. Getting the families clear makes the rest of Class 9 and 10 algebra far easier.
The families of numbers
- ✓ Natural numbers (N): the counting numbers 1, 2, 3, …
- ✓ Whole numbers (W): the naturals plus 0.
- ✓ Integers (Z): whole numbers and their negatives: … −2, −1, 0, 1, 2 …
- ✓ Rational numbers (Q): any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0.
Rational vs irrational
A rational number can be written as p/q. In decimal form it either terminates (like 0.75) or repeats a pattern forever (like 0.333…). An irrational number cannot be written as p/q; its decimal goes on forever with no repeating pattern. Famous examples are √2, √3 and π. Together, the rationals and irrationals make up the real numbers (R) — every point on the number line is a real number.
Numbers on the number line
Every real number has its own point on the number line, and every point is some real number. You can even locate an irrational number like √2 geometrically using a right-angled triangle and a compass — a neat reminder that irrationals are real, exact numbers, not just "messy decimals".
Surds and rationalising the denominator
A root like √2 or √5 that stays irrational is called a surd. You can use the usual laws, for example √a × √b = √(ab). When a surd sits on the bottom of a fraction, we usually rationalise the denominator — multiply top and bottom by a suitable surd so the bottom becomes rational. For instance, 1/√2 = (1 × √2)/(√2 × √2) = √2/2.
Laws of exponents
The chapter also extends the exponent rules to real powers: am × an = am+n, (am)n = amn, and am ÷ an = am−n. These same rules let you handle fractional powers, where a1/2 means √a.
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