CBSE Mathematics Number Systems Notes Set 18

Number Systems : Key Ideas

Kinds of Numbers

1. Natural numbers are 1, 2, 3, ... and whole numbers add zero to this list. Integers include the negatives as well.
2. A rational number can be written as p/q with integers p and q, q not zero. Its decimal form either ends or repeats a block of digits.
3. An irrational number cannot be written as p/q. Its decimal form neither ends nor repeats, as in \( \sqrt2 \) and \( \pi \).
4. Rational and irrational numbers together make the real numbers, and every point of the number line is a real number.
5. The distance between points a and b on the number line is \( |b-a| \).

Decimals and Fractions

1. To convert a pure recurring decimal, multiply by 10 (one repeating digit), 100 (two) or 1000 (three), then subtract the original number.
2. For \( 0.\overline{37} \): \( 100x - x = 37 \), so \( x = \tfrac{37}{99} \).
3. For a mixed recurring decimal such as \( 0.2\overline{37} \), shift twice so that the repeating tails line up, then subtract: \( 1000x - 10x = 235 \).
4. A fraction whose denominator has only the prime factors 2 and 5 terminates; otherwise it recurs.

Surds and Rationalisation

1. Simplify a surd by pulling out perfect powers: \( \sqrt[3]{54} = 3\sqrt[3]{2} \).
2. Only like surds can be added or subtracted, so reduce each to a common radicand first.
3. The conjugate of \( a+\sqrt b \) is \( a-\sqrt b \), and their product \( a^2-b \) is rational. Multiply top and bottom by the conjugate to rationalise.
4. Some square roots of surd expressions are perfect squares, for example \( 5+2\sqrt6 = (\sqrt3+\sqrt2)^2 \).
5. For \( x + \tfrac1x = s \): \( x^2 + \tfrac1{x^2} = s^2 - 2 \), and the same step can be repeated for fourth powers.

Laws of Exponents

1. \( a^m \cdot a^n = a^{m+n} \) and \( (a^m)^n = a^{mn} \).
2. \( a^{-n} = \tfrac{1}{a^n} \) and \( a^{p/q} = \sqrt[q]{a^p} \).
3. To evaluate \( 9^{-3/2} \), write 9 as \( 3^2 \) to get \( 3^{-3} = \tfrac1{27} \).

Worked Examples

1. \( 0.\overline3 = \tfrac13 \), because \( 10x - x = 3 \).
2. \( (3+\sqrt5)(3-\sqrt5) = 9-5 = 4 \).
3. \( \tfrac{1}{\sqrt2+1} = \sqrt2-1 \).