Chapter 1: Number Systems

Numbers used to be simple: $1, 2, 3, \ldots$ for counting, then negatives, then fractions. In Class IX we step back and ask a sharper question , which numbers actually live on the number line? The answer is that the rationals you already know are not enough. Between any two rationals there are infinitely many points like $\sqrt{2}$, $\pi$, and $1.010010001\ldots$ that no fraction can capture. Together with the rationals, these irrational numbers fill up the line completely. The union is the real number system, $\mathbb{R}$, and the whole chapter is about its anatomy.

The hook is decimals. Every rational number is a decimal that either terminates or eventually repeats. The instant a decimal refuses to do either , like $0.101001000100001\ldots$ , it must be irrational. This single dictionary (rationals $\leftrightarrow$ recurring/terminating decimals) is the most useful idea you will carry forward. It powers most of the questions you will see in exams and in olympiads.

You will also learn to place irrationals on the number line using only a ruler, a pencil, and the Pythagoras theorem. Numbers like $\sqrt{2}$ and $\sqrt{3}$ stop being abstract symbols and become honest points you can pin down geometrically. The "successive magnification" idea lets you locate a decimal like $2.665$ as precisely as you wish, by zooming into the number line a digit at a time.

The algebraic half of the chapter is just as important. We add, subtract, multiply and divide irrationals, learn how to rationalise a denominator like $\dfrac{1}{\sqrt{2}+1}$, and discover that the laws of exponents you used for integers , $a^m \cdot a^n = a^{m+n}$ and friends , continue to work even when the powers are fractions. This is what allows expressions like $2^{1/2}$ to be a number at all, namely $\sqrt{2}$.

By the end you should be able to switch fluently between three views of the same number: a decimal, a position on the number line, and an algebraic expression. That fluency is what the rest of high school maths quietly assumes.

What's inside

• Rational and irrational numbers , definitions, examples, and how to find numbers between any two rationals.
• The real number line , locating $\sqrt{n}$ geometrically and zooming in on a decimal.
• Decimal expansions , terminating, non-terminating recurring, and non-terminating non-recurring.
• Operations on real numbers , sums, products and the rules that mix rationals with irrationals.
• Rationalisation , clearing a surd from a denominator using conjugates.
• Laws of exponents for real numbers , extending integer rules to rational exponents.

Key results / Formula card

Keep this card near you while doing the chapter , almost every question is one of these lines applied carefully.