Class 9 Maths Chapter 3: The World of Numbers — Notes, Examples & Practice

The natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$ come from counting. Adding zero gives the whole numbers $\{0, 1, 2, \dots\}$. Zero (śūnya) was a major Indian contribution: it lets the place-value system write any number with just ten digits, because a 0 can hold an empty place, as in 405.

Subtraction like $3 - 5$ needs negative numbers. The integers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ describe debts, floors below ground, and temperatures below freezing.

On the number line, numbers increase to the right. So $-2 > -7$ even though 7 is bigger than 2: $-2$ is to the right of $-7$.

A rational number can be written as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Every integer is rational ($-4 = \dfrac{-4}{1}$), and so is every fraction and every terminating decimal ($0.75 = \dfrac{3}{4}$).

The same rational number has many forms: $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{-3}{-6}$. The standard form has $q > 0$ and no common factor between $p$ and $q$.

The absolute value $|x|$ is the distance of $x$ from 0 on the number line, so it is never negative: $|-5| = 5$, $|3| = 3$, $|0| = 0$. The distance between two numbers $a$ and $b$ is $|a - b|$.

Between any two different rational numbers there is always another one — for example their average $\dfrac{a + b}{2}$. Repeat the idea and you find infinitely many. This is called the density of rational numbers.

To find several at once, write both numbers with a common denominator, then multiply the top and bottom by a bigger number to create room. Between $\dfrac{1}{3}$ and $\dfrac{1}{2}$: $\dfrac{2}{6}$ and $\dfrac{3}{6}$ have nothing between their numerators, but $\dfrac{20}{60}$ and $\dfrac{30}{60}$ have nine numbers in between: $\dfrac{21}{60}, \dfrac{22}{60}, \dots, \dfrac{29}{60}$.

An irrational number cannot be written as $\dfrac{p}{q}$ with integers $p, q$. Examples: $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\pi$. Their decimals go on forever without a repeating block.

**Why $\sqrt{2}$ is irrational (proof by contradiction):** suppose $\sqrt{2} = \dfrac{p}{q}$ in lowest terms. Squaring, $p^2 = 2q^2$, so $p^2$ is even, which forces $p$ to be even. Write $p = 2k$: then $4k^2 = 2q^2$, so $q^2 = 2k^2$ and $q$ is even too. But then $p$ and $q$ share the factor 2 — contradicting 'lowest terms'. So no such fraction exists.

$\pi$, the ratio of a circle's circumference to its diameter, is also irrational. Madhava of Sangamagrama (14th century, Kerala) found an infinite series for it: $\dfrac{\pi}{4} = 1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} + \dots$. Values like $\dfrac{22}{7}$ and 3.14 are only approximations.

Rational and irrational numbers together form the real numbers $\mathbb{R}$. Every real number is a point on the number line, and every point on the line is a real number — there are no gaps.

You can locate $\sqrt{2}$ exactly with a ruler and compass: draw a unit square on the number line from 0 to 1; its diagonal has length $\sqrt{1^2 + 1^2} = \sqrt{2}$. Swing an arc from 0 with that radius onto the line.

Repeating the idea gives the square root spiral: start with a right triangle with legs 1 and 1 (hypotenuse $\sqrt{2}$), then build a new right triangle on that hypotenuse with another leg of 1 (hypotenuse $\sqrt{3}$), and so on: $\sqrt{n + 1} = \sqrt{(\sqrt{n})^2 + 1^2}$. The book also mentions numbers beyond the real line — imaginary numbers such as $\sqrt{-1}$ — which you will study later.

Divide $p$ by $q$ and the decimal either terminates ($\dfrac{3}{8} = 0.375$) or repeats forever ($\dfrac{2}{11} = 0.181818\ldots = 0.\overline{18}$). It must repeat because only $q$ different remainders are possible, so a remainder eventually comes back.

Which fractions terminate? Write $\dfrac{p}{q}$ in lowest terms. The decimal terminates exactly when $q$ has no prime factors other than 2 and 5 (because then the fraction can be rewritten with denominator a power of 10).

Irrational numbers have decimals that never terminate and never repeat. Cyclic numbers are a fun pattern: the repeating block of $\dfrac{1}{7} = 0.\overline{142857}$ just rotates for $\dfrac{2}{7} = 0.\overline{285714}$, $\dfrac{3}{7} = 0.\overline{428571}$, and so on.