Class 9 › Mathematics › Chapter
The World of Numbers
Natural numbers, zero, integers, rational and irrational numbers, real numbers, the number line, density, decimal expansions, converting decimals to $\frac{p}{q}$ and the proof that $\sqrt{2}$ is irrational.
1. The Number System at a Glance
| Set | Symbol | Description |
|---|---|---|
| Natural numbers | $\mathbb{N}$ | Counting numbers $\{1, 2, 3, \dots\}$ |
| Integers | $\mathbb{Z}$ | $\{\dots, -2, -1, 0, 1, 2, \dots\}$: naturals, zero and the negatives |
| Rational numbers | $\mathbb{Q}$ | All numbers $\frac{p}{q}$ with $p, q$ integers, $q \neq 0$ |
| Irrational numbers | $\mathbb{I}$ | Numbers on the number line that are not rational |
| Real numbers | $\mathbb{R}$ | All rational and irrational numbers together |
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$, and irrational numbers lie in $\mathbb{R}$ but outside $\mathbb{Q}$.
- Each set is contained in the next: every natural number is an integer, every integer is rational, every rational is real.
- Rational and irrational numbers have nothing in common: a number is exactly one of the two.
- Whole numbers $= \mathbb{N} \cup \{0\}$ also sit inside $\mathbb{Z}$.
2. Natural Numbers
Natural numbers $\mathbb{N} = \{1, 2, 3, 4, \dots\}$ are the numbers used for counting.
- One-to-one correspondence: matching each object of one collection with exactly one object of another (without using number names). It is the basic idea behind counting and comparing quantities.
- Natural numbers are closed under addition (sum of two naturals is natural) and multiplication, but not closed under subtraction (a smaller number minus a larger one is not a natural number).
- Every natural number $n$ has an additive inverse $-n$ (so that $n + (-n) = 0$), which is why negatives are needed.
3. Zero (Śhūnya) as a Number
Zero: the result of subtracting a number from itself: $a - a = 0$. It is a genuine number on which arithmetic can be done (not just a placeholder for an empty column).
Brahmagupta's Rules for Zero
$$a + 0 = a, \qquad a - 0 = a, \qquad a \times 0 = 0$$
- Adding or subtracting zero leaves a number unchanged.
- Any number multiplied by zero gives zero.
- Division by zero is not defined (this is why $q \neq 0$ in $\frac{p}{q}$).
4. Integers and Brahmagupta's Sign Rules
Integers $\mathbb{Z}$: the natural numbers, their negatives and zero. Positive integers are "fortunes" (dhana), negative integers are "debts" (ṛiṇa). On the number line, positives lie to the right of zero and negatives to the left.
| Rule | Statement |
|---|---|
| Fortune + fortune | Positive + positive = positive |
| Debt + debt | Negative + negative = negative (add the sizes, keep the minus sign) |
| Fortune − 0 / Debt − 0 | Subtracting zero changes nothing |
| Debt × fortune | Negative × positive = negative |
| Debt × debt | Negative × negative = positive |
Sign summary
$$(+)(+) = +,\quad (-)(-) = +,\quad (+)(-) = -,\quad (-)(+) = -$$
- The same sign rules hold for division: like signs give a positive quotient, unlike signs a negative one.
- Subtracting a negative = adding a positive: $a - (-b) = a + b$ (taking away a debt makes you richer).
- Why $(-)\times(-) = +$: multiplying by a negative number means removing that many debts, which increases wealth.
- Integers are closed under addition, subtraction and multiplication, but not under division.
5. Fractions and Rational Numbers
Fraction: a number representing a part of a whole. Each positive fraction has an additive inverse (a negative fraction).
Rational number: any number that can be written as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. The set is denoted $\mathbb{Q}$ (quotient).
Rational number: any number that can be written as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. The set is denoted $\mathbb{Q}$ (quotient).
- Why $q \neq 0$: division by zero has no meaning, so $\frac{p}{0}$ is not a number.
- Sign placement: the minus sign of a negative fraction may sit with the numerator or the denominator: $-\dfrac{a}{b} = \dfrac{-a}{b} = \dfrac{a}{-b}$.
- $\mathbb{Q}$ includes $\mathbb{N}$, whole numbers and $\mathbb{Z}$: any integer $n$ is $\dfrac{n}{1}$.
- Non-unique representation: a rational number has infinitely many forms (equivalent fractions) obtained by multiplying or dividing numerator and denominator by the same non-zero number.
- Lowest terms: $\frac{p}{q}$ is in lowest terms when $p$ and $q$ are co-prime (no common factor other than $1$). By convention, a rational number is written and plotted in lowest terms with $q \gt 0$.
- Equality test: $\dfrac{a}{b} = \dfrac{c}{d}$ if and only if $ad = bc$.
6. Laws of Operations on Rational Numbers
Brahmagupta's Rules (valid for positive and negative rationals)
$$\frac{a}{b} \pm \frac{c}{b} = \frac{a \pm c}{b} \qquad\qquad \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \quad (b, d \neq 0)$$
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \quad (b, c, d \neq 0)$$
- Addition / subtraction: first make the denominators equal (use a common multiple of the denominators), then add or subtract the numerators and keep the common denominator.
- Division means multiplying by the reciprocal of the divisor; the divisor must not be zero.
Commutative
Order does not matter
$\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}$ and $\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}$ (not true for subtraction or division).
Distributive
Multiplication over addition
$p(q + r) = pq + pr$ for rational $p, q, r$.
Identities
0 and 1
$x + 0 = x$ and $x \times 1 = x$; additive inverse of $x$ is $-x$.
Closure
Stays rational
The sum, difference and product of rationals are rational; the quotient is rational provided the divisor is not zero.
7. Rational Numbers on the Number Line; Absolute Value
Plotting $\frac{p}{q}$ ($q \neq 0$, lowest terms)
1. Mark the origin $0$ and integers at equal spacing (right = positive, left = negative).
2. Divide the unit interval (between consecutive integers) into $q$ equal parts.
3. Move $p$ parts from $0$: to the right if the number is positive, to the left if negative.
4. For an improper fraction, write it as a mixed number: it lies between the two integers on either side, and the interval is divided into $q$ parts starting from the lower integer.
Absolute value $|x|$: the distance of $x$ from $0$ on the number line.
• $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x \lt 0$. Hence $|x| \ge 0$ always and $|0| = 0$.
• $|{-x}| = |x|$.
• $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x \lt 0$. Hence $|x| \ge 0$ always and $|0| = 0$.
• $|{-x}| = |x|$.
Distance between two numbers
$$d(a, b) = |a - b|$$
- It is symmetric ($|a - b| = |b - a|$) and never negative.
8. Density of Rational Numbers
Density property: between any two rational numbers, however close, there is another rational number. Hence there are infinitely many rational numbers between any two rational numbers.
Average method
$$a \lt \frac{a + b}{2} \lt b \quad (a \lt b)$$
- The average of two rationals is rational (closure) and lies exactly halfway.
- Repeating the method on the new pair gives more numbers between them, without end.
Why $\frac{a+b}{2}$ lies between $a$ and $b$
1. Let $a \lt b$. Then $\dfrac{a+b}{2} - a = \dfrac{b - a}{2} \gt 0$, so $\dfrac{a+b}{2} \gt a$.
2. Also $b - \dfrac{a+b}{2} = \dfrac{b - a}{2} \gt 0$, so $\dfrac{a+b}{2} \lt b$.
3. Both gaps are equal, so it is the midpoint.
- Common-denominator method: write $a = \dfrac{k_1}{m}$ and $b = \dfrac{k_2}{m}$ with the same denominator. The numbers $\dfrac{k_1 + 1}{m}, \dfrac{k_1 + 2}{m}, \dots$ lie between them. To get $n$ such numbers this way you need $k_2 - k_1 \gt n + 1$; if the gap is too small, multiply numerator and denominator by a larger number to increase $m$.
- Decimal method: for decimal numbers, insert extra digits (for example, extend the decimal places) to create numbers in between.
- Density suggests the rationals might fill the whole number line, but they do not: irrational numbers fill the remaining points.
9. Irrational Numbers
Irrational number: a number on the number line that cannot be written as $\frac{p}{q}$ with integers $p, q$ ($q \neq 0$). Its decimal expansion is non-terminating and non-repeating.
- Origin: in a square of side $1$, the diagonal $d$ satisfies $1^2 + 1^2 = d^2$ (Baudhāyana–Pythagoras Theorem), so $d = \sqrt{2}$, a length that is not a ratio of integers.
- Examples: $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{10}$, $\pi$. The square root of a perfect square is rational (it is a whole number), and the square root of a positive integer that is not a perfect square is irrational.
- $\pi$ (circumference $\div$ diameter) is irrational (proved by Lambert, 1761). Any fraction used for $\pi$ is only an approximation (Āryabhaṭa's $\frac{3927}{1250} = 3.1416$ is an asanna, an approximation). An exact expression needs an infinite sum, such as Mādhava's series: $$\pi = 4\left(1 - \frac13 + \frac15 - \frac17 + \cdots\right)$$ An infinite sum means the value that the partial sums get closer and closer to.
Proof that $\sqrt{2}$ is irrational (Proof by Contradiction)
Method: assume the opposite of what we want to prove and show it leads to a contradiction.
Step 1 (Assume): $\sqrt{2} = \dfrac{p}{q}$ with $q \neq 0$ and $p, q$ co-prime (lowest terms).
Step 2: square both sides: $2 = \dfrac{p^2}{q^2}$.
Step 3: $2q^2 = p^2$.
Step 4: $p^2$ is a multiple of $2$, so $p^2$ is even. If a square is even, the number is even, so $p$ is even: $p = 2k$.
Step 5: substitute: $2q^2 = (2k)^2 = 4k^2$.
Step 6: divide by $2$: $q^2 = 2k^2$.
Step 7: $q^2$ is even, so $q$ is even.
Step 8 (Contradiction): $p$ and $q$ are both even, so they share the factor $2$, contradicting that they are co-prime. Hence the assumption is false and $\sqrt{2}$ is irrational.
- Same method for $\sqrt{n}$: for a prime $n$ (such as $3, 5, 7$) use "if $n$ divides $p^2$ then $n$ divides $p$"; the argument ends with both $p$ and $q$ divisible by $n$. For composite non-square values (such as $10$) the same idea works by looking at the prime factors involved.
- Key fact used: if a prime divides $p^2$, it divides $p$.
10. Constructing $\sqrt{n}$ and Locating it on the Number Line
Fig. Locating $\sqrt{2}$: $OA = AB = 1$, $OB = \sqrt{2}$, and the arc with centre $O$ and radius $OB$ meets the number line at $P$.
Construction of $\sqrt{2}$ with ruler and compass
Step 1: on the number line take $OA = 1$ unit and draw a perpendicular to $OA$ at $A$.
Step 2: on this perpendicular mark $B$ with $AB = 1$ unit and join $OB$. By the Baudhāyana–Pythagoras Theorem, $OB = \sqrt{1^2 + 1^2} = \sqrt{2}$.
Step 3: with centre $O$ and radius $OB$ draw an arc cutting the number line at $P$. Then $OP = \sqrt{2}$, so $P$ represents $\sqrt{2}$.
General method for $\sqrt{n}$ (Square Root Spiral)
- If a segment has length $\sqrt{n-1}$, draw a perpendicular of length $1$ at its end; the hypotenuse joining the origin to the new point has length $\sqrt{(\sqrt{n-1})^2 + 1^2} = \sqrt{n}$.
- Starting with a unit segment and repeating, the hypotenuses are $\sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \dots$ in order, forming the square root spiral.
- To mark $\sqrt{n}$ on the number line, draw an arc of radius $\sqrt{n}$ centred at $O$ to meet the line.
- Irrational numbers can be located exactly by construction; the rest are simply understood to lie on the line.
11. Real Numbers
Real numbers $\mathbb{R}$ = rational numbers $\cup$ irrational numbers. Together they form the continuous, unbroken number line.
- Every real number corresponds to exactly one point on the number line, and every point corresponds to a real number.
- Every length, temperature or physical measurement is a real number.
- Rational numbers alone leave "gaps" (the irrational points); the real line has none.
- Real numbers cannot give a square root of a negative number, so $\sqrt{-1}$ is not real; the imaginary number $i$ ($i = \sqrt{-1}$) was introduced for this and is studied in higher classes.
12. Decimal Expansions
Rational
Terminating
Division ends with remainder $0$, so the decimal stops.
Rational
Non-terminating repeating
Remainder never becomes $0$ but the digits repeat in a block forever. A bar marks the repeating block.
Irrational
Non-terminating non-repeating
No end and no repeating block.
Decimal signature of a number
- Rational $\iff$ terminating or repeating decimal. Irrational $\iff$ non-terminating and non-repeating decimal.
- Do not confuse "non-terminating" with "irrational": a decimal that goes on forever but repeats is still rational.
Why rational numbers repeat
1. In the long division of $p$ by $q$, every remainder is one of $0, 1, \dots, q - 1$.
2. If the remainder is $0$ the decimal terminates. If never $0$, only the $q - 1$ non-zero values are possible.
3. So within at most $q$ steps a remainder must repeat; from then on the same digits recur, giving a repeating block (of length at most $q - 1$).
Predicting terminating or repeating (without dividing)
- Write $\frac{p}{q}$ in lowest terms and prime-factorise $q$.
- Terminating iff $q = 2^m \times 5^n$ (only the primes $2$ and/or $5$). Repeating if $q$ has any other prime factor.
- Number of decimal places (terminating case) $= \max(m, n)$, the larger of the powers of $2$ and $5$.
- Reason: multiply numerator and denominator by a suitable number to make the denominator a power of $10$.
- If a terminating decimal has its last non-zero digit at the $k$th decimal place, it equals $\dfrac{p}{10^k}$ with $p$ not divisible by $10$; in lowest terms the denominator is divisible by $2^k$ or $5^k$ (and has no other prime factors).
Non-uniqueness of decimal representation: every terminating decimal also has an alternative form ending in repeating $9$s, and $0.\overline{9} = 1$.
Proof: let $x = 0.\overline{9}$; then $10x = 9.\overline{9}$; subtracting, $9x = 9$, so $x = 1$.
Proof: let $x = 0.\overline{9}$; then $10x = 9.\overline{9}$; subtracting, $9x = 9$, so $x = 1$.
13. Converting Decimals to $\frac{p}{q}$ Form
| Decimal type | Method |
|---|---|
| Terminating | Write the digits over $10^k$ ($k$ = number of decimal places), then reduce to lowest terms. |
| Pure repeating (block repeats right after the decimal point) | Let $x$ = the decimal. Multiply by $10^n$ ($n$ = number of repeating digits). Subtract the original equation. Solve for $x$. |
| General (mixed) repeating (some non-repeating digits, then a repeating block) | Let $x$ = the decimal. Multiply by $10^m$ ($m$ = non-repeating digits after the point) to get one equation, then by $10^{m+n}$ ($n$ = repeating digits) to get another. Subtract the first from the second and solve for $x$. |
Shortcut formula (derived from the subtraction)
$$x = \frac{A - B}{10^m\,(10^n - 1)}$$
- $A$ = the integer formed by all digits (including the integer part) up to the end of one repeating block, ignoring the decimal point.
- $B$ = the integer formed by the digits up to the end of the non-repeating part, ignoring the decimal point.
- For a pure repeating decimal $m = 0$, so $x = \dfrac{A - B}{10^n - 1}$.
- Why it works: $(10^{m+n} - 10^m)x = A - B$.
- Always reduce the answer to lowest terms. The denominator has $n$ nines followed by $m$ zeros.
14. Cyclic Numbers
Cyclic number: a repeating block of digits whose multiples (by $1, 2, 3, \dots$) are cyclic rearrangements (same digits shifted around a circle) of the block.
- The repeating block $142857$ of $\dfrac{1}{7}$ is the standard example: its multiples by $1$ to $6$ use the same six digits in rotated order.
- Reason: dividing by $7$ the six non-zero remainders $1$ to $6$ all occur before the pattern repeats; the fractions $\frac{2}{7}, \frac{3}{7}, \dots$ have the same block shifted.
- Such patterns are a feature of rational numbers; irrational decimals never loop.
15. Closure Summary
| Set | Addition | Subtraction | Multiplication | Division |
|---|---|---|---|---|
| Natural numbers $\mathbb{N}$ | Yes | No | Yes | No |
| Integers $\mathbb{Z}$ | Yes | Yes | Yes | No |
| Rational numbers $\mathbb{Q}$ | Yes | Yes | Yes | Yes (divisor $\neq 0$) |
16. Quick Revision Sheet
| Concept | Key points |
|---|---|
| Hierarchy | $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$; irrationals are in $\mathbb{R}$ but not $\mathbb{Q}$ |
| Zero rules | $a + 0 = a$, $a - 0 = a$, $a \times 0 = 0$ |
| Sign rules | $(-)(-) = +$, $(+)(-) = -$, $a - (-b) = a + b$ |
| Rational number | $\frac{p}{q}$, $p, q \in \mathbb{Z}$, $q \neq 0$; equal iff $ad = bc$ |
| Laws | $\frac{a}{b} \pm \frac{c}{b}$, $\frac{ac}{bd}$, $\frac{ad}{bc}$; commutative, distributive |
| Absolute value / distance | $|x| \ge 0$; distance $|a - b|$ |
| Density | $\frac{a+b}{2}$ lies between $a$ and $b$; infinitely many in between |
| Irrational | Not $\frac{p}{q}$; non-terminating non-repeating; proved by contradiction |
| Terminating test | Lowest-terms denominator $= 2^m 5^n$; places $= \max(m, n)$ |
| Repeating to $\frac{p}{q}$ | $x = \dfrac{A - B}{10^m(10^n - 1)}$ |
| $0.\overline{9}$ | Equals exactly $1$ |
Common mistakes:
• Saying $\sqrt{81}$ or $0.333\ldots$ is irrational (the first is $9$; the second is $\frac13$, a repeating decimal).
• Treating any never-ending decimal as irrational.
• Forgetting to reduce $\frac{p}{q}$ to lowest terms before testing for terminating decimals.
• Sign slips: $-(-b) = +b$ and $(-a)^2 = a^2$.
• Dividing by zero, or allowing $q = 0$ in a rational number.
• Saying $\sqrt{81}$ or $0.333\ldots$ is irrational (the first is $9$; the second is $\frac13$, a repeating decimal).
• Treating any never-ending decimal as irrational.
• Forgetting to reduce $\frac{p}{q}$ to lowest terms before testing for terminating decimals.
• Sign slips: $-(-b) = +b$ and $(-a)^2 = a^2$.
• Dividing by zero, or allowing $q = 0$ in a rational number.