Chapter 3: The World of Numbers

Chapter 3: The World of Numbers

Detailed Notes and Summary

Introduction

This chapter explains how numbers developed over thousands of years according to human needs. It begins with the need to count objects, then moves to zero, negative numbers, fractions, rational numbers, irrational numbers, real numbers, decimals, cyclic patterns, and finally gives a glimpse of imaginary numbers.

The chapter shows that the world of numbers did not appear all at once. It grew step by step:

Natural Numbers → Integers → Rational Numbers → Irrational Numbers → Real Numbers

3.1 The Dawn of Mathematics: The Human Need to Count

Mathematics began because humans needed to count things in daily life.

Early humans needed to know:

• How many animals were in a herd

• How many days had passed

• How much food was stored

• How many tools or goods were exchanged

Before written numbers existed, people used simple methods like matching one object with another. This is called one-to-one correspondence.

Example

A herder could place one pebble in a pot for each cow going out to graze. In the evening, one pebble was removed for each cow returning. If pebbles remained, it meant some cows were missing.

This idea gave birth to Natural Numbers.

Natural Numbers

Natural numbers are counting numbers.

$$\mathbb{N}=\{1,2,3,4,5,...\}$$

They are used for counting objects.

Important Points

• Natural numbers start from 1.

• They do not include 0.

• They are used for counting.

• They continue forever.

3.1.1 A History Written in Bone

Early humans recorded numbers using tally marks on bones.

Lebombo Bone

The Lebombo Bone was discovered in the Lebombo Mountains between South Africa and Swaziland. It is about 35,000 years old and has 29 carved notches.

It may have been used as:

• A lunar phase counter

• A menstrual calendar

• A way to track time

Ishango Bone

The Ishango Bone was found near the Nile River in the Democratic Republic of Congo. It dates to around 20,000 BCE.

It contains groups of notches, including:

$$11,13,17,19$$

These are prime numbers between 10 and 20.

This shows that early humans had begun to understand number patterns.

Prime Numbers

A prime number is a number greater than 1 that has only two factors: 1 and itself.

Examples:

$$2,3,5,7,11,13,17,19$$

3.1.2 The Indian Context: Trade and Astronomy

As civilisation developed, people needed larger and more organised number systems.

In the Indus Valley Civilisation, cities like Lothal and Harappa used standard weights and measures for trade.

Merchants needed numbers for:

• Counting goods

• Measuring weight

• Exchanging items

• Keeping accounts

Indian thinkers also studied large numbers. In the Vedas, names were given to powers of 10 up to very large numbers.

For example, powers of 10 helped in the development of the decimal number system.

Importance of Powers of 10

Our number system is based on powers of 10.

Examples:

$${10}^1=10{10}^2=100{10}^3=1000{10}^4=10000$$

The development of the Indian place-value system prepared the way for one of the greatest mathematical inventions: zero.

3.2 The Revolution of Śhūnya: When Nothing Became Something

For a long time, people could count objects, but they did not treat “nothing” as a number.

For example, if a person had 5 apples and gave away all 5, earlier people simply thought the person had nothing. But they did not always have a number to represent that nothing.

The Indian concept of Śhūnya, meaning zero or emptiness, changed mathematics forever.

Zero

Zero is written as:

$$0$$

It represents nothing, but in mathematics it is a number with its own rules.

3.2.1 From Philosophy to Mathematics: The Concept of Śhūnyatā

In Indian philosophy, especially in the Upanishads and Buddhist literature, the idea of Śhūnyatā meant emptiness or nothingness.

This philosophical idea later entered mathematics.

Indian thinkers were comfortable with the idea of “nothingness”, which helped them accept zero as a mathematical number.

Thus, the philosophical idea of emptiness became the mathematical idea of zero.

3.2.2 The Bakhśhālī Manuscript and Brahmagupta’s Rules

The Bakhśhālī Manuscript used a dot or bindu to represent zero.

But zero became a proper mathematical number when Brahmagupta gave rules for it in his work Brāhmasphuṭasiddhānta in 628 CE.

Brahmagupta defined zero as:

$$a-a=0$$

This means when a number is subtracted from itself, the result is zero.

Brahmagupta’s Rules for Zero

Rule 1: Adding Zero

$$a+0=a$$

When zero is added to a number, the number remains unchanged.

Example:

$$7+0=7$$

Rule 2: Subtracting Zero

$$a-0=a$$

When zero is subtracted from a number, the number remains unchanged.

Example:

$$9-0=9$$

Rule 3: Multiplying by Zero

$$a\times 0=0$$

When any number is multiplied by zero, the result is zero.

Example:

$$12\times 0=0$$

3.3 Integers: Expanding the Horizon

Brahmagupta also introduced the idea of negative numbers.

He explained numbers using the idea of:

• Fortunes or Dhana: positive numbers

• Debts or Ṛiṇa: negative numbers

For example:

$$+5$$

means a fortune of 5.

$$-5$$

means a debt of 5.

Integers

Integers include positive numbers, negative numbers, and zero.

$$\mathbb{Z}=\{...,-3,-2,-1,0,1,2,3,...\}$$

Important Points

• Positive integers lie to the right of zero on the number line.

• Negative integers lie to the left of zero.

• Zero is neither positive nor negative.

Number Line for Integers

$$-5,-4,-3,-2,-1,0,1,2,3,4,5$$

Moving right means the value increases.

Moving left means the value decreases.

3.3.1 The Arithmetic of Integers

Brahmagupta gave rules for arithmetic with positive and negative numbers.

Rule 1: Fortune + Fortune = Fortune

$$5+4=9$$

Positive plus positive gives positive.

Rule 2: Debt + Debt = Debt

$$(-5)+(-4)=-9$$

Negative plus negative gives negative.

Rule 3: Fortune minus Zero = Fortune

$$7-0=7$$

Rule 4: Debt minus Zero = Debt

$$-6-0=-6$$

Rule 5: Debt × Fortune = Debt

$$(-3)\times 4=-12$$

Negative multiplied by positive gives negative.

Rule 6: Debt × Debt = Fortune

$$(-3)\times (-4)=12$$

Negative multiplied by negative gives positive.

Why is negative × negative positive?

If a debt is removed, your financial condition improves.

Example:

If someone removes 4 debts of ₹3 each, you gain ₹12.

So,

$$(-3)\times (-4)=+12$$

3.4 Filling the Spaces: Fractions and Rational Numbers

Counting numbers and integers are not enough for measuring parts of a whole.

For example:

• Half a cup of milk

• One-third of a field

• Three-fourths of a metre

These are represented by fractions.

Fraction

A fraction represents a part of a whole.

Example:

$$\frac{3}{4}$$

means 3 parts out of 4 equal parts.

Fractions can be positive or negative.

Examples:

$$\frac{3}{4},-\frac{3}{4}$$

Rational Numbers

A rational number is any number that can be written in the form:

$$\frac{p}{q}$$

where $p$and $q$are integers and $q\ne 0$.

$$\mathbb{Q}=\{\frac{p}{q}:p,q\in \mathbb{Z},q\ne 0\}$$

Examples of Rational Numbers

$$\frac{2}{3},\frac{-5}{7},4,-9,0,1.25,0.333...$$

Integers are also rational numbers because they can be written with denominator 1.

Example:

$$5=\frac{5}{1}-10=\frac{-10}{1}$$

Why is $𝐪\mathbf{\ne }\mathrm{𝟎}$?

Division by zero is not defined.

So, a rational number cannot have zero as its denominator.

$$\frac{5}{0}$$

is not defined.

Equivalent Rational Numbers

Rational numbers can have many equivalent forms.

Example:

$$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{10}{20}$$

These fractions represent the same value.

Lowest Form

A rational number is usually written in its simplest form, where the numerator and denominator have no common factor except 1.

Example:

$$\frac{12}{30}=\frac{2}{5}$$

Laws of Rational Numbers

1. Equality of Rational Numbers

Two rational numbers

$$\frac{a}{b}$$

and

$$\frac{c}{d}$$

are equal if:

$$ad=bc$$

Example:

$$\frac{2}{3}=\frac{4}{6}$$

because:

$$2\times 6=3\times 412=12$$

2. Addition of Rational Numbers

If denominators are the same:

$$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$$

Example:

$$\frac{2}{7}+\frac{3}{7}=\frac{5}{7}$$

If denominators are different, first make them equal.

Example:

$$\frac{1}{2}+\frac{1}{3}$$

LCM of 2 and 3 is 6.

$$\frac{1}{2}=\frac{3}{6}\frac{1}{3}=\frac{2}{6}$$

So,

$$\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$$

3. Subtraction of Rational Numbers

$$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$$

Example:

$$\frac{5}{8}-\frac{3}{8}=\frac{2}{8}=\frac{1}{4}$$

4. Multiplication of Rational Numbers

$$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$$

Example:

$$\frac{2}{3}\times \frac{5}{7}=\frac{10}{21}$$

5. Division of Rational Numbers

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$$

where $c\ne 0$.

Example:

$$\frac{2}{3}÷\frac{5}{7}=\frac{2}{3}\times \frac{7}{5}=\frac{14}{15}$$

Properties of Rational Numbers

Closure Property

Rational numbers are closed under:

• Addition

• Subtraction

• Multiplication

• Division, except division by zero

This means the result is again a rational number.

Commutative Property

Addition and multiplication are commutative.

$$a+b=b+aa\times b=b\times a$$

Distributive Property

$$p(q+r)=pq+pr$$

Example:

$$2(3+4)=2\times 3+2\times 42\times 7=6+814=14$$

3.4.1 Representation of Rational Numbers on the Number Line

Rational numbers can be represented on a number line.

Method

To represent:

$$\frac{p}{q}$$

on the number line:

1. Divide each unit interval into $q$equal parts.

2. Move $p$parts from zero.

3. Move right if the number is positive.

4. Move left if the number is negative.

Example 1

To represent:

$$\frac{3}{4}$$

Divide the interval between 0 and 1 into 4 equal parts. Then move 3 parts to the right.

Example 2

To represent:

$$-\frac{3}{4}$$

Divide the interval between 0 and -1 into 4 equal parts. Then move 3 parts to the left.

Example 3

To represent:

$$\frac{9}{4}$$

Convert it into mixed form:

$$\frac{9}{4}=2\frac{1}{4}$$

So it lies between 2 and 3.

Absolute Value of a Rational Number

The absolute value of a rational number is its distance from zero on the number line.

It is written as:

$$\mid x\mid$$

Examples

$$\mid \frac{5}{3}\mid =\frac{5}{3}\mid -\frac{5}{3}\mid =\frac{5}{3}\mid 0\mid =0$$

The absolute value is always non-negative.

$$\mid x\mid \ge 0$$

Distance Between Two Rational Numbers

The distance between two rational numbers $a$and $b$is:

$$\mid a-b\mid$$

Example:

Distance between -4 and 3:

$$\mid 3-(-4)\mid =\mid 7\mid =7$$

3.4.2 The Density of Rational Numbers

Rational numbers are dense.

This means that between any two rational numbers, there is always another rational number.

In fact, there are infinitely many rational numbers between any two rational numbers.

Method to Find a Rational Number Between Two Numbers

Take their average.

If $a$and $b$are two rational numbers, then a rational number between them is:

$$\frac{a+b}{2}$$

Example

Find a rational number between 1 and 2.

$$\frac{1+2}{2}=\frac{3}{2}$$

So,

$$\frac{3}{2}$$

lies between 1 and 2.

Find a rational number between 1 and $\frac{3}{2}$.

$$\frac{1+\frac{3}{2}}{2}=\frac{\frac{2}{2}+\frac{3}{2}}{2}=\frac{\frac{5}{2}}{2}=\frac{5}{4}$$

So,

$$\frac{5}{4}$$

lies between 1 and $\frac{3}{2}$.

3.5 Irrational Numbers

Rational numbers are very dense, but they still do not fill the entire number line.

Some numbers cannot be expressed as a ratio of two integers. These are called irrational numbers.

Irrational Number

An irrational number cannot be written in the form:

$$\frac{p}{q}$$

where $p$and $q$are integers and $q\ne 0$.

Examples

$$\sqrt{2},\sqrt{3},\sqrt{5},\pi$$

Their decimal expansions are non-terminating and non-repeating.

How Irrational Numbers Arise

Consider a square of side 1 unit.

Using the Baudhāyana–Pythagoras Theorem:

$$d^2=1^2+1^2{d}^2=1+1d^2=2d=\sqrt{2}$$

The diagonal of a unit square has length $\sqrt{2}$, which is irrational.

3.5.1 The Proof of Irrationality of $\sqrt{\mathrm{𝟐}}$

We prove that $\sqrt{2}$is irrational using proof by contradiction.

Proof

Assume that $\sqrt{2}$is rational.

So,

$$\sqrt{2}=\frac{p}{q}$$

where $p$and $q$are integers, $q\ne 0$, and $\frac{p}{q}$is in lowest form.

Squaring both sides:

$$2=\frac{p^2}{q^2}$$

Multiplying both sides by $q^2$:

$$2q^2=p^2$$

So, $p^2$is even.

Therefore, $p$is even.

Let:

$$p=2k$$

Substitute this in the equation:

$$2q^2={\left(2k\right)}^{2}2q^2=4k^2$$

Divide by 2:

$$q^2=2k^2$$

So, $q^2$is even.

Therefore, $q$is also even.

Now both $p$and $q$are even, so they have a common factor 2.

But this contradicts our assumption that $\frac{p}{q}$was in lowest form.

Therefore, our assumption is wrong.

Hence,

$$\sqrt{2}$$

is irrational.

3.5.2 Construction of Length $\sqrt{𝐧}$

Irrational numbers like $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, etc., can be constructed geometrically on the number line.

Construction of $\sqrt{\mathrm{𝟐}}$

Step 1

Draw a number line and mark point $O$as 0.

Mark point $A$such that:

$$OA=1$$

Step 2

At point $A$, draw a perpendicular line.

On this perpendicular line, mark point $B$such that:

$$AB=1$$

Step 3

Join $O$to $B$.

Using the Pythagoras theorem:

$$O{B}^2=O{A}^2+A{B}^{2}O{B}^2=1^2+1^{2}O{B}^2=2OB=\sqrt{2}$$

Step 4

With $O$as centre and $OB$as radius, draw an arc cutting the number line at $P$.

Then:

$$OP=\sqrt{2}$$

So, point $P$represents $\sqrt{2}$on the number line.

3.5.3 The Story of Pi $\left(𝛑\right)$and Madhava’s Infinite Series

Another famous irrational number is:

$$\pi$$

Pi is the ratio of the circumference of a circle to its diameter.

$$\pi =\frac{\mathrm{\text{Circumference}}}{\mathrm{\text{Diameter}}}$$

Approximate value:

$$\pi \approx 3.1416$$

Āryabhaṭa’s Approximation

Āryabhaṭa gave the approximation:

$$\frac{3927}{1250}=3.1416$$

He understood that this was only an approximation.

Madhava’s Infinite Series

Mādhava of Sangamagrama discovered an infinite series for $\pi$:

$$\pi =4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots )$$

This shows that irrational numbers cannot be expressed exactly using a single fraction. Instead, they may require infinite processes.

3.6 Real Numbers: Decimals and Cyclic Patterns

Real numbers include both rational and irrational numbers.

$$ℝ=\mathbb{Q}\cup \mathrm{\text{Irrational Numbers}}$$

The real number line is continuous and unbroken.

Every point on the number line represents a real number.

Real Numbers

Real numbers include:

• Natural numbers

• Whole numbers

• Integers

• Rational numbers

• Irrational numbers

Examples:

$$1,0,-5,\frac{3}{4},0.25,\sqrt{2},\pi$$

3.6.1 Rational Decimals: Terminating and Repeating

The decimal expansion of a rational number is always either:

1. Terminating

2. Non-terminating repeating

Terminating Decimal

A terminating decimal stops after a finite number of digits.

Example:

$$\frac{3}{8}=0.375$$

The decimal stops after three digits.

Other examples:

$$0.5,0.25,1.75,3.125$$

Non-Terminating Repeating Decimal

A non-terminating repeating decimal continues forever, but a digit or group of digits repeats.

Example:

$$\frac{5}{11}=0.454545...$$

This can be written as:

$$0.\overline{45}$$

Another example:

$$\frac{1}{3}=0.3333...=0.\bar{3}$$

Predicting the Type of Decimal Expansion

For a rational number:

$$\frac{p}{q}$$

in lowest form, the decimal expansion is terminating if the prime factors of $q$are only 2, only 5, or both 2 and 5.

Examples

$$\frac{3}{20}$$

Prime factorisation:

$$20=2^2\times 5$$

So, it is terminating.

$$\frac{3}{20}=0.15$$

Non-Terminating Repeating Decimal

If the denominator has prime factors other than 2 and 5, the decimal expansion is repeating.

Example:

$$\frac{4}{15}$$

Prime factorisation:

$$15=3\times 5$$

Since 3 is present, the decimal is non-terminating repeating.

Converting Decimals into $\frac{𝐩}{𝐪}$Form

Case 1: Terminating Decimals

Example:

Convert 0.35 into fraction form.

$$0.35=\frac{35}{100}=\frac{7}{20}$$

Case 2: Pure Repeating Decimals

A pure repeating decimal starts repeating immediately after the decimal point.

Example:

$$0.\bar{6}$$

Let:

$$x=0.\bar{6}$$

Multiply by 10:

$$10x=6.\bar{6}$$

Subtract:

$$10x-x=6.\bar{6}-0.\bar{6}9x=6x=\frac{6}{9}=\frac{2}{3}$$

So,

$$0.\bar{6}=\frac{2}{3}$$

Example: Convert $\mathrm{𝟎}\mathbf{.}\overline{\mathrm{𝟒𝟓}}$

Let:

$$x=0.\overline{45}$$

Since two digits repeat, multiply by 100:

$$100x=45.\overline{45}$$

Subtract:

$$100x-x=45.\overline{45}-0.\overline{45}99x=45x=\frac{45}{99}=\frac{5}{11}$$

So,

$$0.\overline{45}=\frac{5}{11}$$

Case 3: General Repeating Decimals

A general repeating decimal has some non-repeating digits before the repeating part.

Example:

$$0.1\bar{6}$$

Let:

$$x=0.1\bar{6}$$

There is one non-repeating digit, so multiply by 10:

$$10x=1.\bar{6}$$

Now multiply again by 10 because one digit repeats:

$$100x=16.\bar{6}$$

Subtract:

$$100x-10x=16.\bar{6}-1.\bar{6}90x=15x=\frac{15}{90}=\frac{1}{6}$$

So,

$$0.1\bar{6}=\frac{1}{6}$$

3.6.2 The Magic of Cyclic Numbers

Some repeating decimals have special repeating blocks called cyclic numbers.

Example:

$$\frac{1}{7}=0.\overline{142857}$$

The repeating block is:

$$142857$$

Now multiply it by numbers from 1 to 6:

$$142857\times 1=142857142857\times 2=285714142857\times 3=428571142857\times 4=571428142857\times 5=714285142857\times 6=857142$$

The same digits appear again and again in a cyclic order.

This is why 142857 is called a cyclic number.

3.6.3 Irrational Decimals: Chaos and Infinity

Irrational numbers have decimal expansions that:

• Never end

• Never repeat

• Have no fixed repeating pattern

Examples:

$$\sqrt{2}=1.414213562373095...\pi =3.141592653589793...$$

These decimals go on forever without repeating.

Non-Uniqueness of Decimal Representations

Some rational numbers can be written in two decimal forms.

Example:

$$1=1.0000...$$

and also:

$$1=0.9999...$$

So,

$$0.\bar{9}=1$$

Proof

Let:

$$x=0.\bar{9}$$

Multiply by 10:

$$10x=9.\bar{9}$$

Subtract:

$$10x-x=9.\bar{9}-0.\bar{9}9x=9x=1$$

Therefore:

$$0.\bar{9}=1$$

3.7 Conclusion: The Never-Ending Journey

The chapter ends by showing that numbers developed because human needs developed.

We moved from counting numbers to zero, negative numbers, fractions, irrational numbers, and real numbers.

The real number line contains all rational and irrational numbers.

But the journey does not end here. There are numbers beyond real numbers, such as imaginary numbers.

The Evolution of Our World of Numbers

1. Natural Numbers

$$\mathbb{N}=\{1,2,3,4,...\}$$

Used for counting.

2. Integers

$$\mathbb{Z}=\{...,-3,-2,-1,0,1,2,3,...\}$$

Include natural numbers, zero, and negative numbers.

3. Rational Numbers

$$\mathbb{Q}=\{\frac{p}{q}:p,q\in \mathbb{Z},q\ne 0\}$$

Include integers and fractions.

Their decimals are terminating or repeating.

4. Irrational Numbers

Cannot be written as $\frac{p}{q}$.

Examples:

$$\sqrt{2},\sqrt{3},\pi$$

Their decimals are non-terminating and non-repeating.

5. Real Numbers

$$ℝ$$

Include both rational and irrational numbers.

They form the complete number line.

Imaginary Numbers: A Glimpse

The chapter also asks:

What is:

$$\sqrt{-1}$$

There is no real number whose square is -1.

So mathematicians introduced a new number:

$$i=\sqrt{-1}$$

This is called an imaginary number.

Imaginary numbers are studied in higher classes and are useful in:

• Electrical engineering

• Quantum mechanics

• Modern technology

Important Definitions

Natural Numbers

Counting numbers starting from 1.

$$1,2,3,4,...$$

Integers

Positive numbers, negative numbers, and zero.

$$...,-2,-1,0,1,2,...$$

Rational Numbers

Numbers that can be written as:

$$\frac{p}{q},q\ne 0$$

Irrational Numbers

Numbers that cannot be written as a ratio of two integers.

Real Numbers

All rational and irrational numbers together.

Absolute Value

Distance of a number from zero.

$$\mid x\mid$$

Dense Property

Between any two rational numbers, infinitely many rational numbers exist.

Terminating Decimal

A decimal that ends.

Example:

$$0.25$$

Repeating Decimal

A decimal in which digits repeat forever.

Example:

$$0.\bar{3}$$

Non-Repeating Decimal

A decimal that never repeats.

Example:

$$\pi =3.141592...$$

Key Formulae and Rules

Rational Number

$$\frac{p}{q},q\ne 0$$

Equality of Rational Numbers

$$\frac{a}{b}=\frac{c}{d}\iff ad=bc$$

Addition

$$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$$

Subtraction

$$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$$

Multiplication

$$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}$$

Division

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}$$

Distance Between Two Numbers

$$\mid a-b\mid$$

Rational Number Between Two Numbers

$$\frac{a+b}{2}$$

Pythagoras Theorem

$${\mathrm{\text{Hypotenuse}}}^2={\mathrm{\text{Base}}}^2+{\mathrm{\text{Height}}}^2$$

Used to construct irrational lengths like $\sqrt{2}$.

Quick Chapter Summary

This chapter explains the growth of numbers from simple counting to advanced number systems. Humans first used natural numbers for counting. Ancient tally marks on bones show that people understood numbers thousands of years ago. In India, trade, astronomy, and the use of powers of 10 helped develop a strong number system.

The concept of Śhūnya, or zero, was one of the greatest mathematical developments. Brahmagupta gave rules for zero and negative numbers. Negative numbers were explained as debts and positive numbers as fortunes.

The chapter then introduces fractions and rational numbers. Rational numbers are numbers that can be written in the form $\frac{p}{q}$, where $q\ne 0$. They can be placed on the number line and are dense, meaning infinitely many rational numbers exist between any two rational numbers.

However, rational numbers do not fill the entire number line. Numbers like $\sqrt{2}$and $\pi$are irrational because they cannot be written as fractions. The proof of irrationality of $\sqrt{2}$is done using contradiction.

Together, rational and irrational numbers form real numbers. Decimal expansions help us identify numbers: rational numbers have terminating or repeating decimals, while irrational numbers have non-terminating and non-repeating decimals. The chapter also discusses cyclic numbers like 142857, which appears in the decimal expansion of $\frac{1}{7}$.

The chapter concludes by showing that mathematics is a never-ending journey. Beyond real numbers, there are imaginary numbers such as $\sqrt{-1}$, which are studied in higher classes.

Most Important Points to Remember

1. Natural numbers are used for counting.

2. Zero was formally developed in Indian mathematics.

3. Brahmagupta gave rules for zero and negative numbers.

4. Integers include positive numbers, negative numbers, and zero.

5. Rational numbers can be written in the form $\frac{p}{q}$, where $q\ne 0$.

6. Rational numbers are dense on the number line.

7. Irrational numbers cannot be written as fractions.

8. $\sqrt{2}$is irrational.

9. $\pi$is also irrational.

10. Rational decimals are terminating or repeating.

11. Irrational decimals are non-terminating and non-repeating.

12. Real numbers include both rational and irrational numbers.

13. The real number line is continuous and unbroken.

14. Cyclic numbers show beautiful repeating patterns.

15. Imaginary numbers go beyond the real number line.