Table of Contents

Number Theory

Natural Numbers

Natural numbers represent the simplest and most natural way of counting. These are the numbers we use every day, like 1, 2, 3, 4, 5, 6, 7, 8, 9, and so on They are always positive and have no fractional or decimal part. If a number doesn’t carry a sign, it is considered positive.

Example: 5 + 5 = 10 is a natural number.
Similarly, 1678 + 1678 = 3356 is a natural number.

4.2 Whole Numbers

Whole numbers came into existence with the introduction of zero. Early humans did not use zero, but over time, its significance grew, and it became part of the counting set. Whole numbers include all natural numbers and zero.

Whole Numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …

Natural Numbers	Whole Numbers
1, 2, 3, 4, 5	0, 1, 2, 3, 4, 5

4.3 Negative Numbers

Negative numbers come into play when we count backwards from zero. These numbers are less than zero and are denoted with a minus sign (-).

Example: The temperature -20°C is 20 degrees below zero.

Key Points:

0 is neither positive nor negative.
Negative numbers decrease as they move farther from zero: -1, -2, -3, …

Decreasing (Negative Numbers)	Increasing (Positive Numbers)
… -4, -3, -2, -1	1, 2, 3, 4, …

4.4 Integers

Integers are a combination of whole numbers and negative numbers, forming a new set:

Integers: …, -4, -3, -2, -1, 0, 1, 2, 3, 4, …

Negative Integers	Zero	Positive Integers
… -3, -2, -1	0	1, 2, 3, …

4.5 Operations

Mathematical operations are processes used to manipulate numbers. The four basic operations are:

Addition
Subtraction
Multiplication
Division

Let’s explore these operations in detail:

4.5.1 Addition

Addition involves combining two or more numbers to find their total. The symbol for addition is + (read as “plus”).

Example: 2 + 6 = 8
Here, 2 and 6 are called addends, and 8 is the sum.

Properties of Addition:

Commutative Law: The order of addition doesn’t matter.
- a + b = b + a
- Example: 4 + 6 = 6 + 4 = 10
Associative Law: Grouping doesn’t affect the result.
- (a + b) + c = a + (b + c)
- Example: (2 + 3) + 6 = 2 + (3 + 6) = 11
Adding two positive numbers gives a positive sum.
- Example: 2 + 5 = 7
Adding two negative numbers gives a negative sum.
- Example: (-2) + (-5) = -7
Adding a positive and a negative number is equivalent to subtraction.

4.5.2 Subtraction

Subtraction means taking one number away from another. The symbol for subtraction is – (read as “minus”).

Example: If you have 5 oranges and take away 2, you’re left with 3 oranges.

Subtraction is represented as:

Minuend – Subtrahend = Difference

Minuend: The number being subtracted from.
Subtrahend: The number to subtract.
Difference: The result.
Example: 8 – 2 = 6
- Here, 8 is the minuend, 2 is the subtrahend, and 6 is the difference.

Properties of Subtraction:

Subtraction is not commutative: a – b ≠ b – a
Subtraction is not associative: (a – b) – c ≠ a – (b – c)
The difference is positive if the minuend is larger than the subtrahend.
- Example: 5 – 3 = 2
The difference is negative if the subtrahend is larger than the minuend.
- Example: 3 – 5 = -2

4.5.3 Multiplication

Multiplication is the repeated addition of a number. The symbol for multiplication is × (read as “times” or “multiplied by”).

Formula: a × b = c

a: Multiplicand
b: Multiplier
c: Product
Example: 3 × 4 = 12
- Here, 3 is the multiplicand, 4 is the multiplier, and 12 is the product.

Properties of Multiplication:

Commutative Law: Changing the order doesn’t affect the product.
- Example: 2 × 6 = 6 × 2 = 12
Associative Law: Grouping doesn’t affect the product.
- Example: (2 × 3) × 6 = 2 × (3 × 6) = 36
Signs of the product:
- (+ × + = +), (- × – = +): Same signs give a positive product.
- (+ × – = -), (- × + = -): Different signs give a negative product.
Any number multiplied by 1 equals itself.
- Example: 5 × 1 = 5
Any number multiplied by 0 equals 0.
- Example: 100 × 0 = 0
Multiplication by powers of 10 shifts the digits.
- Example: 3 × 10 = 30, 5 × 100 = 500

4.5.4 Division

Division is the process of distributing a number into equal parts. The symbol for division is ÷ (read as “divided by”).

Formula: Dividend ÷ Divisor = Quotient + Remainder

Dividend: The number being divided.
Divisor: The number you divide by.
Quotient: The result of division.
Remainder: What’s left after division.
Example: 22 ÷ 2 = 11

Properties of Division:

Division is not commutative: a ÷ b ≠ b ÷ a
Division is not associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
Signs of the quotient:
- (+ ÷ + = +), (- ÷ – = +): Same signs give a positive quotient.
- (+ ÷ – = -), (- ÷ + = -): Different signs give a negative quotient.
The remainder is always positive.

1. Addition

Definition

Addition is the process of combining two or more numbers to get their total or sum. It is represented by the symbol +.

Formula

$a + b = c$

Where

$a$

and b , $c$ is the sum.

Properties of Addition

Commutative Property
- Definition: The order of numbers does not affect the sum.
- Formula: $a + b = b + a$
  
  Example:
  
  $3 + 5 = 8 ,$
  
  $5 + 3 = 8$ $5 + 3 = 8$ .
Associative Property
- Definition: The grouping of numbers does not affect the sum.
- Formula: $(a + b) + c = a + (b + c)$
  
  Example:
  
  $(2 + 4) + 6 = 2 + (4 + 6) = 12$ $(2 + 4) + 6 = 2 + (4 + 6) = 12$ .
Identity Property
- Definition: Adding 0 to any number gives the same number.
- Formula: $a + 0 = a$
  
  a+0=a
- Example: $7 + 0 = 7$
  
  $7 + 0 = 7$ .
Adding Positives and Negatives
- Rule: Adding a positive increases the value; adding a negative decreases it.
- Examples:
  - Positive + Positive: $4 + 6 = 10$
    
    4+6=10
  - Positive + Negative: $7 + (-3) = 4$
    
    $7 + (- 3) = 4$ .

2. Subtraction

Definition

Subtraction is the process of finding the difference between two numbers by removing one from the other. It is represented by the symbol −.

Formula

$a – b = c$

$a - b = c$
Where

$a$

$a$ is the minuend,

$b$

$b$ is the subtrahend, and

$c$

$c$ is the difference.

Properties of Subtraction

Non-Commutative Property
- Definition: Changing the order changes the result.
- Formula: $a – b \neq b – a$
  
  a−b=b−a
- Example: $8 - 5 = 3$
- $5 – 8 = -3$ $5 - 8 = - 3$ .
Non-Associative Property
- Definition: Changing the grouping changes the result.
- Formula: $(a – b) – c \neq a – (b – c)$
  
  (a−b)−c =a−(b−c)
- Example: $(10 - 5) - 2 = 3$
  
  $10 – (5 – 2) = 7$ $10 - (5 - 2) = 7$ .
Subtracting Zero
- Rule: Subtracting 0 from a number leaves it unchanged.
- Formula: $a - 0 = a$
- $a -0 =a$
- Example: $9 – 0 = 9$
  
  $9 - 0 = 9$ .
Subtracting the Same Number
- Rule: Subtracting a number from itself gives 0.
- Formula: $a – a = 0$
  
  a−a=0
- Example: $12 – 12 = 0$
  
  $12 - 12 = 0$ .

3. Multiplication

Definition

Multiplication is the process of adding a number to itself repeatedly a certain number of times. It is represented by the symbol × or ·.

Formula

$a \times b = c$

Where

$a$

$a$ is the multiplicand

$b$

$b$ is the multiplier, and

$c$

$c$ is the product.

Properties of Multiplication

Commutative Property
- Definition: The order of factors does not affect the product.
- Formula: $a \times b = b \times a$
- Example: $3 \times 4 = 12$
- $3 \times 4 = 12$
  
  $3 \times 4 = 12$ and
  
  $4 \times 3 = 12$ $4 \times 3 = 12$ .
Associative Property
- Definition: The grouping of factors does not affect the product.
- Formula: $(a \times b) \times c = a \times (b \times c)$
- Example: $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$
- $(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24$ .
Distributive Property
- Definition: A number multiplied by a sum or difference equals the sum or difference of the products.
- Formula:
  $a \times (b + c) = (a \times b) + (a \times c)$
- $a \times (b – c) = (a \times b) – (a \times c)$
- Example:
  $2 \times (4 + 3) = (2 \times 4) + (2 \times 3) = 8 + 6 = 14$
- $2 \times (4 + 3) = (2 \times 4) + (2 \times 3) = 8 + 6 = 14$ .
Identity Property
- Rule: Multiplying any number by 1 gives the same number.
- Formula: $a \times 1 = a$
- $a \times 1 = a$
- Example: $9 \times 1 = 9$
- $9 \times 1 = 9$ .
Zero Property
- Rule: Multiplying any number by 0 gives 0.
- Formula: $a \times 0 = 0$
  
  $a \times 0 = 0$
- Example: $7 \times 0 = 0$
  
  $7 \times 0 = 0$ .
Sign Rules
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Examples:
  $4 \times 5 = 20$
  
  $4 \times 5 = 20$ ,
  
  $(-3) \times (-2) = 6$ $(- 3) \times (- 2) = 6$ ,
  
  $5 \times (-3) = -15$ $5 \times (- 3) = - 15$ .

4. Division

Definition

Division is the process of splitting a number into equal parts. It is represented by the symbol ÷ or as a fraction.

Formula

$a \div b = c$

Where

$a$

$a$ is the dividend,

$b$

$b$ is the divisor, and

$c$

$c$ is the quotient.

Properties of Division

Non-Commutative Property
- Definition: Changing the order changes the result.
- Formula: $a \div b \neq b \div a$
- Example: $10 \div 2 = 5$
  
  $10 \div 2 = 5$ , but
  
  $2 \div 10 = 0.2$ $2 \div 10 = 0.2$ .
Non-Associative Property
- Definition: Changing the grouping changes the result.
- Formula: $(a \div b) \div c \neq a \div (b \div c)$
- Example: $(12 \div 3) \div 2 = 2$
  
  $(12 \div 3) \div 2 = 2$ , but
  
  $12 \div (3 \div 2) = 8$ $12 \div (3 \div 2) = 8$ .
Identity Property
- Rule: Dividing a number by 1 leaves it unchanged.
- Formula: $a \div 1 = a$
  
  $a \div 1 = a$
- Example: $9 \div 1 = 9$
  
  $9 \div 1 = 9$ .
Division by Zero
- Rule: Division by 0 is undefined.
- Example: $8 \div 0$
  
  $8 \div 0$ is undefined.
Sign Rules
- Positive ÷ Positive = Positive
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Examples:
  $20 \div 4 = 5$
  
  $20 \div 4 = 5$ ,
  
  $(- 12) \div (- 3) = 4$ $15 \div (- 3) = - 5$

1. Exponents

Definition

An exponent indicates how many times a base number is multiplied by itself. It is written in the form

$a^n$

$a^{n}$ , where:

$a$

$a$ is the base, and
$n$

$n$ is the exponent (or power).

Example:

$2^3 = 2 \times 2 \times 2 = 8$

$2^{3} = 2 \times 2 \times 2 = 8$
Here,

$2$

$2$ is the base, and

$3$

$3$ is the exponent.

Key Properties of Exponents

Zero Exponent
- Rule: Any number raised to the power of 0 equals 1.
- Formula: $a^0 = 1$
  
  $a^{0} = 1$ (where
  
  $a \neq 0$ $a \neq = 0$ )
- Examples:
  $5^0 = 1$
  
  $5^{0} = 1$ ,
  
  $10^0 = 1$ $1 0^{0} = 1$ ,
  
  $(-7)^0 = 1$ $(- 7)^{0} = 1$ .
Multiplying Powers with the Same Base
- Rule: Add the exponents.
- Formula: $a^{m} \times a^{n} = a^{m + n}$
- Examples:
  $2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$
  
  $2^{3} \times 2^{2} = 2^{3 + 2} = 2^{5} = 32$ ,
  
  $3^4 \times 3^1 = 3^{4+1} = 3^5 = 243$ $3^{4} \times 3^{1} = 3^{4 + 1} = 3^{5} = 243$ .
Dividing Powers with the Same Base
- Rule: Subtract the exponents.
- Formula: $a^{m} \div a^{n} = a^{m - n= a m − n}$
- $(where$
  
  $m > n$ $m > n$ )
  
  Examples:
  
  $5^4 \div 5^2 = 5^{4-2} = 5^2 = 25$ $5^{4} \div 5^{2} = 5^{4 - 2} = 5^{2} = 25$ ,
  
  $10^6 \div 10^3 = 10^{6-3} = 10^3 = 1,000$ $1 0^{6} \div 1 0^{3} = 1 0^{6 - 3} = 1 0^{3} = 1, 000$ .
Raising a Power to Another Power
- Rule: Multiply the exponents.
- Formula: $(a^m)^n = a^{m \times n}$
  
  $(a^{m})^{n} = a$
- Examples:
  $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$
  
  $(2^{3})^{2} = 2^{3 \times 2} = 2^{6} = 64$ ,
  
  $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$ $(3^{2})^{3} = 3^{2 \times 3} = 3^{6} = 729$ .
Negative Exponents
- Rule: A negative exponent indicates a reciprocal.
- Formula: $a^{-n} = \frac{1}{a^n}$
  
  $a^{- n} = a ^{n} 1$
- Examples:
  $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
  
  $2^{-3} = 2 ^{3} 1 = 8 1$ ,
  
  $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$ $5^{-2} = 5 ^{2} 1 = 25 1$ .
Exponent of 1
- Rule: Any number raised to the power of 1 equals the number itself.
- Formula: $a^1 = a$
  
  $a^{1} = a$
- Examples:
  $7^1 = 7$
  
  $7^{1} = 7$ ,
  
  $15^1 = 15$ $1 5^{1} = 15$ .

Examples of Exponents

$4^2 = 4 \times 4 = 16$

$4^{2} = 4 \times 4 = 16$
$3^4 = 3 \times 3 \times 3 \times 3 = 81$

$3^{4} = 3 \times 3 \times 3 \times 3 = 81$
$10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000$

$1 0^{5} = 10 \times 10 \times 10 \times 10 \times 10 = 100, 000$
$2^{-4} = \frac{1}{2^4} = \frac{1}{16}$

$2^{-4} = 2 ^{4} 1 = 16 1$
$(5^2)^3 = 5^{2 \times 3} = 5^6 = 15,625$

$(5^{2})^{3} = 5^{2 \times 3} = 5^{6} = 15, 625$

Exponent or Power of a Number

The exponent (or power) of a number tells us how many times to multiply the number by itself. For example, in the expression

$6^3$

the ‘3’ is called the exponent or power, and it means multiplying 6 by itself three times:

$6^3 = 6 \times 6 \times 6 = 216$

The exponent helps us represent large numbers or repeated multiplications in a simpler way.

Square of a Number

When the exponent is 2, it is called the square of the number. For example,

$6^2$

means multiplying 6 by itself twice:

$6^2 = 6 \times 6 = 36$

So, we read

$6^2$

as “6 to the power of 2” or “6 squared.” Another example is the square of 5:

$5^2 = 5 \times 5 = 25$

Cube of a Number

When the exponent is 3, it is called the cube of the number. For example,

$4^3$

means multiplying 4 by itself three times:

$4^3 = 4 \times 4 \times 4 = 64$

So, we read

$4^3$

as “4 to the power of 3” or “4 cubed.” Another example is the cube of 5:

$5^3 = 5 \times 5 \times 5 = 125$

Large Numbers with Exponents

Exponents make it easier to write large numbers. For example:

$10^6 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000$

This is a much simpler way of writing large numbers.

Negative Exponents

A negative exponent means dividing 1 by that number raised to the positive exponent. For example,

$2^{-3}$

means:

$2^{- 3} = \frac{1}{2^{3}} = \frac{1}{2 \times 2 \times 2} = \frac{1}{8}$

So,

$a^{-n} = \frac{1}{a^n}$

$a^{- n} = a ^{n} 1$ .

Properties of Exponents

Any number raised to the power of 0 is 1:
$a^0 = 1$ $a^{0} = 1$ Example:

$2^0 = 1, \quad 8^0 = 1, \quad 100^0 = 1$
Multiplying numbers with the same base adds the exponents:
$a^m \times a^n = a^{m+n}$ Example:

$2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$
Dividing numbers with the same base subtracts the exponents:
$\frac{a^m}{a^n} = a^{m-n}$ Example:

$\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$
Raising a number to a power and then raising the result to another power multiplies the exponents:
$(a^m)^n = a^{m \times n}$ Example:

$(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$
Multiplying different bases with the same exponent:
$a^m \times b^m = (a \times b)^m$ Example:

$2^3 \times 3^3 = (2 \times 3)^3 = 6^3 = 216$
A negative base raised to an even power results in a positive number, and raised to an odd power results in a negative number:
$(-a)^m = a^m \quad \text{if } m \text{ is even}$ $(-a)^m = -a^m \quad \text{if } m \text{ is odd}$
Any power of 1 is always 1:
$1^m = 1$ $1^{m} = 1$ Example:

$1^{1000} = 1$
Any power of 0 is always 0:
$0^m = 0$ $0^{m} = 0$ Example:

$0^5 = 0$

Grouping in Exponents

When working with exponents, it’s important to use parentheses to avoid confusion, especially when dealing with negative numbers or multiple operations. For example:

With parentheses:
$(-2)^3 = (-2) \times (-2) \times (-2) = -8$
Without parentheses:
$-2^3 = -(2 \times 2 \times 2) = -8$

Always be careful with how the terms are grouped to ensure the correct result.

2. Square Roots

Definition

The square root of a number is a value that, when multiplied by itself, gives the original number. It is represented by the symbol

$\sqrt{}$

Example:

$\sqrt{16} = 4$

$16 = 4$ because

$4 \times 4 = 16$

$4 \times 4 = 16$ .

Key Properties of Square Roots

Square Root of Perfect Squares
- Perfect squares are numbers like $1, 4, 9, 16, 25, \dots$
  
  $1, 4, 9, 16, 25, \dots$ .
- Examples:
  $\sqrt{9} = 3$
  
  $9 = 3$ ,
  
  $\sqrt{25} = 5$ $25 = 5$ ,
  
  $\sqrt{144} = 12$ $144 = 12$ .
Square Root of a Product
- Rule: The square root of a product is the product of the square roots.
- Formula: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
  
  $= a \times b$
- Example: $\sqrt{16 \times 25} = \sqrt{16} \times \sqrt{25} = 4 \times 5 = 20$
  
  $16 \times 25 = 16 \times 25 = 4 \times 5 = 20$ .
Square Root of a Fraction
- Rule: The square root of a fraction is the square root of the numerator divided by the square root of the denominator.
- Formula: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
  
  $= b a$
- Example: $\sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}$
  
  $25 16 = 25 16 = 5 4$ .
Square Root of 0 and 1
- Rule: $\sqrt{0} = 0$
  
  $0 = 0$ and
  
  $\sqrt{1} = 1$ $1 = 1$

Examples of Square Roots

$\sqrt{49} = 7$

$49 = 7$ (since

$7 \times 7 = 49$ $7 \times 7 = 49$ )
$\sqrt{81} = 9$

$81 = 9$ (since

$9 \times 9 = 81$ $9 \times 9 = 81$ )
$\sqrt{0.25} = 0.5$

$0.25 = 0.5$ (since

$0.5 \times 0.5 = 0.25$ $0.5 \times 0.5 = 0.25$ ).
$\sqrt{2} \approx 1.414$

$2 \approx 1.414$ (irrational square root).

3. Cube Roots

Definition

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is represented by the symbol

$\sqrt[3]{}$

Example:

$\sqrt[3]{27} = 3$

$327 = 3$ because

$3 \times 3 \times 3 = 27$

$3 \times 3 \times 3 = 27$ .

Key Properties of Cube Roots

Cube Root of Perfect Cubes
- Perfect cubes are numbers like $1, 8, 27, 64, 125, \dots$
  
  $1, 8, 27, 64, 125, \dots$ .
- Examples:
  $\sqrt[3]{8} = 2$
  
  $38 = 2$ ,
  
  $\sqrt[3]{64} = 4$ $364 = 4$ ,
  
  $\sqrt[3]{125} = 5$ $3125 = 5$
Cube Root of a Product
- Rule: The cube root of a product is the product of the cube roots.
- Formula: $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$
  
  $3 a \times b = 3 a \times 3 b$
- Example: $\sqrt[3]{8 \times 27} = \sqrt[3]{8} \times \sqrt[3]{27} = 2 \times 3 = 6$
  
  $3 8 \times 27 = 38 \times 327 = 2 \times 3 = 6$ .
Cube Root of a Fraction
- Rule: The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator.
- Formula: $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$
  
  $3 b a = 3 b 3 a$
- Example: $\sqrt[3]{\frac{27}{64}} = \frac{\sqrt[3]{27}}{\sqrt[3]{64}} = \frac{3}{4}$
  
  $3 64 27 = 3 64 3 27 = 4 3$

Examples of Cube Roots

$\sqrt[3]{8} = 2$

$38 = 2$ (since

$2 \times 2 \times 2 = 8$ $2 \times 2 \times 2 = 8$ ).
$\sqrt[3]{125} = 5$

$3125 = 5$ (since

$5 \times 5 \times 5 = 125$ $5 \times 5 \times 5 = 125$ ).
$\sqrt[3]{0.001} = 0.1$

$3 0.001 = 0.1$ (since

$0.1 \times 0.1 \times 0.1 = 0.001$ $0.1 \times 0.1 \times 0.1 = 0.001$ ).
$\sqrt[3]{343} = 7$

$3343 = 7$ (since

$7 \times 7 \times 7 = 343$ $7 \times 7 \times 7 = 343$ )

Comparison Between Square and Cube Roots

Feature	Square Root ()	Cube Root () $3$ 3\sqrt[3]{}
Definition	Value that squares to give the number.	Value that cubes to give the number.
Symbol	$\sqrt{}$	$\sqrt[3]{}$ $3$
Examples	$\sqrt{25} = 5$ $25 = 5$	$\sqrt[3]{27} = 3$ $327 = 3$

Feature

Square Root ()

Cube Root ()

$3$ 3\sqrt[3]{}

Definition

Value that squares to give the number.

Value that cubes to give the number.

Symbol

$\sqrt{}$

$\sqrt[3]{}$ $3$

Examples

$\sqrt{25} = 5$ $25 = 5$

$\sqrt[3]{27} = 3$ $327 = 3$

4.8 Mathematical Expression

A mathematical expression is a combination of numbers, symbols, and operators (such as +, −, ×, ÷, etc.). It represents a mathematical relationship or value. For example,

$1 + 2 + 3$

$1 + 2 + 3$ is a simple mathematical expression.

Note: An expression does not contain an equal sign (=) because it doesn’t represent an equation or comparison, but rather a calculation.

An expression may include various operations such as addition, subtraction, multiplication, and division. For example:

$2 – 3 + 6 + 3 \times 2$

$2 - 3 + 6 + 3 \times 2$

This expression contains subtraction, addition, and multiplication operations.

To group certain parts of an expression together, brackets are used. Brackets are symbols that enclose parts of the expression to indicate that those parts should be treated as a unit. There are four types of brackets:

Vinculum (or bar or line brackets)
Small brackets (or round brackets or parentheses: $()$
Middle brackets (or curly brackets or braces: $\{\}$
Big brackets (or square brackets or box brackets: $[]$

For example

$5 \times (3 + 2) \times (6 – 4)$

4.9 Order of Operations

When an expression involves multiple operations like addition, subtraction, multiplication, and division, it is important to know the correct order in which to perform these operations to get the correct result.

Consider this example:

$2 + 3 \times 4$

If we calculate $2 + 3$

$2 + 3$ first, we get:

$2 + 3 \times 4 = 5 \times 4 = 20$
If we calculate $3 \times 4$

$3 \times 4$ first, we get:

$2 + 3 \times 4 = 2 + 12 = 14$

Clearly, the second approach gives the correct answer.

To determine the correct order of operations, we follow the BODMAS rule, which specifies the sequence of operations:

B: Brackets first (parentheses and other types of brackets)
O: Orders (exponents, powers, square roots, cube roots, etc.)
D: Division
M: Multiplication
A: Addition
S: Subtraction

Thus, the order of operations in an expression is:

Solve the parts in brackets first.
Solve powers and roots (orders) before multiplication, division, addition, or subtraction.
Perform division before multiplication, addition, or subtraction.
Perform multiplication before addition or subtraction.
Perform addition before subtraction.
Finally, perform subtraction.

Example of BODMAS:

For the expression:

$7 + (6 \times 5^2 + 3)$

Start with the brackets:
$6 \times 5^2 + 3$
Solve the powers (orders) first:
$7 + (6 \times 25 + 3)$
Multiply:
$7 + (150 + 3)$
Add:
$7 + 153 = 160$

The final answer is

$160$

By following the BODMAS rule, we ensure that we perform the operations in the correct order to obtain the correct result.

4.10 Divisibility

A number a is said to be divisible by another number b if, when a is divided by b, the remainder is zero. In other words, a can be divided by b without leaving any remainder.

For example:

$12 \div 4 = 3$

$12 \div 4 = 3$ (Remainder = 0, so 12 is divisible by 4)
$15 \div 5 = 3$

$15 \div 5 = 3$ (Remainder = 0, so 15 is divisible by 5)
$14 \div 5 = 2$

$14 \div 5 = 2$ (Remainder = 4, so 14 is not divisible by 5)

4.11 Even Numbers

Even numbers are those that are divisible by 2. This means when an even number is divided by 2, there is no remainder.

For example:

$0 \div 2 = 0$

$0 \div 2 = 0$ (Remainder = 0, so 0 is even)
$2 \div 2 = 1$

$2 \div 2 = 1$ (Remainder = 0, so 2 is even)
$4 \div 2 = 2$

$4 \div 2 = 2$ (Remainder = 0, so 4 is even)
$6 \div 2 = 3$

$6 \div 2 = 3$ (Remainder = 0, so 6 is even)

Thus, the numbers

$0, 2, 4, 6, 8, 10$

$0, 2, 4, 6, 8, 10$ , and so on, are all even numbers.

4.12 Odd Numbers

Odd numbers are those that are not divisible by 2. When an odd number is divided by 2, there is a remainder of 1.

For example:

$1 \div 2 = 0$

$1 \div 2 = 0$ (Remainder = 1, so 1 is odd)
$3 \div 2 = 1$

$3 \div 2 = 1$ (Remainder = 1, so 3 is odd)
$5 \div 2 = 2$

$5 \div 2 = 2$ (Remainder = 1, so 5 is odd)
$7 \div 2 = 3$

$7 \div 2 = 3$ (Remainder = 1, so 7 is odd)

Thus, the numbers

$1, 3, 5, 7, 9, 11$

$1, 3, 5, 7, 9, 11$ , and so on, are all odd numbers.

4.13 Fractions

A fraction is composed of two parts: a numerator and a denominator. The numerator tells us how many parts we have, while the denominator tells us how many parts the whole is divided into.

For example:

If you have 1 apple and you divide it into 2 equal parts, each part is $\frac{1}{2}$

$2 1$ of the apple. Here, 1 is the numerator and 2 is the denominator.

Types of Fractions

Proper Fraction
A proper fraction is a fraction where the numerator is less than the denominator.Examples:
- $\frac{3}{5}$
  
  $5 3$ (Numerator 3 is less than Denominator 5)
  
  $\frac{6}{14}$
  
  $146 $
  
  $\frac{4}{7}$ 47
Improper Fraction
An improper fraction is a fraction where the numerator is greater than the denominator.Examples:
- $\frac{5}{3}$
  
  $3 5$ (Numerator 5 is greater than Denominator 3)
- $\frac{8}{7}$
  
  $7 8$
- $\frac{11}{9}$
  
  $9 11$
Mixed Fraction
A mixed fraction is a combination of a whole number and a proper fraction.Examples:
- $2 \frac{1}{2}$
  
  $2 2 1$ (This means 2 whole parts and
  
  $\frac{1}{2}$ $2 1$ of another part)
- $7 \frac{3}{5}$
  
  $7 5 3$
Unit Fraction
A unit fraction is a fraction where the numerator is 1.Examples: $\frac{1}{2}$

$2 1$
- $\frac{1}{3}$
  
  $3 1$
- $\frac{1}{5}$
  
  $5 1$
Like Fractions
Fractions with the same denominator are called like fractions.Examples:
- $\frac{2}{5}$
  
  $5 2$ and
  
  $\frac{3}{5}$ $5 3$ are like fractions because they both have the denominator 5.
- $\frac{4}{7}$
  
  $7 4$ and
  
  $\frac{1}{7}$ $7 1$
Unlike Fractions
Fractions with different denominators are called unlike fractions.Examples:
- $\frac{1}{3}$
  
  $3 1$ and
  
  $\frac{2}{5}$ $5 2$ are unlike fractions because they have different denominators (3 and 5).
- $\frac{4}{7}$
- $74 and$
  
  $\frac{5}{8}$

By understanding the different types of fractions, we can simplify, add, subtract, multiply, and divide fractions more easily.

4.14 Decimal Numbers

A decimal number consists of two parts: a whole part and a fractional part separated by a decimal point (.).

For example, in the number 21.37:

Whole part: 21
Decimal point: .
Fractional part: 37

This means that the number is read as twenty-one point three seven.

Key Points:

The whole part is the part to the left of the decimal point.
The fractional part is the part to the right of the decimal point.

Examples:

$1.245 = 1 + 0.245$

$1.245 = 1 + 0.245$
$67.867 = 67 + 0.867$

$67.867 = 67 + 0.867$

A whole number can be written as a decimal by adding “.0” after it:

$55 = 55.0$

$55 = 55.0$
$108 = 108.0$ $108 = 108.0$

Leading zeros to the left of a whole number do not change the value:

$0000108 = 108$

$0000108 = 108$

Similarly, trailing zeros after the decimal point in the fractional part do not change the value:

$1.200 = 1.2$

$1.200 = 1.2$

Reading Decimal Numbers:

$21.37$

$21.37$ is read as twenty-one point three seven.
$30.02445$

$30.02445$ is read as thirty point zero two four four five.

Multiplication of Decimal Numbers by Powers of 10

When a decimal number is multiplied by powers of 10, the decimal point shifts to the right. Here’s how it works:

Multiply by 10: Shift the decimal point 1 place to the right.
Multiply by 100: Shift the decimal point 2 places to the right.
Multiply by 1000: Shift the decimal point 3 places to the right, and so on.

Examples:

Decimal Number	Multiplied by	Result
0.3125	x 100	31.25
0.322	x 1000	322

Fractional Representation of a Decimal Number

A decimal number can be expressed as a fraction by following these steps:

Write the decimal number over 1.
Multiply both numerator and denominator by 10 for every digit after the decimal point.
Simplify the fraction.

Example: Convert 0.75 into a fraction:

Write 0.75 as $\frac{0.75}{1}$

$1 0.75$ .
Since there are two digits after the decimal point, multiply both the numerator and denominator by 100: $\frac{0.75}{1} \times \frac{100}{100} = \frac{75}{100}$

$10.75 \times 100100 = 10075$
Simplify the fraction: $\frac{75}{100} = \frac{3}{4}$

$10075 = 43$

Thus,

$0.75 = \frac{3}{4}$

$0.75 = 4 3$ .

Another way:

Write $0.75$

$0.75$ as

$\frac{75}{100}$ $100 75$ .
Simplify the fraction to get $\frac{3}{4}$

$4 3$ .

Recurring Decimal Numbers

A recurring decimal number (or repeating decimal) is a decimal number in which one or more digits repeat indefinitely. It is represented by placing a bar (horizontal line) over the repeating digits.

Examples:

$0.3333…$

$0.3333\dots$ is written as

$0.\overline{3}$ $0. 3$ (repeating 3).
$0.456456456…$

$0.456456456\dots$ is written as – 0.456 $0. 456$ (repeating 456)

4.15 Rational Numbers

A rational number is any number that can be written as a fraction of two integers. This means that if

$p$

$p$ and

$q$

$q$ are integers, then the number

$\frac{p}{q}$

is a rational number, where

$q \neq 0$

$q = 0$ .

Examples of Rational Numbers:

$3$
$3$

$3$ 31\frac{3}{1} $1 3$ , so it is a rational number.
$\frac{7}{8}$ $8 7$ is a rational number because both 7 and 8 are integers.

Note: A rational number can also be written as a decimal, which either terminates or repeats.

4.16 Irrational Numbers

Irrational numbers are numbers that cannot be written as a fraction of two integers. These numbers have non-terminating, non-repeating decimal expansions.

Examples of Irrational Numbers:

$\sqrt{2} = 1.41421356237309504…$

$2 = 1.41421356237309504\dots$ (It cannot be written as a fraction.)

$\pi = 3.1415926535897932384…$

$π = 3.1415926535897932384\dots$ (It cannot be written as a fraction.)

4.17 Real Numbers

The real numbers include:

All the rational numbers (fractions, integers).
All the irrational numbers (such as

Real numbers cover all the numbers we use in daily life, including whole numbers, decimals, fractions, and irrational numbers.

4.18 Factors

Factors of a number are the numbers that divide it exactly (without a remainder). A number can have many factors.

For example:

The factors of 12 are: $1, 2, 3, 4, 6, 12$

$1, 2, 3, 4, 6, 12$ .
The factors of 18 are: $1, 2, 3, 6, 9, 18$

$1, 2, 3, 6, 9, 18$ .

Example: Find all the factors of 18:

Start with $1 \times 18 = 18$

$1 \times 18 = 18$ .
$2 \times 9 = 18$

$2 \times 9 = 18$ .
$3 \times 6 = 18$

$3 \times 6 = 18$ .
The factors of 18 are: $1, 2, 3, 6, 9, 18$

$1, 2, 3, 6, 9, 18$ and their negatives:

$-1, -2, -3, -6, -9, -18$ $- 1, - 2, - 3, - 6, - 9, - 18$ .

4.19 Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Examples of Prime Numbers:

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \dots$

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \dots$
$2 is the smallest and the only even prime number.$

$7$ is prime because it is divisible by only

$1$ and

$7$

4.20 Composite Numbers

A composite number is a natural number greater than 1 that has more than two factors. These numbers are divisible by numbers other than 1 and itself.

Examples of Composite Numbers:

$4$

$4$ is divisible by

$1, 2, 4$ . $6$ is divisible by

$1, 2, 3, 6$
$9$

$9$ is divisible by

$1, 3, 9$

Note: The number 1 is neither prime nor composite.

4.21 Prime Factorization

Prime factorization is the process of writing a number as a product of prime numbers.

Example: Prime Factorization of 18:

Start with $18$
$18 \div 2 = 9$

$18 \div 2 = 9$ , so 2 is a prime factor.
$9 \div 3 = 3$

$9 \div 3 = 3$ , so 3 is a prime factor.
$3 \div 3 = 1$

$3 \div 3 = 1$ , so 3 is another prime factor.

Thus, the prime factorization of 18 is:

$18 = 2 \times 3 \times 3 = 2 \times 3^{2}$

4.14 Decimal Numbers

A decimal number consists of two parts: the whole part and the fractional part, separated by a decimal point.

Example:

In the number 21.37,

21 is the whole part, and
37 is the fractional part.

We can express a decimal number as the sum of its whole and fractional parts:

21.37 = 21 + 0.37
1.245 = 1 + 0.245
67.867 = 67 + 0.867

Important Notes:

Whole Numbers as Decimals: Whole numbers can be written as decimals with a 0 in the fractional part.
- For example, 55 can be written as 55.0 or 108 as 108.0.
Trailing Zeros: You can ignore any trailing zeros after the decimal point.
- For example: 1.200 is the same as 1.2.

Reading Decimal Numbers:

21.37 is read as “twenty-one point three seven”.
30.02445 is read as “thirty point zero two four four five”.

Multiplication of a Decimal Number by Powers of 10

When you multiply a decimal number by powers of 10, the decimal point shifts to the right:

0.3125 × 10 = 3.125
0.322 × 100 = 32.2
0.322 × 1000 = 322.0

Fractional Representation of a Decimal Number

To convert a decimal number into a fraction:

Write the decimal number over 1.
Multiply both the numerator and denominator by 10 for each digit after the decimal point.
Simplify the fraction.

Example: Convert 0.75 into a fraction.

Step 1: Write 0.75 as 0.75/1.
Step 2: Multiply both numerator and denominator by 100 (since there are two decimal places): $\frac{0.75}{1} \times \frac{100}{100} = \frac{75}{100}$
Step 3: Simplify the fraction: $\frac{75}{100} = \frac{3}{4}$

Thus, 0.75 = 3/4.

Recurring Decimal Numbers

A recurring decimal has digits that repeat indefinitely. These numbers are written with a bar over the repeating digits.

Examples:

0.3333… can be written as 0.3̅.
0.456456456… can be written as 0.456̅.

4.15 Rational Numbers

A rational number is any number that can be written as a fraction. If p and q are integers, and q ≠ 0, then the number p/q is a rational number.

Examples:

3 is a rational number because it can be written as 3/1.
1/2 is a rational number.

4.16 Irrational Numbers

An irrational number cannot be expressed as a fraction. These numbers have non-repeating, non-terminating decimal expansions.

Examples:

√2 ≈ 1.41421356237309504… (non-repeating)
π ≈ 3.1415926535897932384… (non-terminating)

4.17 Real Numbers

The set of real numbers consists of both rational numbers (fractions) and irrational numbers.

4.18 Factors

Factors are numbers that you multiply together to get another number. For example, the factors of 12 are:

Example:

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

So, the factors of 12 are 1, 2, 3, 4, 6, 12 (including negative factors: -1, -2, -3, -4, -6, -12).

Example for 18:

Start with 1 × 18 = 18, then try 2 × 9 = 18, 3 × 6 = 18, and so on.

Thus, the factors of 18 are 1, 2, 3, 6, 9, 18 and their negative counterparts -1, -2, -3, -6, -9, -18.

4.19 Prime Numbers

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

Examples:

2 is prime because its only divisors are 1 and 2.
3 is prime because its only divisors are 1 and 3.
5, 7, and 11 are also prime numbers.

The smallest prime number is 2, and it’s the only even prime number.

4.20 Composite Numbers

A composite number is a natural number greater than 1 that has more than two divisors.

Examples:

4 is composite because its divisors are 1, 2, and 4.
6 is composite because its divisors are 1, 2, 3, and 6.
9 is composite because its divisors are 1, 3, and 9.

Note: 1 is neither prime nor composite.

4.21 Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors.

Example:

To find the prime factors of 18:

Start with 18.
Divide by the smallest prime number, 2: $18 ÷ 2 = 9$
Next, divide 9 by the smallest prime number, 3: $9 ÷ 3 = 3$
Finally, divide 3 by 3 again: $3 ÷ 3 = 1$

Thus, the prime factorization of 18 is:

$18 = 2 \times 3 \times 3 = 2 \times 3^2$

4.22 Highest Common Factor (HCF)

The Highest Common Factor (HCF) is the largest factor that two or more numbers share.

Example:

Find the HCF of 12 and 16 using different methods:

Method 1: Factorization

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
Common factors: 1, 2, 4
The HCF is 4.

Method 2: Prime Factorization

Prime factors of 12: 2 × 2 × 3
Prime factors of 16: 2 × 2 × 2 × 2
Common prime factors: 2 × 2 = 4
The HCF is 4.

Method 3: Division Method

Divide the larger number by the smaller one: $16 ÷ 12 = 1 \text{ (remainder } 4\text{)}$
Divide the previous divisor by the remainder: $12 ÷ 4 = 3 \text{ (remainder } 0\text{)}$
When the remainder becomes 0, the divisor (4) is the HCF.

4.23 Twin Prime Pair

Two prime numbers that have a difference of 2 are called twin primes.

Examples:

(3, 5): 5 – 3 = 2, so 3 and 5 are twin primes.
(5, 7): 7 – 5 = 2, so 5 and 7 are twin primes.

4.24 Co-prime Numbers

Two numbers are co-prime (or relatively prime) if their only common factor is 1.

Example:

18 and 25 are co-prime because their factors are:
- 18: 1, 2, 3, 6, 9, 18
- 25: 1, 5, 25
The only common factor is 1.

4.25 Multiples

The multiples of a number are the results of multiplying that number by any integer (except 0).

Example:

The multiples of 3 are 3, 6, 9, 12, 15, …
Similarly, -3, -6, -9, -12, … are also multiples of 3.

4.26 Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers.

Method for Finding LCM:

You can use the HCF to find the LCM using the formula:

$\text{LCM} = \frac{\text{Product of two numbers}}{\text{HCF}}$

Example:

For 4 and 10:

HCF of 4 and 10 = 2
The product of 4 and 10 is 40.
So, $\text{LCM of 4 and 10} = \frac{40}{2} = 20$

$LCM of 4 and 10 = 240 = 20$

This version provides clearer explanations, examples, and steps to help understand the concepts more easily.

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