Arithmetic

Arithmetic

1.1 What is Arithmetic?

Arithmetic is the branch of mathematics that deals with numbers and the basic operations we perform on them.

It is the foundation of all mathematics – from simple calculations in daily life to advanced sciences.

The four fundamental operations are:

Addition (+) → putting together

Subtraction (−) → taking away

Multiplication (×) → repeated addition

Division (÷) → sharing equally

1.2 Types of Numbers

1. Natural Numbers (1, 2, 3, 4, …) – counting numbers.

2. Whole Numbers (0, 1, 2, 3, …) – natural numbers plus zero.

3. Integers (…, −3, −2, −1, 0, 1, 2, 3, …) – positive, negative, and zero.

4. Fractions (½, ¾, 2/5) – part of a whole.

5. Decimals (0.1, 0.25, 3.75).

6. Percentages (%, per hundred, e.g., 50% = 50/100 = 0.5).

1.3 Properties of Numbers

Commutative Property

a + b = b + a

a × b = b × a

Associative Property

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Distributive Property

a × (b + c) = (a × b) + (a × c)

1.4 Addition (+)

Definition: Combining two or more numbers to get a total.

Example: 25 + 17 = 42

👉 Rules:

Adding zero does not change a number. (a + 0 = a)

Order does not matter (commutative).

1.5 Subtraction (−)

Definition: Taking one number away from another.

Example: 53 − 29 = 24

👉 Rules:

Subtraction is not commutative (a − b ≠ b − a).

Subtraction is not associative.

1.6 Multiplication (×)

Definition: Repeated addition.

Example: 6 × 4 = 6 + 6 + 6 + 6 = 24

👉 Rules:

Multiplying by zero always gives zero.

Multiplying by one keeps the number unchanged.

1.7 Division (÷)

Definition: Splitting a number into equal parts.

Example: 20 ÷ 4 = 5

👉 Rules:

Division by zero is undefined.

Division is not commutative.

1.8 Fractions

Proper Fraction: numerator < denominator (e.g., 3/4).

Improper Fraction: numerator ≥ denominator (e.g., 7/3).

Mixed Number: whole + fraction (e.g., 2 ½).

👉 Operations:

Addition/Subtraction: Make denominators equal.

Multiplication: Multiply numerators and denominators.

Division: Multiply by the reciprocal.

1.9 Decimals

Example: 3.75 = 3 + 75/100

Addition and subtraction: line up decimal points.

Multiplication: ignore decimal, then place it back.

Division: shift decimals to make divisor a whole number.

1.10 Percentages

Definition: A fraction out of 100.

Conversions:

50% = 50/100 = 0.5

25% = 25/100 = 0.25

Example: 20% of 250 = (20/100) × 250 = 50

1.11 Order of Operations (BODMAS Rule)

When more than one operation is involved, follow BODMAS:

B – Brackets

O – Orders (powers, roots)

D – Division

M – Multiplication

A – Addition

S – Subtraction

Example:

10 + 2 × (6 − 4)² ÷ 2

= 10 + 2 × (2)² ÷ 2

= 10 + 2 × 4 ÷ 2

= 10 + 8 ÷ 2

= 10 + 4

= 14

1.12 Real-Life Applications of Arithmetic

Shopping: Adding bills, calculating discounts.

Banking: Interest, savings, and loans.

Cooking: Measuring ingredients with fractions.

Business: Profits, losses, and percentages.

📝 Practice Exercises

1. Add: 456 + 789 = ?

2. Subtract: 2025 − 1789 = ?

3. Multiply: 67 × 43 = ?

4. Divide: 980 ÷ 35 = ?

5. Simplify: (12 + 18) × 5 ÷ 10 = ?

6. Convert:

0.85 into percentage

25% into fraction

7. Find 15% of 640.

8. Simplify: 100 − 25 × (4 + 2).

✅ Summary

Arithmetic is the foundation of all math.

It involves numbers and basic operations (addition, subtraction, multiplication, division).

Fractions, decimals, and percentages are essential for real-life problem solving.

The BODMAS rule is followed to solve complex expressions.