In mathematics, mastering the basic operations of addition, subtraction, multiplication, and division is crucial. These operations are the building blocks for more complex mathematical concepts. This section delves into each operation with a focus on integers, fractions, and decimals, emphasizing the correct ordering of operations and the use of brackets.

**Addition**

Addition combines two or more numbers into their total or sum.

**Properties**:**Commutative**: Order doesn't affect the sum, e.g., $3 + 4 = 4 + 3$.**Associative**: Grouping doesn't affect the sum, e.g., $(2 + 3) + 4 = 2 + (3 + 4)$.

**Example: Calculating a Sum**

**Question**: What is the sum of $78$, $45.3$, and $-22$?

**Solution**:

**Subtraction**

Subtraction finds the difference between two numbers, essentially reversing addition.

**Symbols and Terms**: The - symbol denotes subtraction. The number being subtracted is the subtrahend, from the minuend, to get the difference.**Properties**:**Non-Commutative**: The order in subtraction matters, e.g., $5 - 2 \neq 2 - 5$.

**Example: Finding a Difference**

**Question**: What is the difference between $50$ and $32.75$?

**Solution**:

**Multiplication**

Multiplication adds a number to itself a specified number of times, streamlining addition.

**Symbols and Terms**: The × or * symbol is for multiplication. The numbers being multiplied are factors, with the result called the product.**Properties**:**Commutative**: The order of factors doesn't affect the product, e.g., $3 × 4 = 4 × 3$.**Associative**: Grouping of factors doesn't affect the product, e.g., $(2 × 3) × 4 = 2 × (3 × 4)$.

**Example: Multiplying Numbers**

**Question**: Multiply $3.5$ by $2.4$.

**Solution**:

**Division**

Division splits a number into equal parts, the inverse of multiplication.

**Symbols and Terms**: The ÷ or / symbol represents division. The number being divided is the dividend, by the divisor, to get the quotient.**Properties**:**Non-Commutative**: Order matters significantly in division.**Division by Zero**: Undefined.

**Example: Dividing Numbers**

**Question**: Divide $56$ by $0.7$.

**Solution**:

**Order of Operations and Bracket Usage**

Correct operation ordering and bracket usage are pivotal for solving problems accurately. PEMDAS guides the order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

**Brackets**: Operations within are prioritized; nested ones are solved inside out.

**Example**: