Concept Notes 1: The Grammar of Arithmetic — Order of Operations
An arithmetic expression is any meaningful combination of numbers, operational symbols ($+, -, \times, \div$), and grouping brackets that evaluates to a specific numerical value.
1. The Precedence Hierarchy (BODMAS / PEMDAS)
To ensure universal consistency, operations must be performed in the following strict hierarchical sequence:
| Priority Rank | Acronym Letter | Operation Name | Description & Rule |
|---|---|---|---|
| 1st | B | Brackets | Evaluate all expressions inside brackets first (innermost to outermost). |
| 2nd | O | Orders / Of | Exponents, powers, and the "Of" operator (e.g., $\frac{1}{4} \text{ of } 100 = 25$). |
| 3rd | D & M | Division & Multiplication | Equal priority! Evaluated strictly in order from left to right. |
| 4th | A & S | Addition & Subtraction | Equal priority! Evaluated strictly in order from left to right. |
Crucial Rule on Equal Priority: In an expression like $18 \div 3 \times 2$, division and multiplication share equal precedence. We evaluate from left to right: $18 \div 3 = 6 \rightarrow 6 \times 2 = \mathbf{12}$. (Doing multiplication first to get $18 \div 6 = 3$ is a common error!)
2. Types of Brackets and Unpacking Order
When an expression contains nested grouping symbols, evaluate them from the inside out:
- Vinculum (Bar): $\overline{a + b}$
- Round Parentheses: $( \dots )$
- Curly Brackets (Braces): $\{ \dots \}$
- Square Brackets: $[ \dots ]$
Worked Example: Evaluate $45 - [20 + \{18 - (12 - 4)\}]$: 1. Innermost parentheses: $(12 - 4) = 8$. 2. Expression becomes: $45 - [20 + \{18 - 8\}]$. 3. Curly brackets: $\{18 - 8\} = 10$. 4. Expression becomes: $45 - [20 + 10]$. 5. Square brackets: $[20 + 10] = 30$. 6. Outer subtraction: $45 - 30 = \mathbf{15}$.