Brackets
Worked Example
Problem: Simplify the expression 2[3(4 + 1) − 5].
Step 1: Start with the innermost grouping — the parentheses. Evaluate 4 + 1.
2[3(5)−5]
Step 2: Multiply inside the brackets: 3 times 5 equals 15.
2[15−5]
Step 3: Subtract inside the brackets: 15 minus 5 equals 10.
2[10]
Step 4: Multiply the result by the factor outside the brackets.
2×10=20
Answer: The simplified value is 20.
Why It Matters
Brackets serve multiple roles across mathematics. In arithmetic and algebra, they act as a second level of grouping around parentheses, making deeply nested expressions easier to read. In later courses, brackets denote closed intervals such as [2,5], meaning all numbers from 2 to 5 including both endpoints, and they also frame the rows and columns of matrices.
Common Mistakes
Mistake: Confusing brackets [ ] with parentheses ( ) in interval notation.
Correction: A bracket [ or ] means the endpoint is included (closed), while a parenthesis ( or ) means the endpoint is excluded (open). For example, [2,5) includes 2 but not 5.
Related Terms
- Parentheses — Innermost grouping symbols used before brackets
- Set Braces — Curly braces { } used for sets
- Order of Operations — Rules that govern when to evaluate brackets
- Interval Notation — Uses brackets to show closed endpoints
- Matrix — Often enclosed in square brackets