What Are Number Operations?
Number operations are the basic processes used to work with quantities. The four primary operations are addition, subtraction, multiplication, and division. They answer questions like “How many in total?”, “How much more or less?”, “How many groups?” or “How many in each group?”.
The Four Core Operations
- Addition (+): Combines quantities. Example:
7 + 5 = 12. - Subtraction (−): Finds the difference or removes a quantity. Example:
12 − 5 = 7. - Multiplication (×): Repeated addition or equal groups. Example:
4 × 6 = 24. - Division (÷ or /): Splitting into equal groups or measuring how many groups fit. Example:
24 ÷ 6 = 4. Note: division by zero is undefined.
Key Properties
- Commutative (order doesn’t matter): addition, multiplication.
a + b = b + a, a × b = b × a. - Associative (grouping doesn’t matter): addition, multiplication.
(a + b) + c = a + (b + c), (a × b) × c = a × (b × c). - Distributive: Multiplication distributes over addition/subtraction.
a × (b + c) = a×b + a×c. - Identity elements:
a + 0 = a, a × 1 = a. - Inverses:
a + (−a) = 0, for a ≠ 0, a × (1/a) = 1. - Zero property:
a × 0 = 0.
Order of Operations
To evaluate expressions consistently, follow the order of operations (often remembered as PEMDAS/BODMAS):
- Parentheses/Brackets
- Exponents/Orders
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Example: 3 + 6 × (5 − 2)^2 ÷ 3
Compute inside parentheses: 5 − 2 = 3
Exponents: 3^2 = 9
Multiply/Divide left to right: 6 × 9 ÷ 3 = 54 ÷ 3 = 18
Add: 3 + 18 = 21.
Working With Different Number Types
- Whole numbers and integers: Include zero and negatives (integers). Mind sign rules (e.g., negative times negative is positive).
- Fractions: Use common denominators for addition/subtraction; multiply numerators and denominators; divide by multiplying by the reciprocal. Example:
2/3 ÷ 5/6 = 2/3 × 6/5 = 12/15 = 4/5. - Decimals: Line up decimal points for addition/subtraction; count total decimal places for multiplication; move the decimal in the divisor to make it a whole number for division.
- Percents: Convert among forms:
35% = 35/100 = 0.35. Apply operations after converting to fractions or decimals. - Powers and roots: Extend multiplication and inverse operations:
a^m × a^n = a^{m+n}, √a is the number which squared equals a.
Estimation and Mental Math
- Rounding: Simplify numbers to make quick, approximate calculations.
- Compatible numbers: Choose nearby numbers that divide or multiply cleanly.
- Benchmarks: Use known values (e.g.,
10% of a number) to estimate percents.
Division, Remainders, and Algorithms
Division can produce whole-number quotients, decimals, fractions, or remainders. Long multiplication and long division are step-by-step algorithms to compute results accurately when mental math is impractical.
Common Pitfalls
- Ignoring order of operations.
- Sign mistakes with negatives.
- Adding or subtracting fractions without common denominators.
- Misplacing decimal points.
- Attempting to divide by zero.
Why It Matters
Number operations underpin nearly all of mathematics—algebra, geometry, statistics—and appear in real-life contexts like budgeting, measurement, scaling recipes, data analysis, and science.