Arithmetic · 3 min read
Numbers and Their Classifications
The number system is the foundation of every mathematical operation. Knowing how numbers are classified helps you recognize their properties and choose the right operation.
The Real Number System
Each set sits inside the larger set around it; irrational numbers are real but not rational.
| Set | Description | Examples |
|---|---|---|
| Natural numbers | The counting numbers; no zero, fractions, or negatives | 1, 2, 3, 4, … |
| Whole numbers | Natural numbers plus zero | 0, 1, 2, 3, … |
| Integers | Whole numbers and their negatives | …, −3, −2, −1, 0, 1, 2, 3, … |
| Rational numbers | Can be written as a/b with integers a and b, b ≠ 0; decimals terminate or repeat | 1/2, −3/4, 5, 0.75, 0.333… |
| Irrational numbers | Cannot be written as a fraction; decimals never terminate or repeat | π ≈ 3.14159…, √2 ≈ 1.41421…, e ≈ 2.71828… |
| Real numbers | All rational and irrational numbers; every point on the number line | All of the above |
Key Concept
Every integer is rational because it can be written with a denominator of 1 (5 = 5/1). The square root of a perfect square, such as √49 = 7, is rational; the square root of a non-perfect square, such as √10, is irrational.Properties of Numbers
Even and Odd
Even integers are divisible by 2 (…, −4, −2, 0, 2, 4, …); odd integers leave a remainder of 1. Even + even = even, odd + odd = even, even + odd = odd, even × any integer = even, odd × odd = odd.
Prime and Composite
A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. A composite has more than two factors. 2 is the only even prime; 1 is neither prime nor composite.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction, so it is never negative: |5| = 5 and |−5| = 5.
Absolute Value as Distance
−3 and 3 are both 3 units from zero.
Place Value and Expanded Form
In the decimal system each position represents a power of 10. Moving left from the decimal point, positions represent ones, tens, hundreds, and thousands; moving right, they represent tenths, hundredths, and thousandths.
Example: Expanded Form
- 3,456.78 = 3,000 + 400 + 50 + 6 + 0.7 + 0.08
- = 3 × 1,000 + 4 × 100 + 5 × 10 + 6 × 1 + 7 × 0.1 + 8 × 0.01
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