Numbers and Operations · 4 min read
The Real Number System
Classifying Real Numbers
The real number system includes every number that can be located on a number line. Each set below is contained in the next, and every real number is either rational or irrational, never both.
| Set | Definition | Examples |
|---|---|---|
| Natural (counting) numbers | 1, 2, 3, 4, … | 5, 17, 100 |
| Whole numbers | Natural numbers and 0 | 0, 8, 250 |
| Integers | Whole numbers and their opposites | −12, −1, 0, 9 |
| Rational numbers | Numbers that can be written as p/q, where p and q are integers and q ≠ 0; their decimals terminate or repeat | −3, 2/3, 0.25, 0.333… |
| Irrational numbers | Real numbers that cannot be written as a ratio of integers; their decimals neither terminate nor repeat | π, √2, √3, e |
| Real numbers | All rational and irrational numbers | Every point on the number line |
A square root of a whole number is rational only when the number is a perfect square: √49 = 7 is an integer, while √50 is irrational. Simplify before classifying: −√36 ÷ 3 = −2 is an integer, and (√2 + 1)² = 3 + 2√2 is irrational.
- The sum, difference, product, and quotient (nonzero divisor) of two rationals is rational.
- Rational + irrational is always irrational, and a nonzero rational × an irrational is always irrational.
- Irrational + irrational and irrational × irrational can be either: √2 + (−√2) = 0 and √2 · √2 = 2 are rational, while √2 + √3 is irrational.
Properties of Operations
| Property | Addition | Multiplication |
|---|---|---|
| Commutative (order) | a + b = b + a | ab = ba |
| Associative (grouping) | (a + b) + c = a + (b + c) | (ab)c = a(bc) |
| Identity | a + 0 = a | a · 1 = a |
| Inverse | a + (−a) = 0 | a · (1/a) = 1, a ≠ 0 |
| Distributive | a(b + c) = ab + ac and a(b − c) = ab − ac |
Subtraction and division are neither commutative nor associative: 5 − 3 ≠ 3 − 5, and (12 ÷ 6) ÷ 2 = 1 while 12 ÷ (6 ÷ 2) = 4. Mental-math strategies are applications of these properties. Computing 8 × 99 as 8 × 100 − 8 × 1 uses the distributive property; computing 4 × 23 × 25 as (4 × 25) × 23 uses the commutative and associative properties of multiplication.
Order of Operations
Evaluate grouping symbols first (innermost first), then exponents and roots, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication does not come before division, and addition does not come before subtraction; each pair has equal priority.
Signed Numbers and Absolute Value
- Absolute value is distance from zero: |x| = x if x ≥ 0 and |x| = −x if x < 0, so |−5| = 5.
- To add numbers with the same sign, add the absolute values and keep the sign; with different signs, subtract the absolute values and keep the sign of the number farther from zero.
- Subtracting is adding the opposite: 5 − (−3) = 5 + 3 = 8.
- A product or quotient of two numbers with the same sign is positive; with different signs it is negative.
- Useful facts: |ab| = |a||b|, |a/b| = |a|/|b| for b ≠ 0, and |a + b| ≤ |a| + |b| (the triangle inequality).
Comparing and Ordering Real Numbers
Convert every number to a decimal approximation, or compare on a number line. Among negatives, the number with the greater absolute value is the smaller number: −√10 ≈ −3.16 is less than −3.1. For fractions with a common numerator, the larger denominator gives the smaller fraction.
- "Multiplication always makes bigger" fails for factors between 0 and 1: 12 × 0.4 = 4.8 < 12.
- "Division always makes smaller" fails for divisors between 0 and 1: 6 ÷ 0.5 = 12.
- Applying operations strictly left to right, or doing all multiplication before any division, signals a rote memory of PEMDAS rather than the equal-priority rule.
- A counterexample is the efficient way to show a general claim is false; one example cannot prove a claim is always true.
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