Every number on the line
The real numbers are all the numbers that can be placed on a number line: positive, negative, zero, fractions, decimals, roots and special numbers like π. Every point on the line is a real number, and every real number is a point.
The family tree
The real numbers are made of smaller sets, each a subset of the next:
| Set | Symbol | Contains | Examples |
|---|---|---|---|
| Natural numbers | ℕ | counting numbers | 1, 2, 3, … |
| Whole numbers | W | natural numbers and 0 | 0, 1, 2, … |
| Integers | ℤ | whole numbers and their opposites | …, −2, −1, 0, 1, 2, … |
| Rational numbers | ℚ | fractions of integers | −3/4, 0.5, 2, 0.333… |
| Irrational numbers | ℚ̄ | non-terminating, non-repeating decimals | √2, π, e |
| Real numbers | ℝ | rational and irrational | all of the above |
Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real. Irrationals are also real, but separate from the rationals.
Rational vs irrational
| Rational | Irrational |
|---|---|
| Can be written as a/b | Cannot |
| Decimal terminates or repeats | Decimal never ends, never repeats |
| 0.75, 1/3, −8, √9 | √2, √7, π |
Classifying numbers
| Number | Natural | Whole | Integer | Rational | Irrational | Real |
|---|---|---|---|---|---|---|
| 5 | ✓ | ✓ | ✓ | ✓ | ✓ | |
| 0 | ✓ | ✓ | ✓ | ✓ | ||
| −7 | ✓ | ✓ | ✓ | |||
| 2/3 | ✓ | ✓ | ||||
| √16 = 4 | ✓ | ✓ | ✓ | ✓ | ✓ | |
| √5 | ✓ | ✓ | ||||
| π | ✓ | ✓ |
Is everything real?
Not quite. √(−1) is not a real number, because no real number squared is negative. Mathematicians call it i, an imaginary number, studied in later courses.
Density
Between any two real numbers there are infinitely many more. Between 1.4 and 1.5 lie 1.41, 1.414, √2, 1.45, and endlessly more.
Ordering real numbers
Convert to decimals to compare.
Example: Order √10, 3.1, π, 22/7
- √10 ≈ 3.162
- 3.1
- π ≈ 3.1416
- 22/7 ≈ 3.1429
Order: 3.1, π, 22/7, √10.
Worked examples
Example 1: Which set contains every number on the number line?
Real numbers.
Example 2: Classify −0.25.
Rational and real (it’s −1/4).
Example 3: Is √49 irrational?
No: √49 = 7, a natural number.
Common mistakes
- Thinking real numbers are only "normal" whole numbers.
- Calling all square roots irrational.
- Forgetting that integers are also rational (−5 = −5/1).
Try it yourself
- Classify 0.121212…
- Classify √50.
- Name a number that is an integer but not a whole number.
- Order: 1.7, √3, 7/4.
Answers: 1) rational, real 2) irrational, real 3) e.g. −3 4) 1.7, √3 (≈ 1.732), 7/4 (1.75)
Quick recap
- Real numbers = rational numbers + irrational numbers.
- Natural numbers ⊂ whole ⊂ integers ⊂ rational ⊂ real.
- Every real number has a place on the number line.