Sorting numbers into sets
Mathematicians group numbers into sets with special properties. Knowing the sets helps you describe solutions precisely: "x is a positive integer" means something very different from "x is a real number".
The main sets
| Set | Symbol | Description |
|---|---|---|
| Natural numbers | ℕ | 1, 2, 3, … |
| Whole numbers | W | 0, 1, 2, 3, … |
| Integers | ℤ | …, −2, −1, 0, 1, 2, … |
| Rational numbers | ℚ | a/b with a, b integers, b ≠ 0 |
| Irrational numbers | ℚ̄ | real numbers that aren’t rational |
| Real numbers | ℝ | all of the above |
(Some textbooks include 0 in the natural numbers. Canadian school courses usually don’t.)
Set notation
| Set notation | Meaning | Example |
|---|---|---|
| ∈ | is an element of | 5 ∈ ℕ |
| ∉ | is not an element of | −2 ∉ W |
| ⊂ | is a subset of | ℤ ⊂ ℚ |
| { } | the set containing | {2, 4, 6} |
Other useful classifications
| Classification | Meaning | Examples |
|---|---|---|
| Even / odd | divisible by 2 or not | 4 / 7 |
| Prime / composite | exactly two factors / more | 7 / 9 |
| Perfect square | n² for an integer n | 1, 4, 9, 16 |
| Perfect cube | n³ | 1, 8, 27 |
| Positive / negative | > 0 / < 0 | 3 / −3 |
Closure
A set is closed under an operation if the result of the operation is always in the set. This property is called closure.
| Set | + | − | × | ÷ |
|---|---|---|---|---|
| Natural | ✓ | ✗ (3 − 5 = −2) | ✓ | ✗ (3 ÷ 2) |
| Integers | ✓ | ✓ | ✓ | ✗ (1 ÷ 2) |
| Rational | ✓ | ✓ | ✓ | ✓ (except ÷ 0) |
| Irrational | ✗ (√2 + (−√2) = 0) | ✗ | ✗ (√2 × √2 = 2) | ✗ |
This shows why the number system "grew": we needed integers to subtract freely and rationals to divide freely.
Classifying numbers
Example: Classify −6.
- Not natural (negative), not whole
- Integer ✓, rational ✓, real ✓
Example: Classify 0.
- Whole ✓, integer ✓, rational ✓, real ✓
- Even (0 = 2 × 0)
- Neither positive nor negative
Worked examples
Example 1: Which set includes 0 but not −1?
Whole numbers.
Example 2: True or false: every integer is a rational number.
True: any integer n = n/1.
Example 3: Is the set of whole numbers closed under subtraction?
No: 2 − 5 = −3 isn’t whole.
Common mistakes
- Forgetting that every integer is also rational.
- Thinking 0 is positive or not even.
- Calling 1 a prime number. It has only one factor.
Try it yourself
- Name the smallest set containing −12.
- True or false: √25 ∈ ℕ.
- Is ℤ closed under multiplication?
- Classify 1.5 as completely as possible.
- Give two irrational numbers whose product is rational.
Answers: 1) integers 2) true (5) 3) yes 4) rational, real, positive 5) e.g. √3 × √3 = 3
Quick recap
- ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ.
- Use set notation (∈, ⊂) to describe membership.
- Closure tells which operations stay inside a set.