Module 1 · Numbers and number sets · Lesson 1 of 39
Number sets
Natural, whole, integer, rational, irrational and real numbers, and how they nest.
In this lesson · about 12 minutes
- Name the number sets inside the real numbers
- Decide the smallest set a number belongs to
- Read the symbols ℝ, ℚ, ℤ and ℕ and write a set in set notation
When we say "numbers" we usually picture 1, 2, 3, 4, 5, … But this course also uses negative numbers, fractions, decimals and roots. Each kind of number belongs to a number set, and the sets fit inside one another like nesting boxes.
| Set | Symbol | What is in it |
|---|---|---|
| Natural numbers (counting numbers) | ℕ | 1, 2, 3, 4, 5, … |
| Whole numbers | 0, 1, 2, 3, 4, …: the natural numbers plus 0 | |
| Integers | ℤ | …, −3, −2, −1, 0, 1, 2, 3, …: whole numbers and their negatives |
| Rational numbers | ℚ | Anything that can be written as a fraction: 14, −23, 5, −0.25, 0.333… |
| Irrational numbers | ℝ − ℚ | Numbers that can't be fractions: √2 = 1.41421356…, π = 3.14159265… |
| Real numbers | ℝ | All rational and irrational numbers together |
Every natural number is a whole number, every whole number is an integer, and every integer is rational (7 = 71). Rational and irrational numbers don't overlap: together they make up the real numbers.
Set notation
We list the members of a set inside curly braces. For example, North American countries = {Canada, United States, Mexico}, and the natural numbers are ℕ = {1, 2, 3, 4, 5, …}.
Sorting drill
Sort the numbers into their smallest set
Pick the smallest (innermost) set each number belongs to.
- 12
- 0
- −8
- 3/4
- −0.25
- 0.333…
- √2
- π
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