Number Systems
See why familiar number sets grow from one another, how they fit together, and how to choose the most specific set that describes a value.
Why do we need several kinds of numbers instead of one giant bucket?
Each familiar number system answers questions that an earlier system cannot. The sets grow outward, and a value keeps its earlier memberships as it enters larger sets.
Irrational numbers also live inside ℝ, but outside ℚ. They are not another rung containing the rationals.
New operations create answers the old set cannot hold.
Start with counting. Then ask increasingly demanding numerical questions and watch the allowable universe expand.
Count
Counting needs positive whole numbers.
Subtract past zero
Negative whole numbers force us beyond ℕ.
Divide between integers
Fractions force us beyond ℤ.
Fill the remaining gaps
Irrational values join the rationals to fill the real number line.
Smaller sets sit inside larger sets.
A natural number never stops being natural when we view it inside the integers, rationals, or reals. The larger sets add possibilities; they do not erase earlier membership.
If a value belongs to a smaller nested set, it also belongs to every larger set containing it.
Irrational numbers share the real number line with rationals, but they sit outside the rational region.
Calling −4 real is true, but Integer tells us more. Classification usually asks for the most specific standard set.
Now place the value in its smallest set.
The containment model is already built. Your job is to decide how far inward each value can go.
Which set is the most specific description of this value?
Classification describes the value, not every operation you can perform on it.
Repeating does not mean irrational
Terminating and repeating decimals are rational because they can be written as ratios of integers.
Irrational is not closed under addition
Two irrational inputs can produce a rational result. Do not turn a classification into an arithmetic rule.
Zero needs a convention note
Some authors include 0 in ℕ and some begin ℕ at 1. This lesson uses ℕ = {1, 2, 3, …}, so 0 is classified here as an integer.
Transfer checkThree fresh cases without the lesson example in front of you
What is the smallest listed set containing −12?
Later algebra will ask which values are allowed in a domain, whether an operation stays inside a set, and when a problem needs a larger number system. This hierarchy gives those questions a map.