AI Tutor
If you only had counting numbers, you could easily count your apples: 1, 2, or 3. But what happens when you eat every apple, owe a friend two apples, or slice one into pieces?
To describe every possible quantity in the physical world, mathematicians organized all measurements into one master family: the real number system.
How do all these different kinds of numbers fit together?

The Number Hierarchy

Think of the real number system like a set of nesting boxes. Every box belongs inside a larger category, but the whole system splits into two distinct branches: rational and irrational.

The Rational Family

Natural numbers are your basic counting numbers (1,2,3,…1, 2, 3, \dots). If you include zero, you get the whole numbers (0,1,2,3,…0, 1, 2, 3, \dots).
When you add negative counterparts to whole numbers, you get integers (…,βˆ’2,βˆ’1,0,1,2,…\dots, -2, -1, 0, 1, 2, \dots). Integers never have fractional or decimal parts.
A rational number is any number you can write as a ratio ab\frac{a}{b}, where aa and bb are integers and b≠0b \neq 0. In decimal form, rational numbers always either terminate (like 0.750.75) or repeat infinitely (like 0.3‾0.\overline{3}).
Why must every integer also be a rational number? Because any integer nn can be written as n1\frac{n}{1}.
What about decimals that go on forever without ever repeating a pattern?

The Irrational Numbers