Real number classification

In Real Analysis you study real numbers and their properties.

The classification of real numbers (using set notation) goes like this:

• natural numbers are the set of numbers we use for counting, $\mathbb{N}$
• whole numbers are the set of numbers we count with plus the number zero, $\mathbb{Z^+}$
• integers are the set of natural numbers, their opposites, and zero, $\mathbb{Z}$
• rational numbers are the set of numbers that are ratios of integers, $\mathbb{Q}$
• as a caveat, the denominator should be $\neq 0$
• they can be represented as decimal numbers as well (terminating or repeating)
• irrational numbers are the set of numbers that can’t be represented as a fraction, $\mathbb{I}$

There are also a few others but they belong to more advanced treatments of number theory. Because of these definitions, some numbers may belong to multiple groups. Below is a diagram of these categories.

Fig 1. Classifying real numbers

These numbers have a bunch of properties that are useful shortcuts when doing calculations by hand.