Kinds of Numbers: A Map from Natural to Complex

hari@dasarpai.com (Dr. Hari Thapliyaal) — Sun, 04 Oct 2026 00:00:00 +0000

Kinds of Numbers
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Natural → integer → rational → real → complex is the main ladder. Each step adds numbers the previous step could not hold.

Kind	Symbol (usual)	One-line idea	Example
Natural	$\mathbb{N}$	Counting sizes	$1, 2, 3, \ldots$
Whole	—	Natural plus zero	$0, 1, 2, \ldots$
Integer	$\mathbb{Z}$	Whole numbers and negatives	$-3, 0, 7$
Rational	$\mathbb{Q}$	Ratio of two integers	$\tfrac{3}{4}$, $-2$, $0.125$
Irrational	(inside $\mathbb{R}$)	Real, not any exact fraction	$\sqrt{2}$, $\pi$
Real	$\mathbb{R}$	Points on the number line	$-1.5$, $e$, $\sqrt{5}$
Imaginary	—	Real multiple of $i$	$3i$, $-i$
Complex	$\mathbb{C}$	Real part plus imaginary part	$2 + 3i$

Two phrases sound alike but mean different things:

Kinds of numbers — which set a value belongs to ($\mathbb{Q}$, $\mathbb{R}$, …).
Positional number system — how you write a value in a base (decimal, binary, hex).

This article maps the sets first, then points at bases and floating-point formats data work uses every day.