Number sets

Some sets of numbers are so important that they have special names and symbols. These are usually written in bold or blackboard bold, e.g. $\mathbf{Z}$ or $\mathbb{Z}$.

The usual containment chain is $\mathbb{N}\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteq \mathbb{R}\subseteq \mathbb{C}.$ That means every natural number is an integer, every integer is rational, every rational is real, and every real is complex.

$\mathbb{N}$

The natural numbers. Convention varies: some authors use $\{1,2,3,\dots \}$, while others include $0$. State the convention when it matters.

$\mathbb{Z}$

The integers: $\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}.$

$\mathbb{Q}$

The rational numbers:

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$\mathbb{R}$

The real numbers: rational and irrational numbers, including values such as $\sqrt{2}$, $\pi$, and [[Euler’s number|$e$]]. They form the usual continuous number line.

$\mathbb{C}$

The complex numbers: $\mathbb{C}=\{a+ib:a,b\in \mathbb{R},i^2=-1\}.$

Common subsets

• ${\mathbb{R}}^{+}$ often means positive real numbers, though some authors include $0$; check convention.
• Intervals such as $[0,1]$ and $(a,b)$ are subsets of $\mathbb{R}$.
• ${\mathbb{R}}^n$ is the set of $n$-tuples of real numbers, the standard coordinate model for Euclidean spaces.

Choosing the right number set is part of defining a Domain. It changes which operations are valid and whether Inverse functions exist.