Number sets

This page gives short definitions of common sets of numbers.

• \(\mathbb{N}\) = Natural numbers: \(\{0,1,2,3,4,\dots\}\)
• \(\mathbb{Z}\) = Whole numbers (integers): \(\{\dots,-3,-2,-1,0,1,2,3,\dots\}\)
• \(\mathbb{Q}\) = Rational numbers: numbers that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\)
• Irrational numbers: numbers that cannot be written as a fraction \(\frac{a}{b}\) with integers \(a,b\) (for example \(\sqrt{2}\), \(\pi\), \(e\)); equivalently \(\mathbb{R} \setminus \mathbb{Q}\).
• \(\mathbb{R}\) = Real numbers: all rational and irrational numbers (e.g. \(\sqrt{2}\), \(\pi\), \(e\))
• \(\mathbb{C}\) = Complex numbers: numbers of the form \(a + bi\) with \(a,b \in \mathbb{R}\), where \(i^2 = -1\) (includes all real numbers as a subset)

Common symbols

• \(\mathbb{P}\) — often used for probabilities or a probability measure (e.g. \(P(A)\) is the probability of event \(A\))
• \(\mathbb{F}\) — a field in abstract algebra (\(\mathbb{F}_p\))
• \(\mathbb{H}\) — the quaternions (Hamilton's quaternions)
• \(\varnothing\) — the empty set
• \(\in\) — “is an element of” (for example \(x \in A\))
• \(\subseteq\) — “is a subset of” (for example \(A \subseteq B\))
• \(\forall\) — “for all” (universal quantifier)
• \(\exists\) — “there exists” (existential quantifier)