Number sets (ℕ, ℤ, ℚ, ℝ, ℂ)

How to read it

The letters are double-struck, drawn with a doubled stroke (Unicode’s term), and in LaTeX each is written with \mathbb, as in Levin’s book. Unicode encodes all five as letterlike symbols: ℕ (U+2115) “natural number”, ℤ (U+2124) “the set of integers”, ℚ (U+211A) “the set of rational numbers”, ℝ (U+211D) “the set of real numbers” and ℂ (U+2102) “the set of complex numbers”. Each of the first four is a subset of the next: Richard Hammack’s Book of Proof writes ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ. Oscar Levin’s Discrete Mathematics writes ℤ = {…, −2, −1, 0, 1, 2, 3, …}, and NIST’s Digital Library of Mathematical Functions calls ℝ the real line and ℂ the complex plane. The sign x ∈ ℝ, “x is a real number”, uses the element-of sign.

Also searched as: blackboard bold, ℂ, complex numbers symbol, double struck letters, integers symbol, is 0 a natural number, ℕ, natural numbers symbol, ℚ, ℝ, rational numbers symbol, real numbers symbol, what does ℝ mean, ℤ.

Is 0 a natural number?

It depends on the book. OpenStax’s Contemporary Mathematics writes ℕ = {1, 2, 3, …}, the positive counting numbers, and NIST’s Digital Library of Mathematical Functions defines ℕ as the set of all positive integers. Oscar Levin’s Discrete Mathematics takes ℕ = {0, 1, 2, 3, …} and warns that “there is a fairly even split between mathematicians about whether 0 is an element of the natural numbers”. Richard Hammack’s Book of Proof starts at 1. Peano was already marking the difference in 1895, writing N for the positive integers and N₀ for the positive integers and zero (Miller).

Why Z and Q?

According to Miller, Z is for the German Zahlen, “numbers”, and Q for Quotient. Before Bourbaki settled them, books used many other letters: Miller lists Helmut Hasse’s Γ for the integers and Ρ (capital rho) for the rationals in 1926, and van der Waerden’s C and Γ in 1930.