How were the number symbols $\mathbb{N}$, $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{H}$, and $\mathbb{C}$ defined?

Question

How were the number-symbols like $\mathbb{N}$, $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{H}$, and $\mathbb{C}$ defined?

I.e. say, why does $\mathbb{Z}$ represent Integer numbers, and so on?

Answer 1

Turning my comment into an answer then: there are have been various different letters used for each of the number sets listed, and a consensus seems to have only settled in the last fifty years or so. For example, in 1953 the second edition of Birkhoff and MacLane's A survey of modern algebra appeared, which uses $J$ for the integers and $R$ for the rationals.

It's probably also worth noting that the modern appearance of the blackboard-bold letters (e.g. ${\mathbb N}$ that we're all familiar with now) can be attributed to Donald Knuth up to a point, as TeX (which he wrote in the $1970$s) has effected quite a standardisation of mathematical typesetting. Prior to TeX it wasn't at all unusual to see the blackboard-bold letters hand-drawn in a typewritten text, or created by overlapping an $I$ with the appropriate other letter. As Xander Henderson notes in the comments, there are actually two approaches; the other being to overlap the same letter. In MathJax this is achieved by Z\!\!\!Z to give $Z\!\!\!Z$. Xander also notes that the name blackboard bold comes from chalk on blackboards, where to write a bold letter you would turn the chalk sideways to produce thick lines. However, this is tricky (chalk is not very wide and it's hard to grip like that and use it against a vertical surface) and time-consuming, so the compromise of double-striking the letter was reached.

Dedekind appears to have been the first to use ${\mathbb N}$ for the natural numbers, in $1888$, in Was ist und was sollen die Zahlen? Peano also used ${\mathbb N}$ the following year in Arithmetices prinicipia nova methodo exposita and goes on in $1895$ to start one of the bigger irrelevant controversies of mathematics ($0\in {\mathbb N}$?) by using ${\mathbb N}_0$ in Formulaire de mathématiques to indicate that $0$ does not belong to ${\mathbb N}$. Since the German for natural numbers is naturellen Zahlen it's a happy coincidence that the ${\mathbb N}$ works well in English too.

Edmund Landau introduce ${\mathbb Z}$ for Zahlen in $1930$ in Grundlagen der Analysis. Strictly, the German word for integers is ganze Zahlen and Hasse, in $1926$, in Höhere Algebra I uses $\Gamma$, probably because of the g sound for ganze.

The Bourbaki mathematicians then, in the $1930$s use ${\mathbb Z}$ for integers (presumably following Landau since they were mostly French) and ${\mathbb Q}$ for the quotients of integers, or the rationals as we call them now. (Bourbaki's Algébre demonstrates these symbols.)

Dedekind probably gets the credit for ${\mathbb R}$ for the reals as he uses in (in blackletter) in $1872$ in Stetigkeit und irrationale Zahlen -- but to be scrupulous he does also use $R$ for the rationals which isn't the clearest choice of notation.

Hamilton is famous for his work on the quaternions, and ${\mathbb H}$ appears to be named for him (although sometimes it's claimed that the ${\mathbb H}$ stands for Hypercomplex numbers), but I've been unable to track down who so honoured him (or if it was even his own notation). He was working in the early 1800s, so if it's his own notation it would have a good claim to be both the first, and the longest established.

Finally, ${\mathbb C}$ appears to appear for the first time only in $1939$ in Nathan Jacobson's Structure and Automorphisms of Semi-Simple Lie Groups in the Large, Annals of Math. 40 (1939), 755-763 with Bourbaki using it nearly $10$ years later.