# How to express a set of all numbers, real and imaginary, irrational and rational?

The notation for real numbers is $$\mathbb{R}/\mathbf{R}$$, integers: $$\mathbb{Z}/\mathbf{Z}$$, complex numbers $$\mathbb{C}/\mathbf{C}$$, and rational: $$\mathbb{Q}/\mathbf{Q}$$ but is there an agreed-upon way to express all numbers with similar notation?

Basically, if I were told to write a phrase that captured every number in existence, how would I do this?

• All numbers${}$? – Angina Seng Nov 1 '19 at 12:52
• The cimplex numbers $\Bbb C$ includes all the others you mention. $3$ is still a complex number even though it's an integer. Is that what you meant? – Arthur Nov 1 '19 at 12:54
• How would you define "all numbers"? If a small girl somewhere has created their own number system using numbers "one Mississippi", "two Mississippi" etc and calls it the set $\mathbb{M}$, would you include that too? – Matti P. Nov 1 '19 at 12:54
• I don't think there are any other "numbers" left in the usual sense when you have already mentioned $\mathbb{C}$. Though, you could define what "all numbers" means. – Mann Nov 1 '19 at 12:55
• There is no limit as to how complicated numbers can be. Abstract (or modern) algebra was developed since 1800s and it allows to create as complicated sets of numbers as you wish as long as you define + and $\times$ for them and these follow the rules of for example a field en.wikipedia.org/wiki/Field_(mathematics) . The Real and Complex numbers are just the start of a really long and weird journey. – mathreadler Nov 1 '19 at 13:27

As you observed, blackboard bold is a standard font used for successive extensions of number systems: $$\Bbb{N} \subseteq \Bbb{Z} \subseteq \Bbb{Q} \subseteq \Bbb{R} \subseteq \Bbb{C}$$ The set of quaternions, denoted by $$\Bbb{H}$$ in honour of the mathematician W. R. Hamilton, would be the next step. The next extension is the set of octonions, denoted by $$\Bbb{O}$$ and the next one the set of sedenions, denoted by $$\Bbb{S}$$.
You will find many other extensions in the Wikipedia articles on Hypercomplex numbers, Hyperreal numbers and Surreal numbers. The class -- this is no longer a set -- of all surreal numbers is denoted by the symbol $$\mathbf{No}$$. They are the largest possible ordered field: every other ordered field can be embedded in the surreals. Finally, a surcomplex number is a number of the form $$a+bi$$, where $$a$$ and $$b$$ are surreal numbers. Surcomplex numbers form an algebraically closed field (except for being a proper class). See also this question on Mathoverflow.
If you ask me whether there is an ordered set of all numbers, the answer is no, but if you ask me whether there is an ordered class of all numbers, the answer is $$\mathbf{No}$$.