kinds of numbers

1. Start with counting
   $\mathbb{N} \rightarrow$ natural nonnegative 'counting' numbers... 1,2,3... with some ambiguity about whether 0 is included
   Symbols for showing clearly that 0 is not included: $\mathbb{N}_{>0} \text{, } \mathbb{N}^+$

2. Include zero
   $\mathbb{N}_0$ $\rightarrow \mathbb{N}$ with 0 included, also called whole numbers

3. Add negatives
   $\mathbb{Z} \text{ integers } \rightarrow $ whole/natural numbers and their negatives, and 0

4. Allow division
   $\mathbb{Q} \text{ rational } \rightarrow$ numbers that can be written as a fraction e.g. 3/4
   $\mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p,q \in \mathbb{Z},\; q \ne 0 \right\}$

5. Fill in continuous gaps
   irrational $\rightarrow$ numbers that cannot be written as a fraction, e.g. sqrt(2) $\mathbb{R}$ real -> rational U irrational, all can be written on number line

6. Add the imaginary number $i = \sqrt{-1}$
   $\mathbb{C} \text{ complex } \rightarrow a + bi$
   A complex number is imaginary if a = 0, real if b = 0.

Other terms:
computable $\rightarrow$ numbers whose digits can be generated by an algorithm