The number sets N, Z, Q, R

The numbers we use are not all of the same kind. As the demands of calculation grow, we are forced to widen our “numerical universe”: from the naturals to the integers, then to the rationals and finally to the reals.

Definition — The number sets

• $\mathbb{N} = \{0,1,2,3,\ldots\}$   natural numbers
• $\mathbb{Z} = \{\ldots,-2,-1,0,1,2,\ldots\}$   integer numbers
• $\mathbb{Q} = \left\{\dfrac{p}{q} : p\in\mathbb{Z},\; q\in\mathbb{Z}\setminus\{0\}\right\}$   rational numbers
• $\mathbb{R}$   real numbers (they also include the irrationals, such as $\sqrt{2}$ and $\pi$)

Each set is contained in the next: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

The chain of inclusions is visualised with four “boxes” one inside the other: each widening adds new numbers without losing the previous ones.

The number sets, one contained in the other: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$.

Links

Topics: Numbers and operations
Concepts: Number sets · Integer numbers · Natural numbers · Rational numbers · Real numbers