> [!NOTE] Definition (Rational Number) > The set of *rational numbers* is given by $\mathbb{Q}=\left\{ \frac{p}{q} \mid p \in \mathbb{Z} \text{ and } q \in \mathbb{N} \right\}.$ # Properties See [[Constructing Rationals]] for definition of operations on rational numbers. Note that [[The sum of and product of any two rational number is again rational]] also additive and multiplicative inverses are rational. **Comparison with reals:** considering Cauchy sequences of rationals allows a natural [[Constructing Reals|construction of real numbers]] which may be *irrational*. For example, [[Square root of 2 is irrational]] on the other hand [[There is a unique real number that is the square root of 2]]. Specifically, the set $\{ x\in \mathbb{Q} \mid x^{2}< 2 \}$ does not have a tight/ least upper bound although it is bounded above. A key property of reals is that any any bounded set of real numbers has a least upper bound which motivates the use of Dedekind to define real numbers. This property is known as [[Real numbers|completeness]]. Note that [[Any real number can approximated arbitrarily closely by a rational number]] so $\mathbb{Q}$ is a [[Dense]] subset of $\mathbb{R}$.