# Can rational and irrational numbers be negative?

Apr 4, 2016

Yes

#### Explanation:

Yes, rational and irrational numbers can be negative. Te only thing that is desired is that they could be mapped to a place on a real number line. Negative numbers are to the left of $0$ on number line.

By definition, rational numbers are a ratio of two integers $p$ and $q$, where $q$ is not equal to $0$. Hence, if $p$ is negative and $q$ is positive (or vice versa but $q \ne 0$), $\frac{p}{q}$ could be negative.

Examples of negative rational numbers are $- 3.14159$, $- \frac{17}{4}$, $- \frac{2}{3}$ or $- 3. \overline{142857}$ (here $\overline{142857}$ indicates these numbers are repeating infinitely). These are equivalent to $- \frac{314159}{100000}$, $- \frac{17}{4}$, $- \frac{2}{3}$ or $- \frac{22}{7}$ (in form $\frac{p}{q}$).

Similarly there could be negative irrational numbers too like $- \pi$, $\sqrt{- 80}$, $- \sqrt{2}$ etc. These are equivalent to their positive irrational numbers like $\pi$, $\sqrt{80}$, $\sqrt{2}$ but towards left of $0$ on real number line.

Similarly, there could be irrational numbers like $6 - 3 \pi$ which are $3 \pi$ units to the left of $6$, but as this would lie to the left of $0$. Some other such numbers are $1 - \sqrt{3}$, $\sqrt{2} - \sqrt{5}$, $- \sqrt{17} + 2$ and $- 7.239135113355111333555 \ldots . .$. (The last number is non-terminating non-repeating decimal number and hence an irrational number.)