Numbers that cannot be expressed as fractions

Question

What are Numbers that cannot be expressed as Fractions called?

Answer 1

Numbers that can be expressed as a fraction are called rational. Numbers that can't be expressed as a fraction are called irrational. $\sqrt{2}$ is famously irrational, and so are many numbers like $\pi$, $\sqrt[5]{17}$, $e$, $\ln 2$, and so on. There are lots more irrational numbers than rational numbers, but I'll leave you to look up the proof yourself.

Answer 2

Any number that can be expressed as the ratio of integers: (i.e., any number that can be expressed as a fraction whose numerator and denominator are both integers), is called a rational number (as in "ratio"). Any number that cannot be expressed as a ratio of integers (fraction) is called an irrational number, as in "not rational".

For example, we have that $2 = \dfrac 21$ is a rational number, as is $0.3\overline{33} = \dfrac 13.\;$ Both numbers are rational numbers because we can express each as equal to a fraction (with integers for numerator and denominator).

On the other hand, $\sqrt 2, \; \pi,\; \dfrac {\sqrt 3}{2}$ are all irrational numbers: none of them can be expressed as a fraction of integers; i.e. there do not exist any integers $p, q, r, s$ such that $\dfrac pq =\sqrt 2$ or such that $\;\pi = \dfrac rs$.