# Are all fractions rational numbers?

No.

A rational number can be expressed in the form $\displaystyle\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. In other words, it is a fraction whose denominator is not zero, and both the denominator and numerator are integers.

Some fractions, however, may contain a numerator or denominator that is not an integer. Some examples of such fractions are

$\displaystyle\frac{\sqrt{3}}{2}$, $\displaystyle\frac{\pi}{4}$ and $\displaystyle\frac{e}{2}$.

A rational number may be represented in many ways, but it can always be expressed as a fraction. For instance, $10^{-1}$ is a rational number because we can express it as $\frac{1}{10}$. Also, the number $0.333 \cdots$, a repeating decimal, is  a rational number because we can also express it as fraction $\frac{1}{3}$.

## 3 Comments

1. Actually, at Spain we usually call numbers as $\frac{\sqrt(3)}{2}$ ratios (razones), instead of fractions (fracciones). That way we can identify fractions and rational numbers. It’s just semantics, but it simplifies things a lot.

• Guillermo Bautista

True. But, in English, we also have ratios containing integers, so that could be quite confusing. I’m glad it works in your country.

2. Good post, Guillermo. Basically, all rational numbers are fractions, but not all fractions are rational numbers.