Mathematics is built on foundations. To master complex algebra or calculus, one must first have a firm grasp of the building blocks. Today, we are breaking down two essential concepts: how we categorize fractions and the formal definition of a rational number.
Fractions are more than just numbers with a line in the middle; they represent parts of a whole. Depending on their denominators (the number on the bottom), we group them into two categories.
Fractions that have the same denominator are called like fractions. Because they share the same base unit, they are easy to add, subtract, and compare.
Example: $\frac{2}{5}$, $\frac{3}{5}$, and $\frac{4}{5}$ are all like fractions.
If the denominators are different, the fractions are called unlike fractions. To perform operations like addition on these, you usually have to find a common denominator first.
Example: $\frac{3}{4}$, $\frac{5}{6}$, and $\frac{7}{8}$ are all unlike fractions.
As you move further into math, you’ll encounter the term “Rational Number.” While it sounds complex, it actually describes a very broad and familiar group of numbers.
A rational number is any number that can be written in the form:
Where:
$p$ and $q$ are integers (positive or negative whole numbers).
$q$ is not equal to zero ($q \neq 0$) (because you cannot divide by zero).
In mathematics, the entire set of rational numbers is denoted by the symbol $\mathbb{Q}$.
Rational numbers include more than just standard fractions. They also include whole numbers (because they can be written over 1) and negative numbers:
Fractions: $\frac{1}{2}$, $\frac{3}{4}$, $-\frac{5}{11}$
Integers: $1, 0, 2, -3$ (Note: $2$ can be written as $\frac{2}{1}$, so it fits the definition!)
Recall that a rational number is any number that can be written in the form $\frac{p}{q}$. Since zero can be expressed as a fraction with any non-zero integer as the denominator, it fits the definition perfectly.
For example:
Because $0$ can be represented as a ratio of two integers (where the bottom number isn’t zero), it is officially part of the set of rational numbers ($\mathbb{Q}$).
While zero can be the numerator (top number), it can never be the denominator (bottom number). In mathematics, division by 0 is not defined.
If you try to divide a number by zero, the expression loses its physical and mathematical meaning. This is why the formal definition of a rational number always includes the strict condition that $q \neq 0$.