A rational number is often written as a fraction $\frac{p}{q}$. However, for mathematical clarity, we usually want to express it in its “Standard Form.”
A rational number $\frac{p}{q}$ is said to be in standard form if it meets two specific conditions:
The integers $p$ (numerator) and $q$ (denominator) have no common divisor other than 1. This means the fraction is fully simplified.
The denominator $q$ is positive.
Examples of Standard Form:
$\frac{5}{6}$
$-\frac{7}{8}$
$\frac{3}{14}$
If you have a rational number like $\frac{10}{-15}$, it is not in standard form because the denominator is negative and both numbers can be divided by 5.
To convert it:
Find the greatest common divisor (GCD). For 10 and 15, the GCD is 5.
Now, the fraction is simplified, and the denominator is positive.
Comparing fractions can be tricky when they have different denominators. To arrange them in Ascending (smallest to largest) or Descending (largest to smallest) order, follow these steps:
Standardize: Convert all rational numbers to their standard form.
Find a Common Denominator: Ensure all fractions have the same denominator (usually by finding the Least Common Multiple).
Compare Numerators: Once the denominators are identical, the order of the rational numbers is determined solely by the value of their numerators.
Task: Write the rational numbers $\frac{1}{3}, \frac{-2}{9}, \frac{-4}{3}$ in ascending order.
Solution:
First, let’s make all the denominators the same (in this case, 9):
$\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}$
$\frac{-2}{9}$ (already has 9 as the denominator)
$\frac{-4}{3} = \frac{-4 \times 3}{3 \times 3} = \frac{-12}{9}$
Now we compare the numerators: -12, -2, and 3.
Clearly, $-12 < -2 < 3$.
Therefore, the ascending order is:
Which corresponds to the original numbers: