Rational Numbers

 

 

$$ \left\{ \frac{3}{4}, \frac{5}{8}, \frac{4}{6} \right\} $$

Let’s order these numbers from smallest to largest.

 

 

Example:

 

$$ \left\{ \frac{3}{5}, \frac{2}{6}, \frac{4}{7} \right\} $$

 

Let’s order these numbers.

If we equalize the numerators of the fractions at their least common multiple; $\text{LCM}(3, 2, 4) = 12$. From here:

 

\[\begin{array}{l l } {\displaystyle\frac{3}{5} \cdot \displaystyle\frac{4}{4} = \displaystyle\frac{12}{20}} \end{array}\]
\[\begin{array}{l l } {\displaystyle\frac{2}{6} \cdot \displaystyle\frac{6}{6} = \displaystyle\frac{12}{36}} \end{array}\]
\[\begin{array}{l l } {\displaystyle\frac{4}{7} \cdot \displaystyle\frac{3}{3} = \displaystyle\frac{12}{21}} \end{array}\]

then

$${\frac{12}{20} >\frac{12}{21} > \frac{12}{36} \Rightarrow \frac{3}{5}> \frac{4}{7} >\frac{2}{6} }$$ is ordered this way.

 

Where \(\displaystyle{\frac{a}{b} } \quad \) and \( \quad \displaystyle{\frac{c}{d} } \) are two positive fractions,

\[\begin{array}{l l } \text{if } {\displaystyle\frac{a}{b} \div \displaystyle\frac{c}{d}} > 1 \quad \text{then} \quad {\displaystyle\frac{a}{b} >\displaystyle\frac{c}{d} } \end{array}\]

\[\begin{array}{l l } \text{if } {\displaystyle\frac{a}{b} \div \displaystyle\frac{c}{d}} < 1 \quad \text{then} \quad {\displaystyle\frac{a}{b} <\displaystyle\frac{c}{d} } \end{array}\]

\[\begin{array}{l l } \text{if } {\displaystyle\frac{a}{b} \div \displaystyle\frac{c}{d}} = 1 \quad \text{then} \quad {\displaystyle\frac{a}{b} =\displaystyle\frac{c}{d} } \quad \text{holds.} \end{array}\]
*(Note: Corrected a subtle textbook typo where the text repeated a less-than sign instead of an equality check inside the final condition).*

 

Example:

 

Let’s examine the fractions \( \displaystyle{\frac{2}{3} } \quad \) and \( \quad \displaystyle{\frac{3}{4} } \).

\[\begin{array}{l l } {\displaystyle\frac{2}{3} \div \displaystyle\frac{3}{4} =\displaystyle\frac{2}{3} \cdot \displaystyle\frac{4}{3} =\displaystyle \frac{8}{9} < 1} \end{array}\]

Since the result is less than 1, the dividend fraction \(\displaystyle {\frac{2}{3} } \quad \) is smaller. That is, \(\displaystyle{\frac{2}{3} < \frac{3}{4} } \).

 

 

5. For positive fractions where the difference between the numerator and the denominator is equal, as the numbers in the numerator and denominator grow larger; the value of proper fractions increases, while the value of improper fractions decreases.

\[\begin{array}{l l }
{\displaystyle\frac{3}{8} <\displaystyle\frac{4}{9} <\displaystyle\frac{7}{12}< \displaystyle\frac{8}{13} \cdots} \\ {\displaystyle\frac{5}{3} >\displaystyle\frac{7}{5} >\displaystyle\frac{9}{7} >\displaystyle\frac{10}{8} \cdots }\\
\end{array}\]

 

Warning: When comparing negative numbers, they are first ordered without considering their signs. Finally, all numbers are multiplied by -1, which reverses the direction of the inequality.

 

Example:

 

\[\begin{array}{l l } a= {-\displaystyle\frac{11}{10}} , b= {-\displaystyle\frac{101}{100}} , c= {-\displaystyle\frac{1001}{1000}} \end{array}\]

Accordingly, let’s determine the relationship between the numbers a, b, and c.

 

Example: