Comparing Numbers Using Reciprocals

Property For any positive numbers $a$ and $b$, the direction of an inequality is reversed when we take the reciprocal of both sides. If $a b$, then $\frac{1}{a} < \frac{1}{b}$.

Examples Example 1 To compare $A = \frac{100}{101}$ and $B = \frac{101}{102}$, we compare their reciprocals. $\frac{1}{A} = \frac{101}{100} = 1 + \frac{1}{100}$ $\frac{1}{B} = \frac{102}{101} = 1 + \frac{1}{101}$ Since $\frac{1}{100} \frac{1}{101}$, we have $\frac{1}{A} \frac{1}{B}$, which implies $A < B$. Example 2 Compare $a = \frac{\sqrt{5}}{\sqrt{5}+1}$ and $b = \frac{\sqrt{6}}{\sqrt{6}+1}$. $\frac{1}{a} = \frac{\sqrt{5}+1}{\sqrt{5}} = 1 + \frac{1}{\sqrt{5}}$ $\frac{1}{b} = \frac{\sqrt{6}+1}{\sqrt{6}} = 1 + \frac{1}{\sqrt{6}}$ Because $\sqrt{5} < \sqrt{6}$, we know $\frac{1}{\sqrt{5}} \frac{1}{\sqrt{6}}$. Therefore, $\frac{1}{a} \frac{1}{b}$, so $a < b$.

Explanation When comparing two fractions that are very close to 1, it can be easier to compare their reciprocals instead. By taking the reciprocal of each number, you can often express them in the form $1 + \epsilon$, where $\epsilon$ is a small fraction. Comparing the sizes of these small fractions allows you to determine the larger reciprocal. Remember that if the reciprocal of one number is larger, the original number itself is smaller.