Lesson 63: Rational Numbers, Non-Terminating Decimals, and Percents and Fractions with Negative Exponents

New Concept

We'll explore how rational numbers create non-terminating decimals and percents, learning why fractions are often the most precise form for calculations.

What’s next

Next, you’ll tackle worked examples on converting, comparing, and calculating with these numbers, plus simplifying expressions with negative exponents.

Property

A rational number converted to a decimal either terminates (ends) or repeats forever. A repeating decimal is shown with a bar, like $\frac{1}{11} = 0.\overline{09}$.

Examples

$\frac{2}{3}$ is the repeating decimal $0.\bar{6}$, which rounds to $0.667$.
To compare $\frac{4}{5}$, $83\frac{1}{3}\%$, and $0.83$, convert them to decimals: $0.800$, $0.833$, $0.830$.
The fraction $\frac{1}{8}$ is a terminating decimal, $0.125$.

Explanation

That repeating bar means the decimal goes on forever! For precise math, it’s best to use the original fraction in your calculations. Only round your final answer if needed, otherwise you might introduce small but sneaky errors into your work!

Property

To handle a negative exponent on a fraction, flip the fraction to its reciprocal and make the exponent positive.

Examples

$(\frac{1}{3})^{-2}$ becomes $3^2$, which equals $9$.
$(\frac{1}{2})^{-3}$ becomes $2^3$, which equals $8$.
$(\frac{3}{2})^{-1}$ becomes $(\frac{2}{3})^1$, which equals $\frac{2}{3}$.

Explanation

Think of a negative exponent as a secret 'flip' command! When a fraction gets this command, it does a somersault—the bottom number goes to the top and the top goes to the bottom. The exponent then becomes positive and happy.

Property

To convert a percent with a fraction into a simple fraction, write it over 100, change the mixed number to an improper fraction, and then simplify the resulting complex fraction.

Examples

$16\frac{2}{3}\% = \frac{16\frac{2}{3}}{100} = \frac{50/3}{100} = \frac{50}{300} = \frac{1}{6}$
$41\frac{2}{3}\% = \frac{125/3}{100} = \frac{125}{300} = \frac{5}{12}$
$66\frac{2}{3}\% = \frac{200/3}{100} = \frac{200}{300} = \frac{2}{3}$

Explanation

Don't get tangled up calculating with a percent like $16\frac{2}{3}\%$! The pro move is to convert it into a clean, simple fraction first. This makes multiplication way easier and keeps your answers super accurate by avoiding repeating decimals.