Converting Decimals and Fractions
A rational number is any number than can be written as a ratio of two integers, such as \(\frac{4}{5}\) or \(\frac{14}{658}.\) Rational numbers can be written in decimal form or fraction form.
Introduction
Integers, terminating decimals, and repeating decimals can be written in fractional form. Likewise, all fractions can be written in decimal form.
Given \( \sqrt{4}, 3, \pi , \sqrt{5}, 0.333......\), which of the numbers can be written in fractional form?
\(\sqrt{4}=2\) is an integer and can be written in fractional form as \(\frac{2}{1}.\)
3 is also an integer which can be written in fractional form as \(\frac{3}{1}.\)
\(\pi=3.1415926......\) goes on forever without a certain number or numbers repeating, so it can't be written in fractional form. You might have seen \(\pi\) written as \(\frac{22}{7}\), but that is only an approximation and thus accurate only to the tenth decimal place.
\(\sqrt{5}=2.236......\) goes on forever without certain numbers repeating, so it can't be written in fractional form.
Even though \(0.333......\) is non-terminating, it is a repeating decimal in \(3\), whose fractional form is \(\frac{1}{3}\).
Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert \(\frac{1}{4}\) to a decimal, we divide 1 by 4 to get the quotient 0.25.
\[\frac{1}{4}=0.25\]
What is the decimal representation of \(\frac{1}{6}?\)
Performing long division when \(1\) is divided by \(6\), the quotient doesn't have a terminating point in that the 6 goes on forever. So, to represent a repeating number, we use a hyphen on the top: \[\frac{1}{6}=0.1\overline{6}. \]
What is the decimal representation of \(\frac{1}{5}?\)
Performing long division when 1 is divided by 5, we get the quotient to be 0.2. Thus,\[\frac{1}{5}=0.2.\]
Convert the fraction \( \frac3{5000} \) into decimal representation.
Converting Decimals to Fractions
To write an integer as a fraction, make the denominator one. For example, the number \(9\) can be written in fractional form as \(\frac{9}{1}\).
To write a terminating decimal as a fraction, write the decimal number into an equivalent fraction by making the numerator the number itself and the denominator 1. For example, if the number is \(0.6\), it becomes \(\frac{0.6}{1}\). After that, multiply both the numerator and the denominator with a number that takes the decimal place of the numerator into the last place. So multiply by 10 \(\left(\frac{0.6\times 10}{1 \times 10}\right)\) to get \(\frac{6}{10}\). Simplify to get \(\frac{3}{5}\).
What is the fractional representation of \(0.623?\)
We have
\[0.623=\frac{0.623}{1}.\]
Multiply both the numerator and the denominator by 1000, to move the decimal point of the numerator to the zero's place:
\[\frac{0.623\times1000}{1\times 1000}=\frac{623}{1000},\]
which can't be simplified any further. \(_\square\)
Which of the following is equal to \(0.54\)?
To write a repeating decimal, such as 0.343434..., as a fraction, we can begin by setting it equal to a variable: \[0.\overline{34} = x.\]
Our decimal repeats after the hundredths place, so we can write another equation, multiplying both sides of our original equation by 100: \[34.\overline{34}... = 100x.\]
Subtracting the first equation from the second, we get \(34 = 99x, \) or \(x = \frac{34}{99}.\)
What is the fractional representation of \(0.\overline{8}?\)
We have
\[0.\overline{8}=x.\]
Multiply both sides of the equation by 10:
\[8\overline{8}=10x.\]
Subtracting the first equation from the second, we have \(8 = 9x\) and \(x=\frac{8}{9}.\)
\[\begin{align*} 0.111111\ldots &= \frac{1}{9} \\ \\ 0.121212\ldots &= \ ? \end{align*}\]
You can read about other methods for converting repeating decimals to fractions here.