Converting Between Fractions and Decimals

The fraction bar means division. For integers $p$ and $q$ with $q$ not zero, the fraction $\frac{p}{q}$ is the quotient $p \div q$: the result of dividing the numerator $p$ (the top number) by the denominator $q$ (the bottom number). Here $q$ is the divisor, the number you divide by. It must be nonzero, because division by zero is undefined. The numerator may be zero: $\frac{0}{9} = 0 \div 9 = 0$, because zero divided by a nonzero number is zero.

Carrying out the division gives one of two kinds of decimal:

• A terminating decimal has a last digit, such as $0.75$.
• A repeating decimal never ends, and one digit or block of digits repeats forever, such as $0.333\ldots$

Write a repeating decimal with a bar over the repeating digits only: $0.333\ldots = 0.\overline{3}$ and $0.1666\ldots = 0.1\overline{6}$. The bar shows exactly which digits repeat, so the decimal still names the exact value.

In that example, a remainder of $0$ ended the division. What happens when the remainder never reaches $0$?

When you divide a whole number by a positive integer $q$, every remainder is a whole number from $0$ to $q - 1$, so only $q$ different remainders are possible. If the remainder is never $0$, some remainder must show up a second time within $q$ steps. Past the decimal point, each step uses only the remainder left by the step before. So once a remainder comes back, the same steps happen again in the same order, and the same digits repeat forever.

That is why the decimal form of a rational number always terminates or repeats. A decimal that neither terminates nor repeats cannot come from dividing an integer by a nonzero integer, so it is not rational. Such a number is called irrational.

Now go the other way, from a decimal to a fraction. A negative decimal is $-1$ times a positive decimal, so convert the positive decimal and attach the negative sign at the end.

For a terminating decimal, place value does the work. The place of the last digit gives the denominator: tenths give $10$, hundredths give $100$, and thousandths give $1000$. Each of these is a power of ten: a number in the list $10$, $100$, $1000$, and so on, where each is ten times the one before.

Take $0.36$. Its last digit is in the hundredths place, so it is $36$ hundredths, or $\frac{36}{100}$. Next, write the fraction in lowest terms, which means the numerator and the denominator share no common factor greater than $1$. Their greatest common factor, the largest whole number that divides both, is $4$, so divide both by $4$. This does not change the value, because you are dividing the fraction by $\frac{4}{4}$, which equals $1$. The same is true for any nonzero number in place of $4$.

A repeating decimal has no last digit, so place value cannot give you a denominator. Use the repetition instead.

Multiplying by $10$ moves every digit one place to the left. Suppose the repeating block starts right after the decimal point. Multiply by $10$ once for each digit in the block. One whole block moves in front of the point, and the digits after the point stay exactly the same. Now you have two numbers with identical digits after the point. Subtract the original number from the new one, and those matching digits subtract to zero. A whole number is left, and one short equation gives the fraction.

The takeaway: a fraction and a decimal are two ways to write one rational number. To turn $\frac{p}{q}$ into a decimal, divide the integer $p$ by the nonzero integer $q$. The result terminates or repeats. To go back, read a terminating decimal by place value, or use multiply-and-subtract on a repeating decimal, and then write the fraction in lowest terms. Keep the value exact along the way: a bar marks repeating digits, and $\approx$ marks a rounded value.

Next, you will evaluate numerical expressions involving rational numbers, solve word problems with rational number operations, and find and correct errors in those operations.