CHANGING DECIMALS TO FRACTIONS

Consider again the place values in the base ten number system.
If we move from left to right, notice that the place value is successively divided by ten:

place values to the right of the decimal point

This pattern continues, after first putting a decimal point to the right of the ones place:

Be certain to notice the difference between hundred and hundredth.
Hundred is to the left of the decimal point, and hundredth is to the right of the decimal point.

The place values are ‘mirrored’ about the ones place, adding ‘th’ to the right of the decimal point:

mirroring about the ones place

The first place to the right of the decimal point has place value [beautiful math coming... please be patient] $\displaystyle\,\frac{1}{10^1} = \frac{1}{10}\,$.

The second place to the right of the decimal point has place value [beautiful math coming... please be patient] $\displaystyle\,\frac{1}{10^2} = \frac{1}{100}\,$.

In general, the $\,n^{\text{th}}\,$ place to the right of the decimal point has place value [beautiful math coming... please be patient] $\displaystyle\,\frac{1}{10^n}\,$.

A base ten number that uses a decimal point is called a decimal.
Thus, [beautiful math coming... please be patient] $\,2.5\,$ and [beautiful math coming... please be patient] $\,0.0003\,$ and [beautiful math coming... please be patient] $\,3.0\,$ are called decimals, but [beautiful math coming... please be patient] $\,3\,$ (no decimal point) is not called a decimal.

In particular, notice that the word decimal has to do with the name being used for a number, not the number itself!
The numbers [beautiful math coming... please be patient] $\,3.0\,$ and [beautiful math coming... please be patient] $\,3\,$ live at the same place on a number line; they are equal; they are the same number.
However, [beautiful math coming... please be patient] $\,3.0\,$ is a decimal, whereas [beautiful math coming... please be patient] $\,3\,$ is not a decimal.

If there are no digits to the left of the decimal point, then it is good practice to put a zero in the ones place.
That is, write   [beautiful math coming... please be patient] $\;0.02\;$   (with zero in the ones place), not   [beautiful math coming... please be patient] $\;.02\;$   (with nothing in the ones place).

To read decimals aloud, start by using the prior rules for reading the part to the left of the decimal point.
Read the decimal point as and.
Only the right-most place value is used for reading the part to the right of the decimal point, as illustrated in the following examples:

Notice that the word and should ONLY be used for the decimal point.
Resist the temptation to insert the word and anywhere else!

Reading a decimal like [beautiful math coming... please be patient] $\,972.28936\,$ following the rules above gets a bit tedious:
nine hundred seventy-two and twenty-eight thousand nine hundred thirty-six hundred-thousandths. (Yuck!)
Thus, it is often read as nine hundred seventy-two point two, eight, nine, three, six.
That is, say point to represent the decimal point, and then just read each digit, separately, that follows the decimal point.

The number [beautiful math coming... please be patient] $\,0.237\,$ can be viewed as [beautiful math coming... please be patient] $$ 2\cdot\frac{1}{10} + 3\cdot\frac{1}{100} + 7\cdot\frac{1}{1000} $$ or can alternately be viewed as [beautiful math coming... please be patient] $$ 237\cdot\frac{1}{1000} = \frac{237}{1000}\;. $$

Recall that in a fraction [beautiful math coming... please be patient] $\displaystyle\,\frac{N}{D}\,$, the top ($\,N\,$) is called the numerator and the bottom ($\,D\,$) is called the denominator.

For example, in the fraction [beautiful math coming... please be patient] $\displaystyle\,\frac{23}{100}\,$, the numerator is [beautiful math coming... please be patient] $\,23\,$ and the denominator is [beautiful math coming... please be patient] $\,100\,$.

To go from a a decimal to a fraction, you use the right-most place value to determine the correct denominator;
the entire number (without the decimal point) becomes the numerator.
In particular, the number of zeros in the denominator is the same as the number of places to the right of the decimal point.

EXAMPLES:
[beautiful math coming... please be patient] $0.0013 = \frac{13}{10000}\;$;
four places to the right of the decimal point; four zeros in the denominator
[beautiful math coming... please be patient] $23.107 = \frac{23107}{1000}\;$;
three places to the right of the decimal point; three zeros in the denominator
[beautiful math coming... please be patient] $0.72 = \frac{72}{100}\;$;
two places to the right of the decimal point; two zeros in the denominator

If you're rusty on fractions, don't worry—they will be reviewed in future sections, starting here.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Multiplying and Dividing Decimals by Powers of Ten

 
 

In this web exercise, you will practice renaming decimals as fractions.
Don't simplify any fractions: leave $\,0.4\,$ as $\,\frac{4}{10}\,$, instead of reducing it to $\,\frac{2}{5}\,$.

Use a forward slash ( $\;/\;$ ) to indicate a fraction:   for example, $\,0.032\,$ gets renamed as $\,32/1000\,$.

Do NOT insert commas in any numbers:   type in $\;32/1000\;$, not $\;32/1,000\;$.

Use the last reported place value for your new name, even if it is zero:
for example, report $\;0.50\;$ as $\;50/100\;$, not $\;5/10\;$.

Change to a fraction:
    
(an even number, please)