**rational**number is, naturally, any number that can be represented as a

**ratio**. That is to say, it can be written as a fraction. Showing that a

**Whole number**or an

**Integer**is rational is trivial: just place the number in the numerator of a fraction with a denominator of 1.

But what about numbers which are written as decimals?

Every middle schooler remembers the joys of converting fractions to decimals (and to *percents*, but that's a post for a different day). But what about converting them back to fractions?

**Terminating** decimals are a cinch. Take a number like *.375*, which is properly pronounced *three hundred seventy five thousandths*. (It's usually pronounced "point three seven five" by lazy people, including some educators, including myself. Shame on me.) They can easily be turned into a fraction where the denominator is a power of 10.

There are three decimal places, so if you multiply the decimal by 10^{3}, you have the numerator 375 and 1000 as the denominator: 375 / 1000, which simplifies to 3/8, which is obviously a rational number.

*How can we convert repeating decimals into fractions?*

**Repeating** decimals are a little trickier. Sure, you may recognize **0.7777777...** and know that it's **7/9**, or even **0.36363636...** is **4/11**. But what a strange number like 0.378378378...?

There is a simple procedure for it, involving a little *Algebra*.

Let x = 0.378378378378... And then do the following calculations:

__x = 0.378378378378....__

999x = 378.00000000000....

x = 378 / 999

*which simplifies to*14 / 37

By multiplying and subtracting, the repeating pattern of the decimal can be removed, leaving a multiple of x that will always be one less than a power of 10. Dividing by the coefficient transforms the value into a fraction (which may or may not be reducible).

This can be done for any repeating decimal. So any repeating decimal can be written as a fraction, i.e., a ratio of two integers and is, therefore, a rational number.

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