February 04, 2003
One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements, Number 12

#### Repeating Decimals

(Not a problem or a puzzle; just an amusement.)

 1/1 1.0 1/2 = 0.5 1/3 = 0. 1/4 = 0.25 1/5 = 0.2 1/6 = 0.1 1/7 = 0. 1/8 = 0.125 1/9 = 0. 1/10 = 0.1 1/11 = 0. 1/12 = 0.08 1/13 = 0. 1/14 = 0.0 1/15 = 0.0 1/16 = 0.0625 1/17 = 0.[0.0588235294117647] 1/18 = 0.0 1/19 = 0.  1/20 = 0.05

But after doing all this work I then googled...

Posted by DeLong at February 04, 2003 08:05 PM | Trackback

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It's probably more an English problem than a math problem, but on this page it states

"As a matter of fact, every fraction that has an exact decimal equivalent consists only of powers of 1/2, or the product of 1/5 times a power of 1/2 or some multiple of these fractions (such as 3 times 1/4 = 3/4 = 0.75); there is no other way for an exact decimal equivalent to exist."

The key line is "product of 1/5 times a power of 1/2" which should have instead stated "a power of 1/5 times a power of 1/2". Consider 1/25 = 0.04.

An important detail, considering that it's the main point of the web page.

Posted by: Ben Vollmayr-Lee on February 5, 2003 05:49 AM

You missed my favorite: 80/81

Posted by: Charles on February 5, 2003 06:23 AM

Ben Vollmayr-Lee points out that the prime factors of ten, namely 2 and 5, matter when you consider the "does it repeat or terminate?" question. In another base, you'd get a different set of non-repeaters.

In binary fractions, for instance, even tenths are repeating. 1/10decimal = 0.0binary (hoping I did it right.) This is a peril for the novice accounting programmer, who soon learns that pennies keep disappearing if (s)he represents amounts in units of dollars.

Posted by: John Aspinall on February 5, 2003 07:09 AM

It's fairly easy to generate any repeating decimal you want-- for example,

0.345345345345... = 345/999

Homework problem: find the general formula.

Posted by: Matt on February 5, 2003 07:43 AM

Matt, you're giving away the store. Call 0.345_345_ instead 115/333.

-Brian

Posted by: Brian on February 5, 2003 12:03 PM

Charles:

It's my favorite tooPat Metheny, Dewey Redman, Michael Brecker, Jack DeJohnette, and Charlie Haden!

Posted by: David on February 6, 2003 12:44 AM
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