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Fascinating Fractions

A fraction, such as 2/3, can also be thought of as a division calculation, namely 2 divided by 3. The answer to a division is usually expressed as a decimal fraction, so that:

2/3 = 0.66666666 ...

This is a simple pattern: the 6 keeps repeating forever. Other fractions, such as 617/500 are a little more interesting:

617/500 = 1.234

This time the decimal fraction stops and has an interesting series of digits in it: 1, 2, 3, 4. And some fractions repeat:

152/333 = 0.456 456 456 ...

All the Same

Looking at :

2/3 = 0.66666666 ...

can you tell me what fraction would be the same as 0.33333333 ... ? Yes, it would be half of 2/3 which would be one third:

1/3 = 0.33333333 ...

So what about 0.111111....? This would be one third of 0.333333... which is one third of one third, or one ninth:

1/9 = 0.11111111 ...

Can you now complete the blanks in this table:

Fraction=Decimal Fraction
1/9=0.11111111...
/ =0.22222222...
1/3=0.33333333...
/ =0.44444444...
/ =0.55555555...
2/3=0.66666666...
/ =0.77777777...
/ =0.88888888...
/ =0.99999999...

The last line is a bit of a trick, since 0.99999... carried on for ever is mathematically the same as 1.00000... or just 1.

The same ... with gaps

Let's try another fraction:

1/99 = 0.0101010101 ...

What fraction has a decimal expansion of 0.02020202020 ...? What about 0.080808080808...? And how about 0.10101010101 ...? (notice that there is a missing first "0" after the decimal point.) The answer to this one is ten times 1/99 or 10/99.

Other series

Here are some fractions with surprises in their decimal expansions. You can use a calculator, but it only gives a few places and the last place is often rounded, spoiling the pattern. SO here are some divisions to practice on if you want to see the full pattern!

Fractionits decimal expansion
1/490.02 04 08 16 32 ..
1/4990.002 004 008 016 032 ...
500/4991.002 004 008 016 032 ...
1/4999 ?
1/9801 ?
100/9801?
10100/9801 ?
1/998001 ?
1/999800010.00000001002 004 008 016 032 ...
101/970299 ?
1001/997002999 ?

If you continue the decimals on far enough, the patterns eventually get scrambled. For instance:

1/49 = 0.02 04 08 16 32 65 30 61 ..!!

What has happened is that the following sum has been computed:

 
 1/49=   
  0.     
    02      
      04         
        08
          16
            32
              64
               128
                 256
                   512
                     ...
------------------------
  0.0204081632653061...
 

Each number is double the previous one (they are the powers of 2), but only moved two places to the right. Once the numbers get into 3 figures (and more), then they start ot overlap and the sum "scrambles" the pattern. But, as we have seen, that pattern really is still there! Try expanding the other patterns above until their patterns get scrambled too.