Calculating Decimal Equivalents of Fractions

by Bill Price

1. Division

When we convert a fraction into a decimal, the numbers that we obtain to the right of the decimal point represent successive attempts at dividing a unit into powers of ten equaling that section of the unit that the fraction represents.

A decimal is an attempt, using only successive slices of a given unit into tenths, hundredths, thousandths, and so on, to equal a non-decimal slice of the same unit.

When we say that 1/7 = .142857142857…, we are saying the following:

1/7 represents the first of seven equal slices of a unit. To represent this same slice in units of ten, we first mark off ten equal intervals on the unit, and see how many we can fit into the 1/7. By doing so we find that we can fit 1/10 into the 1/7, but not 2/10. Therefore we begin building our decimal value with the number 1.

Beginning from the 1/10 position, we now divide the following 1/10 into ten equal sections, each representing 1/100 of the original unit. We now determine how many of these 1/100’s we can fit into the remaining space before we exceed the 1/7 mark. In so doing we find that we can fit a maximum of 4. The 5th 1/100 exceeds the 1/7 mark. Therefore our next decimal digit is 4.

From this 4th 1/100 position there is still space that needs to be filled before we can reach the 1/7 mark. Since we are working with decimals, i.e., successive divisions of powers of 10, we now attempt to fill in this remaining space with as many 1/1000ths as we can without exceeding the mark. We find that we can fit in a maximum of 2. (Remember that the 1/1000 is calibrated as a one thousandth part of the original unit, not a one thousandth part of the remaining interval to the mark). Therefore our next decimal digit is 2.

Note that with 1/7 this is a never-ending job. We never quite reach the exact spot marked by 1/7 and must continue the process indefinitely with increasingly smaller decimal slices of the unit. For this example, it just so happens that the numbers of slices required follow the repeating pattern 142857.

Decimal digits in fractions of all prime numbers follow a distinct cyclical pattern unique to primes which is described in detail in an essay entitled Repeating Decimals in Fractions of Prime Numbers. For fractions of non-prime integers we find a variety of repeating digit patterns. Some of these fractions end in repeating zeroes, specifically fractions for integers whose prime factors consist entirely of 2 and/or 5 (the prime factors for 10, the base of our numbering system).

To obtain the decimal value of a fraction, the most straightforward method is the division process described above. However, it is interesting to note that this same decimal value can be derived by a completely different method.

2. Addition

Decimal values of all fractions in the form 1/n, where 10^(m-1) < n < 10^m, can also be obtained by adding an infinite series of decimal fractions with powers of 10^m-n in the numerator.

In the specific example of 1/7, the repeating decimal value .142857 can be obtained from the sum of the following infinite series of decimals:

1/10 + 3/100 + 9/1000 + 27/10000 + 81/100000 + 243/1000000 …

Note that the numerators are successive powers of 3 (which is 10-7), beginning from 3⁰. The denominators are successive powers of 10, beginning from 10¹.

.1
.03
.009
.0027
.00081
.000243
.0000729
.00002187
.000006561
.0000019683
.00000059049
.000000177147
.0000000531441
.00000001594323
.000000004782969
.0000000014348907

By adding together an infinite series of decimals according to the above pattern we obtain the same repeating decimal value .142857...

For fractions that have denominators greater than 10 (but less than 100), we see the same process, but with 100 instead of 10 in the above pattern.

The fraction 1/98 is a very graphic example. The decimal value of 1/98 is:

.010204081632653061224489795918367…

At the beginning of this sequence of digits we see successive powers of 2 cascading to the right, each shifted by two decimal places, (i.e., 02, 04, 08, 16, 32, etc.). This decimal value of 1/98 is obtained by added together the following infinite sequence of decimal fractions:

1/100 + 2/10000 + 4/1000000 + 8/100000000 + 16/10000000000 + 32/1000000000000 …

Note that the numerators are successive powers of 2 (i.e., 100 – 98), beginning from 2⁰. The denominators are successive powers of 100, beginning from 100¹.

If we examine 1/998 we see the same thing occurring, but for 1000 instead of 100:

1/998 = .0010020040080160320641282565130261…

This decimal value can be obtained by adding the following sequence of decimal fractions:

1/1000 + 2/1000000 + 4/1000000000 + 8/1000000000000 + 16/1000000000000000 + 32/1000000000000000000 …

We can generalize the above observations into the following equation:

If n and m are positive integers, and 10^(m-1) < n < 10^m , then

Special thanks to fellow amateur mathematician Andy Williams of Wales who in March 1999 first brought this method to my attention.

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