Irrational Number - Decimal Expansions

Decimal Expansions

The decimal expansion of an irrational number never repeats or terminates, unlike a rational number. Similarly for binary, octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases.

To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m, only m remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats.

Conversely, suppose we are faced with a repeating decimal, we can prove that it is a fraction of two integers. For example, consider:

Here the repitend is 162 and the length of the repitend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repitend. In this example we would multiply by 10 to obtain:

Now we multiply this equation by 10r where r is the length of the repitend. This has the effect of moving the decimal point to be in front of the "next" repitend. In our example, multiply by 103:

The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000A matches the tail end of 10A exactly. Here, both 10,000A and 10A have .162162162 ... at the end.

Therefore, when we subtract the 10A equation from the 10,000A equation, the tail end of 10A cancels out the tail end of 10,000A leaving us with:

Then

(135 is the greatest common divisor of 7155 and 9990). 53/74 is a quotient of integers and therefore a rational number.