Why cant irrational be represented in decimal form?

Answer 1

A decimal number is either

• terminating,
• repeating, or
• non-terminating and non-repeating.

The last of these can never be written out since it has infinitely many digits with no way of abbreviating them.

A terminating decimal number is one with at most n digits after the decimal point (where n is finite). Such a number is equivalent to a rational number whose denominator is 10^n. The number is therefore rational.

For example, 0.23568 = 23568/100000. It can be simplified but that is not relevant here.

A number with a decimal representation with a recurring string of length m (where m is finite) is a rational number whose denominator is a 10^k multiple of 10^m-1 or 99...9 (m times). Here k is related to when the repeating sequence starts.

For example, consider 1.23142857142857... (with the 6 digit sequence 142857 repeating)

Let f = 1.23142857142857...

then 100f = 123.142857142857...

and 100f*1000000 = 123142857.142857142857...

So that 999999*100f = 123142857-123 which is an integer.

f is, therefore a ratio of two integers and so is a rational number.

So that only leaves the Irrational Numbers for infinitely long decimal representations with no recurrence.

Answer 2

They don't stop.