TITLE: why is PI considered irrational if it can be expressed as ratio of circumference to diameter?
QUESTION [14 upvotes]: Pi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where both a,b are integers & hence Pi is irrational. 
However as far as i know a ratio of two decimals can always be expressed as a ratio of two integers by adding 0s after the deciaml. eg- 56.89 / 23 can always be written as 5689/2300 . So we should also always be able to express C/D as integers . Why then is Pi irrational ?
Can't we always measure the diameter and circumference accurately and express them as integers in the ratio ?

REPLY [3 votes]: The question seems to come down to this: "Can't we always measure the diameter and circumference accurately …?"
Suppose you are given a circle whose circumference is either
$C = 10.123456789 12345678 1234567 12345678 123456789 23456789 1$ or
$C = 10.123456789 12345678 1234567 12345678 123456789 23456789 2$.
How would you go about measuring the circumference to determine which of the two
equations is true?
In practice you cannot even measure an object in real life to such accuracy
that you can give its size an exact finite decimal expansion with the
certainty that it cannot be any other finite decimal.
The alternative is to use mathematics to determine what $C$ should be, ideally,
for some given $D$. If you start with a finite decimal $D$ and
do the mathematics correctly, however, you will never reach the last digit of $C$,
so you'll never be able to use the "add zeros after the decimal" trick.