why is PI considered irrational if it can be expressed as ratio of circumference to diameter? 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 ?
 A: There are two possible answers, depending on which of two possible points is confusing you:


*

*It isn't true that a ratio of any two "decimals" can be expressed as a ratio of two integers, if by "decimal" you really do mean any decimal expansion at all. The problem is basically that some decimal expansions are infinite, so your idea of multiplying by a power of ten to clear the decimal point doesn't always work. It only works for finite decimal expressions.

*If you already know that your idea only works for finite decimal expansions, maybe you just don't realize that not only can you never find a circle such that both $C$ and $D$ are integers, you actually can't even find a circle such that both $C$ and $D$ have finite decimal expansions.
A: A real number is rational, if it is the ratio of two integers. Otherwise it is called irrational.
If $D$ is the diameter of circle, then $C= \pi D$ is it's circumference. If $\pi$ is irrational and $D$ rational, then $C$ is irrational. If it were rational, then $C/D = \pi$ would be rational. 
"Turning the argument around": We know, that the circumference of a circle with rational diameter is always irrational. Therefore $C/D = \pi$ is irrational. If it were rational, then $C = \pi D$ would be too.
So regarding your last question:
No! You cannot measure both diameter AND circumference of a circle with arbitrary accuracy. At least one of them is irrational and therefore does not have a representation as the ratio of two integers.
A: 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.
A: The problem is if you have a number of form $C/D$ where any of $C$ or $D$ have infinite decimal expansion you cannot do the trick with adding zero - actually you have too many digits, and you can only approximate $C/D$ by replacing $C$ and $D$ by some values that are close to $$C and $D$ respectively but have finite decimal expansion.
A: It should come as no surprise that our /accurate/ approximate measurements of circumference and diameter can give us no more than an /accurate/ approximation of pi.
