Irrational quantity

If you're going to insist that all numbers only exist as in counting or measurement, then all numbers are inherently irrational. You would say that 5 is not simply 5, but an approximation of 5: 5.000... Physical representations of numbers are of course imperfect and subject to measurement and counting, but that doesn't mean that numbers don't exist outside of that. 5 and pi are both values/ratios (the ratio of 5/1 is no different from the ratio of circumfrance to diameter - you don't have to actually measure that lengths of the circumfrance and diameter - they can be determined through calculus) that exist irrespective of being measured. We apply math to reality, we don't learn math from reality. The equation for the area of a circle did not come from someone looking at a physical circle that he had measured and saying, "I believe that the equation A=pi*r^2 happens to work quite well." It was determined using logic. Although the mathematical definition of a cirlce might have been conceived of in observation, the definition does not lie in its appearence but in its mathematical definition (graphed by: ((X-h)^2 + (Y-k)^2 = r^2)). Ideas can exist in of themselves, which, as I've stated, has been proven in Idealism.

You are quite right to say that any number is irrational, if we are not told when to stop dividing, calculating, etc. Mathematics needs a stop sign, otherwise 20 divided by 5 could be 4.000..etc. We have informal methods for completeing a calculation, such as 'I have what I need, now I can stop', but that is not really integrated in the uniform body of maths, if there is such a uniform body.

If calculus can carry on without the concepts of circle and line, and yet provide answers to fit them, then all well and good. But does not the 'pi' of calculus have something of the circle about it? - not the circle of perception perhaps, but the circle of topology?