EXACT measurements of a circle. Possible?

Question

I understand simple circle equations like c=pi×d and a=pi×(rr) (sorry keyboard doesn't have pi or exponents) if pi is irrational with an infinitely long decimal, doesn't that make it impossible to exactly measure either c or d or a or both? I understand for practical terms we use 3.14 as an approximation. Does this mean all measurements derived from pi are approximations. I can imagine a diameter being exactly say 1 meter. Does that mean that the circumference will have intimate decimals too, ie impossible yo measure EXACTLY. My idea is that either d or c can be exactly measured but not both. Please feel free to confirm my ideas or drop knowledge on me. I am a layman so please nothing too crazy

Answer 1

I think you are confusing "can be measured" with "is a rational number". No length can be measured exactly as measuring is a physical process with real world limitations. All measurements, derived from $\pi$ or not, are approximations. You are correct that as $\pi$ is irrational, at least one of the radius and circumference of a circle must be irrational. In mathematics, if we say the diameter of a circle is $1$ unit, the circumference is exactly $2\pi$ units. The fact that this number is irrational does not make it approximate.

Answer 2

Mathematical arguments are about concepts in our heads, not about measurements made with yardsticks or tape measures. If we talk about the number $\pi$, the diameter $d$, or the circumference $c$ of a circle we consider $\pi$, $d$, or $c$ as real numbers, i.e., as elements of the system ${\mathbb R}$. As such they are defined, given, or to be computed with "infinite precision". The elements of ${\mathbb R}$ can be integer, rational, or irrational. An integer number can be encoded as $\pm$ a finite decimal number without loss of information, and a rational number can be encoded as a fraction ${p\over q}$ with $p\in{\mathbb Z}$, $q\in{\mathbb N}_{\geq1}$. It is true that $\pi$ or $\sqrt{2}$ cannot be represented by such means in finite terms, but they are well defined as elements of ${\mathbb R}$ with "infinite precision" via certain formulas, e.g., $$\pi:=4\int_0^1\sqrt{1-x^2}\>dx\>,\qquad \sqrt{2}:={\rm the}\bigl\{x\in{\mathbb R}\bigm| x>0\>\wedge\>x^2=2\bigr\}\ .$$

For daily life these things are not much of a problem since we can always work with finite decimal (in particular: rational) approximations to any real number we can think of, and arrive at good numerical results. This comes from the fact that the basic operations in ${\mathbb R}$ ($+$, $-$, $\cdot$, and $:$) are continuous.