How did they figure out that the $\pi$ will result in an non repeating infinite decimal value?

Question

Why can't π be expressed as a fraction?

Going by the above question, if it is not possible to accurately determine the circumference or diameter of a circle, how did they figure out that the ratio will result in an infinite non repeating value?

Answer 1

Because one can prove that $\pi$ is irrational.

Answer 2

There is a brilliant proof of Ivan Niven that $\pi$ is not rational. Short reproduction below:

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Suppose $\pi = \frac{a}{b}$. Define polynomials

\begin{align} f(x) &= \frac{x^n(a-bx)^n}{n!} \\ F(x) &= \sum_{k=0}{n} (-1)^k f^{(2k)}(x) \end{align}

where $f^{(0)} = f$ and $f^{(i)}$ is the $i$-th derivative of $f$. Moreover, $f^{(i)}(0)$ is an integer, and $f^{(i)}(\pi) = f^{(i)}(\frac{a}{b}-\pi)$ is too. Hence, $F(\pi)$ and $F(0)$ are integers, however

\begin{align} F(\pi) + F(0) &= (F'(x) \sin x - F(x)\cos x)\Big|_0^\pi \\ &= \int_0^\pi \frac{\mathrm{d}}{\mathrm{d}x}(F'(x) \sin x - F(x)\cos x ) \ \mathrm{d}x\\ &= \int_0^\pi F''(x) \sin x - F(x)\sin x \ \mathrm{d}x \\ &= \int_0^\pi f(x) \sin x\ \mathrm{d}x, \end{align}

but

$$0 < f(x)\sin x < \frac{\pi^na^n}{n!}$$

is positive, but can be arbitrarily small for sufficiently large $n$, which leads to contradiction.

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I hope this helps ;-)

Answer 3

Because a real number has repeating decimal value if and only if it is rational (see this and this). Since $\pi$ is irrational (see this and this) it follows that it has a non repeating infinite decimal value.

Answer 4

What was so special about C/D as compared to any other fraction?

That fraction is special because it has many practical uses. There are physical situations where measuring diameter is easier than measuring circumference. Then, if you know the value of $\pi$, you can calculate a circumference (or an area) without actually measuring it. So, people were interested in this magic ratio that allowed them to compute circumference from a known diameter. In early times, they didn't know that this magic number was irrational. Only later did people invent the concept of "irrational", and the proof that $\pi$ is irrational.

I suspect that early users of $\pi$ would not even have cared whether or not it's rational, because rational approximations were quite adequate for their purposes. They weren't proving theorems, they were measuring the sizes of things like wheels and barrels, etc.