How to turn irrational decimals into simple fractions?

By definition an irrational number cannot be represented as a fraction.

However, 10/7 is an approximation for √2 because (10/7) ²=100/49≈2. In fact (100-2) / 49=2.

So 100 (1-1/50) / 49=2 and (10/7) √ (1-1/50). This expands to an infinite series (10/7) (1-1/100 + ...).

If we just take the first two terms we get: (10/7) (99/100) = 990/700=99/70=1.41428 ...

(99/70) ²=9801/4900≈2. This is closer to 2. In fact (9801-1) / 4900=2.

From this we have another series: (99/70) √ (1-1/9801) = (99/70) (1-1/19602) approximately.

So we get (99/70) (19601/19602) = 1940499/1372140=1.414213 ...

So we get better fractional equivalents but we never get exactly the right one.