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.