decimal expansion and square roots given a decimal expansion
$$ 0.345345345... $$
then we can recover the fraction from this decimal expansion , in this case $ \frac{345}{999} $
my question is if for a non-periodic decimal expansion
$$ 0.451235132535126535126164616146462462 $$ we can find two numbers a and b integers so
$$ a+ \sqrt{b} = 0.451235132535126535126164616146462462 $$ or if there is a theroem which make it an impssible task.
 A: If you suspect your number is of this form, you can look at its continued fraction.  Anything of the form $a + \sqrt{b}$ with $a$ and $b$ rational will have an eventually  periodic continued fraction, and you can recover $a$ and $b$ from that.  For example, your $0.451235132535126535126164616146462462$ has continued fraction
$$[0; 2, 4, 1, 1, 1, 2, 9, 6, 1, 262, 4, 2, 13, 2, 4, 1, 1, 37, 1, 1, 88, 1, 1, 1, 4, 1, 4, 2, 2, 1, 2, 6, 2, 1, 8, 1, 4, 1, 1 \ldots]$$
i.e.
$$ 0 + 1/(2 + 1/(4 + 1/(1 + 1/\ldots $$
and this gives no sign of becoming periodic. 
A: We can't do this for transcendental numbers. If we could do this for a transcendental number $\alpha$, then $\alpha$ would be a root of a polynomial of the form $$x^2-2ax+a^2-b,$$ which is impossible.
A: There's a lot of decimal places after $46462462\ldots$ in your question.  If you only want a finite number of decimal places to match, then yes, it is always possible to find integers $a,b$ such that $a + \sqrt{b}$ agrees with some fixed irrational number to any given precision (the higher the precision, the larger $a$ and $b$ might have to be).  The reason that's true is that the sequence $\{\sqrt{n}\}$ increases to infinity but slowly, in the sense that $\sqrt{n+1} - \sqrt{n} \to 0$.   Any sequence with these two properties will have its fractional parts dense in $[0,1)$.
On the other hand, if you want the two sides to be exactly equal, then this is usually not possible, because there are only countably many pairs of integers and uncountably many real numbers.  A very nice feature of quadratic irrationals is that they may be identified by a periodic continued fraction, much like a rational number can be identified by a periodic decimal expansion.
In this sense you can compute continued fraction coefficients to convince yourself that a given sequence of digits has the form $a + \sqrt{b}$, with the caveat that one can only compute finitely many digits, and any finite number of digits might appear to match $a + \sqrt{b}$ (see the first paragraph) only to disagree in the next digit.
