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The Rationality/Irrationality
of the Sum of the Square Roots
of Two Rational Numbers

Let P and Q be two rational numbers. Consider R=√P + √Q. R is irrational unless P and Q are both squares of rational numbers.

Assume the contrary; i.e., that R is rational. Then squaring both sides of the equation R=√P + √Q gives

R² = P + 2√(PQ) + Q
and hence
√(PQ) = (R²−P−Q)/2

The right-hand side (RHS) is necessarily a rational number. If (PQ) is not a square of a rational number then there is a contradiction.

However, even if (PQ) is the square of a rational number R may not be rational. Suppose Q is equal to P. Then R=2√P. If P is not the square of a rational number then R is necessarily irrational.

Suppose (PQ) is the square of a rational number but P is not the square of a rational number and P<Q. This means that in order for (PQ) to be the square of a rational number Q=PS², where S is a rational number. Thus R=√P(1+S). This means R is the product of an irrational number and a rational number and thus is irrational.

Thus for all cases except P and Q both being squares of rational numbers R=√P+√Q is irrational.


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