The proof is by the method of reductio ad absurdum.
We start by assuming that 5 + sqrt(3) is rational.
The set of rational numbers is close under addition (and subtraction)
So, subtracting 5 (or adding -5) gives another rational number.
That is 5 + sqrt(3) - 5 is rational
Therefore sqrt(3) must be rational.
That means that it can be expressed in the form p/q where p and q are co-prime integers.
Thus sqrt(3) = p/q.
This can be simplified to 3*q^2 = p^2
Now 3 divides the left hand side (LHS) so it must divide the right hand side (RHS).
That is, 3 must divide p^2 and since 3 is a prime, 3 must divide p.
That is p = 3*r for some integer r.
Then substituting for p gives,
3*q^2 = (3*r)^2 = 49*r^2
Dividing both sides by 3 gives q^2 = 3*r^2.
But now 3 divides the RHS so it must divide the LHS.
That is, 3 must divide q^2 and since 3 is a prime, 3 must divide q.
But then we have 3 dividing p as well as q which contradict the requirement that p and q are co-prime.
The contradiction implies that sqrt(3) cannot be rational.
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The square root of 3 is an irrational number
The square root of -3 is an imaginary number and the square root of 3 is an irrational number that can't be expressed as a fraction
The square root of 3 is an irrational number
Sometimes it is and sometimes it isn't. The square root of 2 and the square root of 3 are both irrational, as is their product, the square root of 6. The square root of 2 and the square root of 8 are both irrational, but their product, the square root of 16, is rational (in fact, it equals 4).