proof!! the sum of irrational an a rational number?

Question

prove that the sum of square rootz of 3 + 1 is an irrational number

Answer 1

suppose sqrt(3) + 1 were rational, then can be written as a/b. Then we have sqrt(3) = a/b - 1 = (a-b)/b, which is rational. But we know that sqrt(3) is irrational, so the initial assumption was not valid, and sqrt(3) + 1 is irrational.

Answer 2

I'll show that the sum of a rational and an irrational number is irrational.

Let `a' be irrational, and let `b' be rational.

If a + b were rational, then we would know that a + b and b were rational numbers. But the difference of two rational numbers is rational, so this would mean that (a + b) - b is rational; that is, a is rational.

However, we know a is *not* rational. So it must be that a + b is not rational either.

In particular, then, sqrt(3) + 1 is not rational (because it is the sum of an irrational number and a rational number).

Answer 3

supposing that the sum is rational irrationa+rational=?rational so rational -rational =? irrational which is wrong as the aggregate of rational numbers is closed with respect o the operation of sum(sustraction)

Answer 4

LEMMA : x be irrational, n be rational > 0 , then that x*n is irrational.

( suppose not thus x*n = a/b then x = a/(b*n) BOOM x is rational )

x + a ; with x is irrational a rational, then x+a is irrational ?
use lemma + same technique.