Use a proof by contradiction to show that the square root of 3 is national you may use the following fact: for any integer kirke is a multiple of thek is a multiple of 3. hint: the proof is very similar to the proof that is inational 5. use a direct proof to show that the product of a rational number and an integer must be a rational number 6 use a proof by contradiction to show that the sum of an integer and animational number must be irrational

1. Let us proof that √3 is an irrational number, using reductio ad absurdum. Assume that where   and are non negative integers, and the fraction is irreducible, i.e., the numbers and have no common factors.

Now, squaring the equality at the beginning we get that

(1)

which is equivalent to . From this we can deduce that 3 divides the number , and necessarily 3 must divide . Thus, , where is a non negative integer.

Substituting into (1), we get

which is equivalent to

.

Thus, 3 divides and necessarily 3 must divide . Hence, where is a non negative integer.

Notice that

.

The above equality means that the fraction is reducible, what contradicts our initial assumption. So, is irrational.

2. Let us prove now that the multiplication of an integer and a rational number is a rational number. So, , which is equivalent to say that where   and are non negative integers. Also, assume that . So, we want to prove that . Recall that an integer can be written as

.

Then,

.

Notice that the product is an integer. Thus, the fraction is a rational number. Therefore, .

3. Let us prove by reductio ad absurdum that the sum of a rational number and an irrational number is an irrational number. So, we have is irrational and .

Write and let us suppose that is a rational number. So, we get that

.

But the subtraction or addition of two rational numbers is rational too. Then, the number must be rational too, which is a clear contradiction with our hypothesis. Therefore, is irrational.