Question

Show that 2-√3 is an irrational number given that √3 is an irrantional​

Answer 1

Step-by-step explanation:

Proof by contradiction: suppose for the sake of contradiction that 2−3–√ is rational. Let x=2−3–√ , thus is follows that x=ab , without loss of generality a and b have no common factors.

First consider 1x . Clearly 1x=2+3–√ .

We are assuming that x is rational, and thus 1x is rational too; it is equal to ba .

Now we subtract 1x from x . The difference of two rational numbers is rational. In this case, the difference is equal to a2−b2ab .

By computing this difference, we find that 2−3–√−(2+3–√)=−23–√ .

Lemma 1: 3–√ is irrational. Proof: suppose for the sake of contradiction that 3–√ is rational, then 3–√=ab , where a and b have no common factors. Then 3b2=a2 , so a is divisible by 3, so a=3n , so b2=3n2 , so b is divisible by 3, contradicting that a and b have no common factors.

By Lemma 1, we know that −23–√ is irrational, thus reaching a contradiction.

Since the assumption that 2−3–√ is rational leads to a contradiction, we know that it is irrational.

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