3. Prove that (V2+ v3) ¢ Q. You can use Prob. 2's result if needed. You may also use the fact, without proof,that the sum, difference, product, and quotient of two rational numbers also are rationals.

To prove 2+3∉ℚ That is 2+3 is irrational number Since the sum of two irrational numbers is…

Answered: 3. Prove that (v2+ v3) ¢ Q. You can use…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 91RE

See similar textbooks

Related questions

Question

3. Prove that (V2+ v3) ¢ Q. You can use Prob. 2's result if needed. You may also use the fact, without proof,
that the sum, difference, product, and quotient of two rational numbers also are rationals.

Expert Solution

Step 1

To prove $(\sqrt{2} + \sqrt{3}) \notin ℚ$

That is $(\sqrt{2} + \sqrt{3}) i s i r r a t i o n a l n u m b e r$

Since the sum of two irrational numbers is irrational so we shall prove that $\sqrt{2} a n d \sqrt{3} a r e i r r a t i o n a l n u m b e r s$ separately.

Let us assume that $\sqrt{2} i s r a t i o n a l n u m b e r$ with p and q as co prime integers and $q \neq 0$

we have,

$\sqrt{2} = \frac{p}{q} s q u a r i n g b o t h s i d e s 2 q^{2} = p^{2}$

$\Rightarrow p^{2} i s a n e v e n n u m b e r t h a t d i v i d e s q^{2} \Rightarrow p i s a n e v e n n u m b e r t h a t d i v i d e s q L e t p = 2 x 2 q^{2} = 4 x^{2} q^{2} = 2 x^{2} \Rightarrow q^{2} i s a n e v e n n u m b e r t h a t d i v i d e s x^{2} \Rightarrow q i s a n e v e n n u m b e r t h a t d i v i d e s x$