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…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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,

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

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