Question

prove that 3 + root 5 is irrational​

Answer 1

Step-by-step explanation:

first assume 3 + ✓5 is rational

therefore. 3+✓5 = p/q

✓5 = p/q - 3

now ✓5 is irrational and p/q-3 is rational

this contradicts our supposition

therefore 3 + ✓5 is irrational

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Answer 2

ANSWER:

• 3+√5 is an Irrational number.

GIVEN:

• Number = 3+√5

TO PROVE:

• 3+√5 is an Irrational number.

SOLUTION:

Let 3+√5 be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

$\implies \: 3 + \sqrt{5} = \dfrac{p}{q} \\ \\ \implies \: \sqrt{5} = \dfrac{p}{q} - 3 \\ \\ \implies \: \sqrt{5} = \dfrac{p - 3q}{q}$

Here:

• (p-3q)/q is rational but √5 is Irrational.
• Thus our contradiction is wrong.
• 3+√5 is an Irrational number.

NOTE:

• This method of proving an Irrational number is called contradiction method.
• In this method we first contradict a fact and then we prove that our supposition was wrong.