Question

Prove that √5 is an irrational number and hence show that 2 - √5 is also a irrational number.​

Answer 1

Answer:

Let's prove this by the method of contradiction-

Say, √5 is a rational number. ∴ It can be expressed in the form p/q where p,q are co-prime integers.

⇒√5=p/q

⇒5=p²/q² {Squaring both the sides}

⇒5q²=p² (1)

⇒p² is a multiple of 5. {Euclid's Division Lemma}

⇒p is also a multiple of 5. {Fundamental Theorm of arithmetic}

⇒p=5m

⇒p²=25m² (2)

From equations (1) and (2), we get,

5q²=25m²

⇒q²=5m²

⇒q² is a multiple of 5. {Euclid's Division Lemma}

⇒q is a multiple of 5.{Fundamental Theorm of Arithmetic}

Hence, p,q have a common factor 5. this contradicts that they are co-primes. Therefore, p/q is not a rational number. This proves that √5 is an irrational number.

For the second query, as we've proved √5 irrational. Therefore 2-√5 is also irrational because difference of a rational and an irrational number is always an irrational number.

Step-by-step explanation:

let x=2-root 5 is a rational number

therefore by the property of rational number our assumption is wrong hence2-root 5 is irrational number

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Answer 2

Answer:

Step-by-step explanation:

To Prove that √5 is irrational

Let us assume that √5 is rational.

Therefore √5 = p/q, Where P and Q are Co - Primes and Q ≠ 0

√5 = p/q

√5q = p

Squaring on Both sides, We get

5q² = p²

By Theorem : If P divides a², Then P also divides a, We get that 5 divides p² and also it divides p.

Take p Value as 5k..

5q² = (5k)²

5q² = 25k²

q² = 5k²

By Theorem : If P divides a², Then P also divides a, We get that 5 divides q² and also it divides q.

Since P and Q have 5 as a common Factor,  it contradicts our assumption that P and q are Co - Primes...

Therefore , √5 is irrational.

Now We Have to Prove That 2 - √5 is Irrational

2 - √5 = p/q, Where P and Q are Co - Primes and Q ≠ 0

2 - √5 = p/q

- √5 = p/q - 2

- √5 = p - 2q / q

Since √5 is Irrational and p - 2q / q is Rational , It Contradicts Our Assumption...

Therefore , 2 - √5  is Irrational

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