Question

Prove that (2+√5) x 2 is not a rational number

Answer 1

Solutions 

⇒A square root of a number is a value that can be multiplied by itself to give the original number. To solve the problem/equation (2+√5) × 2 we have to see if the number √5 is an irrational number. To prove that square root of 5 is irrational, we will use a proof by contradiction.  In mathematics a contradiction is a statement that goes against an assumption. 

⇒ To show that √5 is an irrational number, we will assume that it is rational.Then, we need to find a contradiction when we make this assumption. If we are going to assume that √5 is rational, then we need to understand what it means for a number to be rational. A rational number is any numbers that can be written as a fraction. In other words, you can rewrite the number so it will have a numerator and a denominator. 

 ⇒ Square root of 5 is rational, it can be written as the ratio of two numbers as shown below:

√5 =  $\frac{x}{y}$

⇒ Our next step is to square both sides of the equation 

√5 = $\frac{x^2}{y^2}$

⇒ Multiply both sides by y²

√5 = $\frac{x^2}{y^2}$ × y² 

We get 5 × y² = x²

⇒ Since 5 × y² is equal to x², 5 × y² and x² must have the same number of prime factors

We just showed 

x² has an even number of prime factors

y² has also an even number of prime factors

5 × y² will then have an odd number of prime factors.

⇒ The number 5 counts as 1 prime factor, so 1 + an even number of prime factors is an odd numberof prime factors

⇒5 × y² is the same number as x². However, 5 × y² gives an odd number of prime factor while x² gives an even number of prime factors

⇒ This is a contradiction since a number cannot have an odd number of prime factors and an even number of prime factors at the same time

⇒ The assumption that square root of 5 is rational is wrong. Therefore, square of 5 is irrational. 

⇒ Since 5 is  irrational the statement (2+√5) × 2 is irrational. 

Answer 2

Answer:

To Prove (2 + √5)2 is not a rational number, we just need to Prove that √5 is an irrational number.

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.