Question

prove the following are irrational numbers
1) 13+25√2
2) 3-√5
3) 1/2+√3
4)√6+√2​

Answer 1

Answer:

1. 13+25√2

Hey !

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prove 13 + 25√2 is irrational.

Let us consider in contradiction that 13 + 25√2 is rational.

let us assume 13+25√2 = a / b

25√2 = a/ b - 13

√2 = a/b -13/25

√2 = 25a - 13b / 25 b

√2 = 25 ( integer ) - 13 ( integer ) / 25 ( integer )

{ integer / integer = p/q }

we know that a number in the form of p/q is rational.

So , 13+25√2 is rational.

But we know that √2 is irrational .

Hence our assumption is wrong.

So , 13+25√2 is irrational

Hence , proved .

2. 3-√5

Let

3 - √5 be a rational number

.°. 3 + √5 = p/q [ where p and q are integer , q ≠ 0 and q and p are co - prime number ]

=> √5 = p/q - 3

=> √5 = p - 3q/q

we know that p/q is a rational number.

.°. √5 is also a rational number.

This contradicts our assumption.

.°. 3 - √5 is an irrational number.

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3. 1/2+√3

Let 1/2+√3 be a rational number.

A rational number can be written in the form of p/q where p,q are integers.

1/2+√3=p/q

√3=p/q-1/2

√3=(2p-q)/2q

p,q are integers then (2p-q)/2q is a rational number.

Then √3 is also a rational number.

But this contradicts the fact that √3 is an irrational number.

So,our supposition is false.

Therefore,1/2+√3 is an irrational number

4. √6+√2

let √6+√2 be rational

this implies (√6+√2)^2 is rational.

this implies 6+2+2√12= 8+2√12 is rational.

but √12 is irrational.

this implies 2√12 is irrational. so, 8+2√12 is irrational.

so our assumption was wrong.

hence √6+√2 is irrational.

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