Question

prove that
$2 + \sqrt{3}$
is an irrational number
​

Answer 1

Answer:

Step-by-step explanation:

Euclid's proof starts with the assumption that √2 is equal to a rational number p/q.

√2=p/q

Squaring both sides,

2=p²/q²

The equation can be rewritten as

2q²=p²

From this equation, we know p² must be even (since it is 2 multiplied by some number). Since p² is an even number, it can be inferred that p is also an even number.

Since p is even, it can be written as 2m where m is some other whole number. This is because the definition of an even number is it can be written as 2 multiplied by a whole number. Substituting p=2m in the above equation:

2q²=p²=(2m)²=4m²

or

2q²=4m²

Dividing both sides of the equation by 2:

q²=2m²

By the same reasoning as before, q² is an even number (since it is written as 2 multiplied by some number). So q is an even number. Let q=2n where n is some whole number. We had assumed √2 to be equal to p/q. So:

√2=p/q=2m/2n

By canceling 2 in the numerator and the denominator of the Right hand side,

√2=m/n

Answer 2

Answer given above.

Kindly check.

Hope it helps ☺