Question

Prove that 3 is an irrational number.​

Answer 1

Answer:

Step-by-step explanation:

3 is a rational number because it can be written in the form of p/q where q is not equal to zero.

If you meant √3 then this is the answer,

Lets assume that √3 is a rational number

So we can write it in the form of p/q where q≠0 and p and q are co-prime.

So lets say that √3 = p/q

Square on both sides,

3 = p²/q²

So 3q² = p² (i)

So now we know that p² is divisible by 3.

If the square of a positive integer is divisible by another positive integer,

then its root would also be divisible by that positive integer.

So p is divisible by 3. Let the quotient be x. Then 3x = p

Now square on both sides we get, 9x² = p²

Now from (i) we know that p² = 3q²

So 9x² = 3q²

So now we get,

3x² = q²

Now since q² is divisible by 3, q would also be divisible by 3.

So p and q are both divisible by 3 which means that they p/q is not rational.

They are irrational

hence, proved