Prove that 2/ square root 12is irrational
To prove that 2/√12 is irrational, we need to assume that it is rational and derive a contradiction.
Let's assume that 2/√12 is rational. By definition, a rational number can be expressed as a fraction p/q, where p and q are integers and q is not equal to 0.
We can write 2/√12 as p/q, where p and q have no common factors:
2/√12 = p/q
Now, let's square both sides of the equation to eliminate the square root:
(2/√12)^2 = (p/q)^2
4/12 = (p/q)^2
Simplifying the left side gives us:
1/3 = (p/q)^2
Now, let's consider the right side of the equation. If (p/q)^2 is rational, it means that p/q must also be rational, because the square of a rational number is rational.
Let's write p/q as a simplified fraction, where p and q have no common factors:
p/q = a/b
Substituting a/b into the equation, we have:
1/3 = (a/b)^2
1/3 = a^2/b^2
b^2 = 3a^2
Since b^2 is divisible by 3, b must also be divisible by 3. Let b = 3k, where k is an integer.
Substituting this back into the equation gives us:
(3k)^2 = 3a^2
9k^2 = 3a^2
3k^2 = a^2
So a^2 is divisible by 3, which means that a must also be divisible by 3. Let a = 3m, where m is an integer.
Substituting this back into the equation gives us:
3k^2 = (3m)^2
3k^2 = 9m^2
k^2 = 3m^2
Now, we have shown that if p/q is rational, then both p and q are divisible by 3. However, this contradicts our initial assumption that p and q have no common factors, because now we know they are both divisible by 3.
Therefore, our assumption that 2/√12 is rational must be incorrect. Hence, 2/√12 is irrational.
This proof demonstrates the contradiction that arises from assuming the rationality of 2/√12 and establishes its irrationality.