Let us assume that 2√3 is rational
then 2√3= p/q (where p and q are co prime )
√3= 2p/q
As pand q are co prime 2p/q is a rational number
therefore√3 is rational but this contradicts the fact that√3 is irrational
this contradiction has arised due to our wrong assumption
hence , 2√3 is a irrational number

Proving 2√3 is an Irrational Number
1. Assume 2√3 is Rational
- Let's assume that 2√3 is a rational number, which means it can be expressed as a ratio of two integers, a/b where a and b are integers and b is not equal to 0.
2. Express 2√3 as a Rational Number
- If 2√3 is rational, then it can be written as a/b = 2√3. Squaring both sides, we get (a/b)^2 = 12.
- This simplifies to a^2 = 12b^2.
3. Derive a Contradiction
- This implies that a^2 is a multiple of 12. Therefore, a must be a multiple of 12.
- Let's say a = 12k, where k is an integer.
4. Substitute a = 12k into the Equation
- Substituting a = 12k back into the equation a^2 = 12b^2, we get (12k)^2 = 12b^2.
- Simplifying, we get 144k^2 = 12b^2, which further simplifies to 12k^2 = b^2.
5. Derive a Contradiction Again
- Similarly, this implies that b^2 is a multiple of 12. Therefore, b must also be a multiple of 12.
6. Contradiction
- However, if both a and b are multiples of 12, they have a common factor of 12, contradicting our assumption that a and b are coprime in a/b.
7. Conclusion
- Since assuming 2√3 is rational leads to a contradiction, we can conclude that 2√3 is an irrational number.