Proof that Root 3 is Irrational:

- Definition of irrational numbers: An irrational number is a number that cannot be expressed as a ratio of two integers (i.e., a fraction).
- Assume Root 3 is rational: Let's assume that Root 3 is a rational number. That is, it can be expressed as a fraction of two integers, p and q, such that Root 3 = p/q. We can also assume that p and q have no common factors (i.e., they are in their simplest form).
- Simplifying the equation: Squaring both sides of the equation, we get 3 = p^2/q^2. Multiplying both sides by q^2, we get 3q^2 = p^2.
- Implication of p^2: This implies that p^2 is a multiple of 3. Therefore, p must also be a multiple of 3 (since the square of any non-multiple of 3 is not a multiple of 3).
- Implication of p: If p is a multiple of 3, then we can express p as 3k, where k is an integer. Substituting this into the equation, we get 3q^2 = (3k)^2 = 9k^2. Dividing both sides by 3, we get q^2 = 3k^2.
- Implication of q^2: This implies that q^2 is also a multiple of 3. Therefore, q must also be a multiple of 3.
- Contradiction: However, this contradicts our assumption that p and q have no common factors. If both p and q are multiples of 3, then they have a common factor of 3.
- Conclusion: Therefore, our assumption that Root 3 is rational leads to a contradiction. Hence, Root 3 is irrational.