To prove that $\sqrt{2}$ is irrational by contradiction, follow these steps:

Assumption for Contradiction

Assume that $\sqrt{2}$ is rational. By definition, a rational number can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0$, and the fraction is in simplest form (i.e., $\gcd(p, q) = 1$).

Thus: $\sqrt{2} = \frac{p}{q}.$

Step 1: Square Both Sides

Squaring both sides of the equation: $2 = \frac{p^2}{q^2}.$

Multiply through by $q^2$ to eliminate the fraction: $p^2 = 2q^2.$

Step 2: Implication About $p^2$

Since $p^2 = 2q^2$, this shows that $p^2$ is even (because it is two times an integer, $2q^2$).

Step 3: Implication About $p$

If $p^2$ is even, then $p$ must also be even (because the square of an odd number is odd). Therefore, we can write $p = 2k$, where $k$ is an integer.

Step 4: Substitute $p = 2k$ Into the Equation

Substitute $p = 2k$ into the equation $p^2 = 2q^2$: $(2k)^2 = 2q^2.$

Simplify: $4k^2 = 2q^2.$

Divide through by 2: $2k^2 = q^2.$

Step 5: Implication About $q^2$

This shows that $q^2$ is also even. If $q^2$ is even, then $q$ must also be even.

Step 6: Contradiction

If both $p$ and $q$ are even, then they have a common factor of 2. This contradicts the assumption that $\frac{p}{q}$ is in simplest form ($\gcd(p, q) = 1$).

Conclusion

The assumption that $\sqrt{2}$ is rational leads to a contradiction. Therefore, $\sqrt{2}$ is irrational.

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Would you like further clarification or examples of similar proofs?

Related Questions

1. Can this proof be generalized for $\sqrt{n}$, where $n$ is not a perfect square?
2. How can you use a similar proof to show that $\sqrt{3}$ is irrational?
3. What are the properties of rational and irrational numbers that make this proof valid?
4. Can you explain the importance of the assumption that $\gcd(p, q) = 1$?
5. How does this proof demonstrate the power of proof by contradiction?

Tip

When proving irrationality, always verify that the initial assumption of the number being rational is well-defined, ensuring no logical inconsistencies in the argument.