Introduction
To prove that the square root of 2 (√2) is an irrational number, we will use a method called proof by contradiction.
Assumption
- Assume that √2 is a rational number.
- This means that it can be expressed as a fraction a/b, where:
- a and b are integers
- b is not zero
- a and b have no common factors (they are in simplest form).
Squaring Both Sides
- From our assumption, we have:
- √2 = a/b
- Squaring both sides gives:
- 2 = a²/b²
- Rearranging leads to:
- a² = 2b²
Analyzing the Equation
- This implies that a² is an even number (since it equals 2 times another integer).
- If a² is even, then a must also be even (as the square of an odd number is odd).
Letting a = 2k
- Let a = 2k for some integer k.
- Substituting back gives:
- (2k)² = 2b²
- This simplifies to:
- 4k² = 2b²
- Dividing both sides by 2 leads to:
- 2k² = b²
Conclusion
- This shows that b² is also even, meaning b is even as well.
- Since both a and b are even, they share a common factor of 2, contradicting our initial assumption that they are in simplest form.
- Therefore, √2 cannot be expressed as a fraction, proving that it is an irrational number.