Elaborate on the proof by contradiction technique and apply it to demonstrate that the square root of 2 is an irrational number.
user
Sure, let me explain the proof by contradiction and apply it to demonstrate that the square root of 2 is an irrational number.

Proof By Contradiction:
This is a type of logical deduction that works through assuming the opposite of what is being proved. If the opposite leads to a contradiction, then the original assumption was false, and the original statement must be true.

For instance, consider proving that the square root of 2 is an irrational number. Here is how we might apply the proof by contradiction:

1. **Assumption**: Suppose that the square root of 2 is a rational number.

2. **Step 1**: If it were a rational number, it could be expressed as a fraction. That is, let x(a/b) be the positive square root of 2, where a and b have no common factor, other than 1.

3. **Step 2**: Multiplying the above extraction x by x, we have:

     x^2 = 2       (by virtue of the property of extracted square root)

     =>   a^2/b^2 = 2.

4. **Step 3**: Now, multiplying the above fraction as numerator over b over denominator 2, we get,

     a^2 = 2 * b^2

5. **Step 4**: Cross-Multiplying we get,

     a = ± b*√2

6. **Step 5**: If b is even, then b*√2 is always an even number. Consequently, if b is even, then a, the numerator, should be even too, contradicting the fact that we chose a and b to have no factors except 1.

7. **Step 6**: If b is an odd number, then b*√2 is also odd, but then a too should be odd, contradicting that a and b have no common factors.

8. **Step 7**: If b is 1, then a = ± sqrt(2). When we take the square, both of them become 2.

9. **Contradiction and Conclusion**: Therefore it is impossible for both a and b to exist, and therefore an's square root of 2 is incommensurable or rational.

10. **Conclusion**: Therefore, we can affirm the original claim, which is square root of 2 is an incommensurable equivalent or an irrational number.

Therefore, the square root of 2 is an irrational number.
assistant
Elaborate on the proof by contradiction technique and apply it to demonstrate that the square root of 2 is an irrational number.
user
Proof by contradiction, also known as reductio ad absurdum, is a technique used in mathematics to prove the truth of a proposition. The method involves assuming that the proposition is false and then deriving a logical contradiction from this assumption. When a contradiction is obtained, it demonstrates that the original assumption (that the proposition is false) must be incorrect, and thus the proposition itself must be true.

Now, let's use proof by contradiction to show that the square root of 2 is an irrational number. An irrational number is a real number that cannot be expressed as a ratio of integers, meaning it cannot be written as a simple fraction. Our goal is to prove that √2 is not a rational number.

Assume the opposite: √2 is rational.

By definition, if √2 is rational, it can be expressed as a ratio of integers. So, there exist integers a and b, such that √2 = a/b, where a and b share no common factors other than 1 (i.e., a and b are coprime).

We can rewrite the equation as:

√2 = a/b

Squaring both sides, we get:

2 = a²/b²

Now, multiply both sides by b² to obtain:

2b² = a²

This equation implies that a² is an even number because it is equal to 2 times something (b²). It's a well-known fact that only even numbers squared result in even numbers. Therefore, a must be an even number too.

If a is even, then a can be written as 2k for some integer k:

a = 2k

Substitute a in our original equation:

2b² = (2k)²

2b² = 4k²

Now, divide both sides by 2:

b² = 2k²

This equation shows that b² is also an even number because it is equal to 2 times something (k²). Just as before, this implies that b must be an even number too.

Now we have shown that both a and b are even, which contradicts our initial assumption that a and b were coprime (sharing no common factors other than 1). Therefore, our initial assumption that √2 is a rational number must be false.

In conclusion, √2 is an irrational number. This proof demonstrates the power of the proof by contradiction technique, which can effectively prove that certain statements are true by showing that the opposite leads to an inconsistency or contradiction.
assistant