Mathematical Proof Lecture

Proof of Irrational Numbers

Assumptions and Logic

1. Initial Assumption:

   • Assume that 6 + √2 is rational.
   • If it is rational, it can be expressed in the form a/b, where a and b are integers.

2. Derivation and Rationality:

   • Rewrite 6 + √2 as a/b.
   • This implies √2 = (a - 6b)/b, which means √2 is rational.
   • Contradicts the fact that √2 is irrational.
   • Therefore, 6 + √2 must be irrational.

3. **Conclusion:

   • The contradiction arises from the wrong assumption that 6 + √2 is rational.
   • Hence, 6 + √2 is irrational.

Second Example: 5 + √3

1. Initial Assumption:

   • Assume that 5 + √3 is rational.
   • Express it as a/b where a and b are integers.

2. Derivation and Rationality:

   • Rewrite 5 + √3 = a/b.
   • This implies √3 = (a - 5b)/b, which contradicts √3 being irrational.

3. Conclusion:

   • Contradiction arises due to our wrong assumption that 5 + √3 is rational.
   • Therefore, 5 + √3 is irrational.

Third Example: 3 + 2√7

1. Initial Assumption:

   • Assume that 3 + 2√7 is rational and can be written as a/b.

2. Derivation and Rationality:

   • Rewrite 3 + 2√7 = a/b.
   • This implies 2√7 = a/b - 3 and therefore √7 = (a - 3b)/(2b), which is rational.
   • Contradicts the fact that √7 is irrational.

3. Conclusion:

   • Contradiction arises due to the assumption that 3 + 2√7 is rational.
   • Therefore, 3 + 2√7 is irrational.

Careful Proof Analysis

1. **Steps:

   • Always express numbers in the form a/b where a and b are coprime integers.
   • Derive an expression that would make the irrational number rational, leading to a contradiction.

2. Example: 5 - 7√2

   • Assume 5 - 7√2 is rational: express it as a/b.
   • This results in 7√2 = (a - 5b)/b.
   • Concludes that √2 would be rational, contradicting its known irrationality.
   • Thus, 5 - 7√2 is irrational.

Summary and Methodology

• These proofs generally follow the same format: assume the number in question is rational, derive a form that leads to a contradiction with a known irrational number, and conclude that the original assumption was wrong.

Key Takeaways

• Understand the proofs by carefully following each step and recognizing the contradiction.
• Always remember the properties of rational and irrational numbers.