Lecture Notes: Understanding Integers, Rational Numbers, and the Irrationality of Square Root of 2

Introduction to Integers

• Definition: Numbers without fractional parts, e.g., 8.
• Contrast with Fractions: 0.8 has a fractional component, not an integer.

Rational Numbers

• Definition: Numbers expressed as a ratio of two integers.
• Example: 0.8 can be written as 8/10 (8 divided by 10).

Rational vs. Irrational Numbers

• Concept of Irrational Numbers: Numbers not expressible as a simple fraction.
• Example: Diagonal of a square with sides of 1 unit length.
  • Using the Pythagorean theorem: (c^2 = 1^2 + 1^2 = 2)
  • (c = \sqrt{2}), which is not rational.

Properties of Even and Odd Numbers

• Even Numbers: Divisible by 2, e.g., 2, 4, 6. Defined as (2 \times \text{integer}).
• Odd Numbers: Not divisible by 2, e.g., 1, 3, 5. Defined as (2 \times \text{integer} + 1).

Squaring Even and Odd Numbers

• Even Squared: Remains even, ((2c)^2 = 4c^2 = 2 \times 2c^2).
• Odd Squared: Remains odd, ((2c+1)^2 = 4c^2 + 4c + 1).

Squaring Rational Numbers

• Multiplication of Fractions: ((\frac{a}{b})^2 = \frac{a^2}{b^2}).
• Verification through multiplicative inverse: Ensures correctness through properties of multiplication and division.

Reducing Fractions to Lowest Terms

• Process: Finding shared factors and dividing.
• Example: 4/6 reduced to 2/3 (co-prime factors only).

Proof of the Irrationality of (\sqrt{2})

• Method: Proof by contradiction.
• Assumption: (\sqrt{2} = \frac{a}{b}), where (a) and (b) are integers.
• Steps:
  1. ((\frac{a}{b})^2 = 2) implies (a^2 = 2b^2).
  2. (a^2) is even, thus (a) is even (let (a = 2c)).
  3. Rewriting gives (4c^2 = 2b^2) implies (b^2) is even, so (b) is even.
  4. Conclusion: Both (a) and (b) are even, sharing a common factor of 2.
  5. Contradiction: (a) and (b) cannot be co-prime.

Conclusion

• Result: (\sqrt{2}) is irrational.
• Insight: Using pure logic and mathematics to discover properties of numbers.

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• Key Messages: Mathematics allows us to understand the universe through logic; importance of proofs and definitions in understanding number properties.