14. Lecture -14
The chapter explores various mathematical concepts through proof techniques such as induction and strong induction. It demonstrates the relationships between arithmetic and geometric means, binary representations of integers, the definition of a celebrity in a party context, and irrationality proofs for numbers like √2. Additionally, it provides methods for counting diagonals in polygons and emphasizes the importance of clear definitions and logical reasoning in mathematics.
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Sections
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What we have learnt
- Arithmetic mean is greater than or equal to geometric mean for any collection of positive real numbers.
- Every positive integer can be expressed as a sum of distinct powers of two, demonstrating its unique binary representation.
- A celebrity in a social context is defined as someone known by everyone else but does not know anyone back, and there can be at most one celebrity.
- The square root of 2 is irrational, which can be proven using induction.
- The total number of diagonals in a polygon can be derived and expressed with a formula.
Key Concepts
- -- Induction
- A proof technique that establishes the truth of a statement for all natural numbers by showing it holds for a base case and that if it holds for an arbitrary case, it holds for the next case.
- -- Celebrity
- A guest in a party who is known by all guests and knows no one in return, which has implications on social network dynamics.
- -- Rational Number
- Any number that can be expressed as the quotient of two integers, where the denominator is not zero.
- -- Diagonals of a Polygon
- Line segments that connect non-adjacent vertices of a polygon, with a formula for counting them derived through induction.
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