The Beauty of Mathematics: A New Course

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The Beauty of MathematicsA New Course

• Joe Conrad
• Solano Community College
• November 21, 2008
• jconrad_at_solano.edu

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Introduction

• CA AA/AS degree requirements
• Intermediate Algebra?
• Math is beautiful!
• Sample of topics

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• Vertices 4
• Regions 2
• Segments 4
• Vertices 6
• Regions 4
• Segments 8

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• Vertices Regions Segments 2

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• From http//cache.eb.com/eb/image?id63271rendT
  ypeId4

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Eulers Polyhedral Formula

• If a solid is bounded by planes, then
• Vertices Faces Edges 2
• V F E 2
• or
• V E F 2.

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• Picture from http//www.ics.uci.edu/eppstein/jun
  kyard/euler/

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• Leonhard Euler (1707 1783)

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• A perfect number is a natural number which is
  equal to the sum of its proper divisors.
• An abundant number is a natural number which is
  greater than the sum of its proper divisors.
• A deficient number is a natural number which is
  less than the sum of its proper divisors.

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Theorems

• All prime numbers are deficient.
• 6 and 28 are perfect.
• Greeks 496 and 8128 are also perfect.

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• Euclid noticed

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Theorems

• Euclid If is prime, then
  
• is perfect.

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• Prime numbers of the form
• are called Mersenne primes.
• For n to work it must be prime.
• By Eulers time it was known that
• n 2, 3, 5, 7, 13, 17 and 19 worked.
• Euler showed that 29 didnt work, but 31 did work.

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• Euclid-Euler Theorem N is an even perfect
  number if and only if
• where is a prime number.

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What has happened since?

• There are only 46 Mersenne primes, hence perfect
  numbers, known. The largest uses an value of n
  43,112,609 which generates a perfect number with
  almost 26 million digits.
• Great Internet Mersenne Prime Search (GIMPS)
• Unknown Are there infinitely many perfect
  numbers?
• Unknown Are there any odd perfect numbers?

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What is ?

• Clearly it is not a natural number.
• Can it be a fraction?
• If so, then for some natural numbers m and
  n. In this case, . Can we find any
  natural numbers with this property?

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• These are the pairs (m, n) that come close
• (1, 1), (3, 2), (7, 5) and (17, 12)
• These pairs produce the following fractions
• Do you notice a pattern?

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• Will these ever equal ? In other words, if
  we square these fractions will we ever get 2?

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Other topics

• Pythagorean Theorem
• History of ?
• Regular polyhedra
• Quadratic formula and higher order equations
• The real numbers are uncountable.
• Logarithms
• Complex numbers