Sanghoon Lee & Theo Smith Honors 391A: Mathematical Gems Prof. Jenia Tevelev March 11, 2015

HOW DOES INDUCTION WORK? 1.) Base Case: Show the First Step Exists 2.) Induction Hypothesis: Assume a Step Exists 3.) Inductive Step: Show that we can take the next step Strong v. Simple Distinction in the Hypothesis: Weak: Assume the step you are on exists Strong: Assume all the steps below you and the one you are on exists

PROOF BY INDUCTION DOES NOT USE INDUCTIVE REASONING Induction: Reasoning from a part to a whole. Using premises I conclude that an outcome is likely

A HISTORY OF INDUCTION Proof By Induction has existed for a long time Euclid used induction to prove the infinitude of primes (Euclid’s Elements, Book IX, Proposition 20) Francesco Maurolico ( ) The first to use a formal proof by induction A Sicilian astronomer and scientist Blaise Pascal ( ) French Mathematician, physicist, and philosopher Used a more explicit statement of proof by induction to prove theorems about his triangle Was aware of and referenced Maurolico Pierre de Fermat ( )

PIERRE DE FERMAT Grew up of Basque origin in France and was homeschooled Studied law at the University of Orléans French lawyer and mathematician Named to Criminal Court in 1638 Researched mathematical concepts as hobby Secretive; he was reluctant to publish his works He had other mathematicians challenge his work (Leibniz, Euler) His failure to interest other mathematicians in Number Theory credits his secrecy Known for his research in Number Theory Particularly interested in divisibility and primality Formulated the Principle of Infinite Descent, a form of Inductive Proof Used Principle of Infinite Descent extensively in his research

PRINCIPLE OF INFINITE DESCENT “Let P be a property that natural numbers may or may not possess. If an assumption that a natural n 0 has property P leads to the existence of a smaller natural n 1 < n 0 that also satisfies P, then no natural number has that property.”  A recursive method to prove by contradiction  Find the minimal counterexample that will debunk the assumption  Several conditions must be satisfied before this principle can be used:  The natural numbers must be well ordered  There are only a finite number of natural numbers that are smaller than any given one

APPLICATIONS FOR P.I.D.

LIMITATIONS OF INDUCTIVE PROOFS The Horse Problem-Using Induction to Prove All Horses are the same color 1.) Base Case: A Group of one horses is the same color 2.) Induction Hypothesis: Assume any group of n horses is the same color 3.) Inductive Step: Any group of n+1 horses is the same color

WORKS CITED "thought." Encyclopaedia Britannica. Encyclopaedia Britannica Online Academic Edition. Encyclopædia Britannica Inc., Web. 08 Mar Bussey, W. H. "The Origin of Mathematical Induction." The American Mathematical Monthly 24.5 (1917): Print ction.pdf “A Mathematical Foundation of Computer Science” David Mix Barrington (CS250)

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