Mathematical Induction By: Emily Elrod and Lora Morris

History 1623 – 1662 2nd to use “induction” More modern proof Francesco Maurolico 1494 – 1575 First to use “induction” Sketchy Proof Published Works Blaise Pascal 1623 – 1662 2nd to use “induction” More modern proof

Maurolico’s Proof1 By Proposition 13 [this proposition asserts that n2 + (2n+1) = (n + 1)2] the first square number 1 added to the following odd number 3 makes the following square number 4; and this second square number 4 added to the third odd number 5 makes the third square number 9; and likewise the third square number 9 added to the fourth odd number 7 makes the fourth square number 16; and so successively to infinity the result is demonstrated by repeated application of Proposition 13. 1. Burton, 464

Pascal’s Proof2 In modern notation, Pascal proved Although this proposition has an infinite number of cases, I will give a rather short demonstration by supposing two lemmas. The first one, which is self-evident, is that this proposition occurs in the second base; because it is apparent that  is to  as 1 is to 1 [that is, for n=1, The second one is that if this proposition is true for an arbitrary base, it will necessarily be true in the next base. From which it is clear that it will necessarily be true in all bases, because it is true in the second base by the first lemma; hence by means of the second lemma, it is true in the third base hence in the fourth base, and so on to infinity. It is therefore necessary only to prove the second lemma. 2. Burton, 465-466

Was it always called ‘mathematical induction’? John Wallis 1616 – 1703 placed a name on this type of logical thinking: “reasoning by recurrence” Augustus De Morgan 1806 – 1871 First to use the terminology ‘mathematical induction’ (1838)

Flaw For every Proof: Induction. Assume for some k that Then Hence the formula holds for k + 1. Conclusion. Therefore, by the principle of mathematical induction, the formula holds for all

What Does an Induction Proof have to have? 1. Proposition 2. Basis for the proof, where we determine if P(1) is true. 3. Induction, where we suppose the formula holds for k and consider k+1. 4. Conclusion

The type of proofs we encountered Summation Ex. Find and prove the formula for Inequality Prove the inequality for appropriate choice of n. Divisibility Prove that , for all n ≥ 1, is divisible by 6.

Exercises for the class. Work together to prove the following by Induction. Write what you have on the board for the last two minutes. You have a total of 10 minutes to work. #1 FRONT ROW Prove the inequality for appropriate choice of n. # 2 BACK ROW Prove that, for all n ≥ 1, is divisible by 6.

Exercise #1

Exercise #2

Challenging Exercise Prove that, for all , is divisible by 5.

Challenging Exercise

Any Questions?

References Burton, D.M. (2007). The history of mathematics: an introduction. New York, NY: McGraw Hill.