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Mathematical Induction U.U.D.M. Project Report 2020:21 Mathematical Induction Hanna Wedin Examensarbete i matematik, 15 hp Handledare: Inger Sigstam Examinator: Veronica Crispin Quinonez Juni 2020 Department of Mathematics Uppsala University Abstract Mathematical Induction is used in all fields of mathematics. In this thesis we will do an overview of mathematical induction and see how we can use it to prove statements about natural numbers. We will take a look at how it has been used in history and where the name mathematical induction came from. We will also look at different types of induction, weak and strong induction. You can also do induction on other types of structures, like the length of propositions. 1 Contents 1 Introduction 3 1.1 History . 4 1.2 Paradoxes with induction . 5 2 Mathematical Induction 5 2.1 Axiom . 6 2.2 Examples . 6 2.2.1 The sum of i ......................... 6 2.2.2 The sum of i2 ......................... 7 3 Strong Induction 8 3.1 Examples . 9 3.1.1 Example . 9 3.1.2 The Fibonacci sequence . 10 3.1.3 Fundamental Theorem of Arithmetic . 10 3.1.4 The sum jk .......................... 10 4 Induction on other structures 12 4.1 Examples . 13 5 More Examples 15 5.1 The Ackermann function . 15 5.1.1 Example 1 . 15 5.1.2 Example 2 . 16 5.1.3 Example 3 . 16 5.1.4 Example 4 . 17 5.2 Linear algebra . 18 5.3 Binomial coefficients . 18 5.3.1 Binomial . 18 5.3.2 The Binomial Theorem . 19 5.4 Calculus . 20 5.4.1 The General Power Rule . 20 5.4.2 The Leibniz rule . 21 5.4.3 Taylor's theorem . 22 6 Conclusion 24 2 1 Introduction Mathematical induction has a big influence in mathematics. It is a way to prove mathematical statements about natural numbers. You start learn about math- ematical induction and the principle of induction in the later upper secondary school in Sweden. You also learn about induction in the university if you study mathematics. The principle of Mathematical Induction consist of three steps: 1. Base case, show that it holds for the first value. 2. Induction step: Here you assume that the statements holds for a random value, and then you show that it also holds for the value after that. 3. Conclusion, because the statement holds for the base and for the inductive step, it is true for every value. You can think of induction in an illustrating way, think of a ladder. In the Base case we check that we have a first step to step on. Then in the induction step we go to an arbitrary step p and then we show that if we go to p there is a step after, p + 1. So if we go to an arbitrary step on the ladder we want to show that we can lift our leg and go to the next step. Why does this work? Because if our base case is p and then we can show that we can go to the second step p + 1, if we now let the second step be p we can show that we can go to the third step, p + 1, and then we can let the third step be p, and so it goes on and therefore we have show that we can go to any step on the ladder. Figure 1: A ladder, an illustration of induction. You can also think of mathematical induction like a domino effect. The 3 base step is that the first domino will fall, and then if any domino fall then the domino after also will fall. So all dominoes will fall. [4] 1.1 History Mathematical induction has been used in mathematics way back in history. Some people think that even Euclid used induction when he proved that there are infinitely many primes, even though there is no evidence that he used it, but some writers think that he implied it without being stated directly. Some also thinks that Plato and Pappus used induction but there are no evidence that they used it[3]. The Italian mathematician Francesco Maurolico (1494 − 1575) did a nonin- ductive proof concerning the sum of the first n natural numbers. Even though Maurolico did a noninductive proof there are people who think that Pascal got his inspiration for the induction principle from Maurolico, when Pascal in the 16th century showed by induction what the sum of the first n natural numbers is [3]. In [3] it says that sometimes induction is compared to the method of ex- haustion, like Calivieres principle where he calculates the area of a circle. There he draws polygons that fits in the circle, because he knew how to calculate the area of a polygon, and then he draws larger and larger polygons, until he has drawn a polygon which is almost in the same size as the circle. The similarities with induction and the method of exhaustion is that you start with a guess, and to prove your guess you do infinitely many iterations which follows from earlier steps. There are some proofs that are used with the method of exhaustion that can be translated into an inductive proof. There was an Egyptian called ibn al-Haytham (969-1038) who used inductive reasoning to prove the formula for n X n 1 1 1 i4 = + n n + (n + 1)n − : 5 5 2 3 i=1 Levi ben Gerson (1288-1344), used mathematical induction, he called the method \rising step by step without end". Comparing to how we are used to use induc- tion where we first do the base case and then the induction step to show that it hold for n to n + 1, Levi started with the induction step and then he showed for the base. He used induction to show that (1 + 2 + ··· + n)2 = n3 + (1 + 2 + ··· + (n − 1))2: Levi also did an inductive proof where he went from n to n − 1 [5]. As you can see mathematicians in history have used mathematical induction and inductive reasoning for a long time, but there were no one who had named this method yet. According to [2] the first ones who started to name induction was the Englishman John Wallis (1616 − 1703) and the Swiss Jacob Bernoulli (1655 − 1705). Wallis used a kind of induction called incomplete induction to 4 Pn 2 2 find the ratio between i=1 i and n (n + 1). Wallis incomplete induction both got bad and good criticism. Bernoulli was one of the ones who gave Wallis bad criticism and he introduced the principle argument from n to n + 1. Bernoulli criticized Wallis but he also thought that even though Wallis used incomplete induction it was a start to induction. Bernoulli showed the Binomial theorem with the argument when you go from n to n + 1. Georg Simon Kl¨ugel(1739 − 1812) explained the weakness of Wallis induc- tion in his dictionary, he also explains Bernoullis proof from n to n + 1. Then in England Thomas Simpson (1710 − 1761) used the n to n + 1, but neither did he give it a name. In the early part of the nineteenth century George Peacock, 1830, used induction, and the n to n + 1 argument and he called it \demonstrative induction". In 1833 Augustus De Morgan suggested a new name \successive in- duction", but in the end of his article he used \mathematical induction". Both the names \demonstrative induction" and \mathematical induction" are used by writers. But then the \demonstrative induction" get disused. In the USA induction was used but not called by its name, in Europe, the name \mathe- matical induction" was used. The Italian mathematician Giuseppe Peano (1858 − 1932) formulated the axiom system we call the Peano's axiom in 1889. With Peano's axiom we can construct all the natural numbers, and one of his axiom is the one we call the Induction axiom. You will see the axiom in section 2.1. There are a lot more of well-known mathematicians who has used mathe- matical induction, if you want to read more about it you can look in the history section in [3]. There are a lot more sources where you can find more information if you are interested. 1.2 Paradoxes with induction There are some paradoxes with induction and inductive reasoning. [3] has some fun examples of different paradoxes. One example is about a teacher who told his students that next week they will get an unprepared test. And the students think that if they don't have had the test on Thursday night they cannot have it on Friday because then they will know that the test must be on Friday and it will not be unprepared. So the test cannot be on Friday, by the same argument the test cannot be on Thursday, and either on Wednesday, or Tuesday or Monday. So they will not have a test next week. Another example is about being bald, if you don't have any hair on you head you are bald but if you only have one hair you will still be bald, so if you only have one hair more you will still be bald, therefore everyone is bald. 2 Mathematical Induction The most common induction is induction over integers. This is the one you start learn in school and use to prove simpler theorems. It is easy to think that math- ematical induction is something you only use in logic and algebra, but the fact is 5 that you use mathematical induction in almost all fields of mathematics.
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