What makes a math theorem important

An introduction to mathematical proofing using full induction

Table of Contents

short version

Preface

thanksgiving

List of abbreviations

Symbol directory

1. Introduction

1.1 Task and objective
1.2 Structure of the thesis

2Mathematics: Science, Language and Key Skills.
2.1Mathematics: A Science
2.2 Mathematics: One language
2.3Mathematics: A Key Competency

3 Introduction to mathematical reasoning
3.1 Why is mathematical proof necessary
3.2 How is a mathematical proof structured?
3.3 What is the goal of a mathematical proof

4 Mathematical Evidence Procedure
4.1 The direct proof
4.2 The indirect proof
4.3 Proof by complete induction

5 Possible uses of complete induction
5.1 Area of ​​application 1: Sum and product values
5.2 Area of ​​application 2: Divisibility of natural numbers
5.3 Area of ​​application 3: Other

6 Complete induction, pros and cons.

7 Conclusion and outlook

bibliography

attachment

APeano axioms

Proof Bernoulli inequality

List of abbreviations

Figure not included in this excerpt

Symbol directory

Figure not included in this excerpt

short version

Mathematics knowledge is not only useful and necessary for mathematicians and scientists; for economists too, mathematics is the language in which they describe and explain many of their models and phenomena. Essential components of mathematics are propositions and proofs. Erstein's consistent proof turns a sentence into a sentence and gives it general validity. There are three basic methods of proof in mathematics,

- the direct evidence
- the indirect proof
- and the proof by induction.

The latter is used for various problems of the form “applies to all natural numbers”. It consists of an induction start and an induction step, followed by the actual proof. He has to follow strict formal criteria in order to receive general recognition.

Keywords: mathematical proofs, complete induction, mathematics as language

Abstract

Mathematical knowledge is not only useful and necessary for mathematicians and scientists. Since mathematics areunderstood as a language as well they are beneficial for economists too. Additionally, it is the language in which they describe and explain their models and phenomena. An essential part of mathematics are theorems and proofs, as only a proof turns a theorem into a theorem and makes it appreciated and accepted by the mathematical community. There are three kinds of proof methods,

- the direct proof,
- the indirect proof
- and the proof by mathematical induction.

The latter method is being applied for issues containing numbers in the form of "for all natural numbers is". This method of proof consists out of an induction basis and an induction step in which the actual proof is made. This is subjected to strictly defined criteria that have to be followed in order to reiceive general appreciation.

Keywords: mathematical proof methods, mathematical induction, mathematic as language

Preface

Mathematics is a science that involves a lot more than moving numbers and formulas around. It acts as a language across various scientific disciplines and enables models and phenomena to be described in a generally understandable and clearly defined way, to show relationships and ultimately to prove them. An integral part of this science is proof. Various authors such as Arens et al. (2010, p. 22) also describe this as the core and essence of mathematics. Unfortunately, this discipline of mathematics is neglected in various degree programs. Mathematical proof can teach its users more than just mathematics. For example, when carrying out a proof, one is forced to argue flawlessly and stringently logically, which is an important qualification not only in mathematics. Of course, nobody expects an economist to have as deep knowledge of mathematical relationships as that of a trained mathematician. Basic knowledge, even in an abstract discipline such as providing evidence, helps not only to apply concepts stubbornly, but also to question them critically and to be able to adapt them to other situations.

Therefore, this term paper aims to show the basic principles of proof. It is not the aim of this elaboration to be able to apply the procedures to complex problems after reading it. This requires more in-depth mathematical knowledge than what is taught in schools and economics courses. This term paper is intended to help ensure that the mathematics lecture is no longer perceived as a hurdle, but as an important part of an economics degree.

thanksgiving

A few words of thanks should also be said at this point. First, of course, I would like to thank my professor, who gave me the opportunity to write this term paper. I would also like to thank my good friend Manuel, who provided me with numerous textbooks and with whom I was able to talk shop about complete induction. Finally, a big thank you goes to my good friend Tobias, who did the editing.

Thomas Wessinger

Straubenhardt, May 23, 2016

1. Introduction

At the beginning of the elaboration, the task and the aim of the work should be described. In addition, the structure of the thesis should also be briefly discussed.

1.1 Task and objective

A solid basic understanding of mathematics is an absolute necessity, also for students of economics. Mathematics represents an interdisciplinary language in which models can be formulated and defined. Economics are also one of these disciplines. Theorems and proofs represent two fundamental aspects of mathematics, knowledge of which offers enormous advantages in terms of understanding concepts.

These elementary components of mathematics are not part of the curriculum in the Bachelor's degree in International Management at the Karlsruhe University of Applied Sciences - Technology and Business. However, knowledge of the methods of mathematical proofing, especially of the method of complete induction, is a qualification that is advantageous not only within mathematics, but also beyond.

This makes it necessary to convey this basic knowledge and to show the advantages associated with this knowledge. The aim of this elaboration is to convey this basic knowledge as an introduction to mathematical proofs, especially using the example of complete induction.

1.2 Structure of the thesis

First of all, mathematics itself should be dealt with and its understanding presented as a demonstrative science. It should be explained why mathematics is a language used by various sciences. In addition, it is shown why mathematics is an important competence. This is followed by an introduction to mathematical reasoning, explaining why proofs are necessary, how they are structured and what goals they pursue. In a further step, the individual proof methods are to be defined and briefly explained. These are direct and indirect methods of proof, and full induction proof. A brief example is given for each method, with a closer look at the full induction method. This is followed by another chapter on the areas of application of this method of proof. Finally, a critical examination of the induction proof follows.

2 Math: Science, Language, and Key Skills.

At the beginning of the elaboration, mathematics in general is discussed. The understanding of mathematics is emphasized as a demonstrative science. It is shown that mathematics represents a cross-cultural linguistic framework in which natural science and parts of the humanities define and explain models and findings. Finally, this chapter explains why a well-founded mathematical understanding is a key qualification that is not only necessary in the context of mathematics.

2.1 Mathematics: A Science

Mathematics is a separate scientific discipline such as Arens et al. (2010, p. 5). It is not a natural science like physics or chemistry, for example, because these deal with objects of human perception. Mathematics, on the other hand, focuses on objects of human thought itself and tries to create connections in them. In contrast to the natural sciences, mathematics does not have to worry about reality, since its statements are about certainties that neither exist nor can exist in the natural sciences, as Langemann et al. (2016, p. 23).

However, science and math often go hand in hand. Essential mathematical knowledge was gained by the fact that researchers observed scientific phenomena and described them using mathematical relations. This fact is borne out by history, since for many centuries great mathematicians were also great scientists and vice versa, as Arens et al. (2010, p. 4) aptly. Archimedes of Syracuse and Galileo Galilei can be named here as examples.

Mathematics was not always perceived as a science in itself, nor did it see itself as such. At the beginning of the 19th century a process of change occurred in this way of thinking and mathematics began to see itself as an independent science. First researchers such as Cauchy and Weierstraß no longer observed scientific phenomena, but worked purely mathematically, as was the case with Arens et al. (2010, p. 4).

Today, as Arens et al. (2010, p. 4) also point out that mathematicians, natural scientists and engineers are independent scientific disciplines. However, they are united by mathematics, since it functions as the language in which they can formulate and justify their results. This fact indicates that mathematics is a Represents language that is understood and used by various scientific disciplines, as can be read in Brunner (2014, p. 22). This will be discussed in more detail in the following section.

2.2 Mathematics: One Language

Mathematics is the language used by various natural and human sciences, such as economics, to describe and justify models and phenomena, such as Arens et al. (2010, p. 5). Mathematics can be used to scientifically describe results and make them plausible. That is why mathematics represents a cross-cultural language and, according to Brunner (2014, p. 22), has a cultural and communicative meaning. The formal-symbolic language used in mathematics and the uniform use of the axiomatic set of rules on which it is based are independent of the culture that uses them. This makes it universally understandable for the description and explanation of models and phenomena from different scientific disciplines across cultures, as can also be read in Brunner (2014, p. 22).

Since mathematics and thus also its language follow strict rules and it is based on provable, logical and irrefutable sentences and axioms, it is possible to acquire and train competencies that go beyond pure mathematics itself. This will be discussed in the next section.

2.3 Mathematics: A Key Competency

Mathematics is not only important for mathematicians. This science affects all people because it pervades all areas of life and has arrived almost everywhere, so that it affects everyone directly or at least indirectly. It is used in telecommunications, navigation, industry, medicine, space travel and also in opinion polls because it is a language that crosses scientific disciplines, as Arens et al. (2010, p. 4 f.).

A solid understanding of mathematics is also indispensable for an economist. Many arithmetic operations are carried out by software solutions, for example, as they have significant advantages in speed and precision compared to humans. However, as Arens et al. (2010, p. 5) when writing software never blindly trust and always critically question the results and check them for plausibility. For this it is necessary to understand the invoices in the background in order to be able to understand them.

As an example, web analytics tools can be cited here, which can carry out many invoices and are a great help in everyday professional life, for example in a marketing department. Nevertheless, it is essential to question the results and make readjustments, which is difficult to do without a general understanding of the underlying mathematical logic. There are numerous other examples and situations in which mathematics is also used in economics. Countless models from business administration and economics are structured and described in the language of "mathematics".

Mathematics is more than just thinking in abstract formulas. Mathematics is a very communication-intensive discipline, as Langemann et al. (2016, p. 12 f.), As it is based on strictly logical arguments that are primarily devised in human language. They go on to explain that reasoning is one of the most important areas of mathematics. The ability to reason is a key skill that is absolutely necessary in professional life, where it will often be a matter of reasoning and convincing. That is where math is a very good teacher, as she forces her user to proceed with logical arguments. This also applies in particular to evidence such as Langemann et al. (2016, p. 25 ff.).

The ability to recognize the essentials of a problem and to be able to apply what has been learned to it by recognizing commonalities that are important for the solution is called abstraction. This ability is of great importance in scientific work and beyond. Mathematics is also a good teacher for this skill, because it is an indispensable part of mathematical thinking and working, as Arens et al. (2010, p. 5). It is further stated there that it is not the purpose of simply learning and applying solution schemes by heart, since, as Langemann et al. (2016, p. 186) added, cannot help in most situations and it is then necessary to consider a problem on the basis of a fundamental understanding in order to find a solution.

This was only a small selection of competencies that can be acquired by working with mathematics and transferred to other areas apart from mathematics. Certainly there are other examples that cannot and should not be dealt with further in this elaboration, as this would go beyond the scope of a housework. Rather, it was about showing that mathematics is not a chore of an economics degree, but an important part of it that is of great use during and after a degree.

3 Introduction to Mathematical Proof

In the previous chapter it was explained that mathematics is a science in its own right. In this chapter, this understanding is expanded to the effect that mathematics is a proving science, as, for example, in Arens et al. (2010, p. 14) and Langemann et al. (2016, p. 25). It is also understood as such, since it is constituted by evidence, as Brunner (2014, p. 12) writes. This is also evident from the fact that sentences are an essential element of mathematics, which only become a sentence through a proof, as in Arens et al. (2010, p. 14 ff.) Can be read.

Before going into the following with the various proof procedures of mathematics and with the procedure of complete induction in particular, a general introduction to the mathematical proof should be given. The aim is to explain why mathematical proofs are necessary, how a mathematical proof generally works and should be structured, and what the goal of a mathematical proof is.

3.1 Why is mathematical proof necessary

At the beginning of this section, we will briefly explain the importance of proving in mathematics and what a mathematical proof actually is. This makes it clear why proofs are extremely important or even essential in this science, as numerous authors of mathematical literature have pointed out.

A mathematical proof is the derivation and argumentation of a true statement that is recognized as such, based on already proven sentences or axioms, based on contradiction-free and formal requirements.

Mathematical proofs are absolutely necessary and a central part of mathematics because, according to Arens et al. (2010, p. 22) represent the “core and essence” of science and at the same time are its most important and most demanding activity, as Brunner (2015, V) explains. In addition, according to Brunner (2014, p. 12), evidence is also a carrier of knowledge, strategies and methods and, thanks to its universally valid axiomatic set of rules, has a culture-spanning character, through which this science makes its findings accessible to all people regardless of their cultural origin.

Theorems and proofs, as Arens et al. (2010, p. 14) write the central components of mathematics. The sentence is the component that functions as a tool and central content, while the proof represents the component that turns the sentence into a sentence, as Brunner (2014, p. 17) writes.[1]

In mathematics, a sentence is to be understood as a true statement that can be derived from true statements or traced back to axioms, such as, for example, this et al. (2016, p. 13) write[2]In addition to the fact of truth and provability, a sentence must also have far-reaching effects on mathematics and its application, as Arens et al. (2010, p. 22).

To clarify, the Pythagorean Theorem can be cited: Its statement is not only valid for a special problem, but represents a true statement for all right triangles and is used in many scientific disciplines. In contrast, the true statement “3 <4” does not represent a sentence. It is true and provable, but does not have the scope to be considered a sentence, as in Arens et al. (2010, p. 22) is argued.

Especially from the perspective of the user, also from other scientific disciplines, it is often sentences that are used in problems and models. It is important here that not only the sentence itself can be applied; there should also be a basic understanding of why a sentence may be used, i.e. how it was derived, as for example Langemann et al. (2016, p. 186). According to Langemann et al. (2016, p. 25) the difference between mathematics and simple arithmetic. That is why it is important to have a basic understanding of proofs, even if one is not dealing with pure mathematics, but only their application in special cases.

3.2 How is a mathematical proof structured?

After this brief overview of the place of proofs in mathematics and the emphasis on their importance, this section is intended to provide a general introduction to how a proof works and should be structured Criteria are pointed out.

First of all it should be mentioned briefly that a proof can generally be done in two ways,

- deductive (from general to specific) or
- inductive (from specific to general),

as defined by Brunner (2014, p. 7 ff.), which will be discussed in more detail later.

A mathematical proof is according to Theobald et al. (2016, p. 7) a sequence of necessarily mathematically correct conclusions from which general validity can ultimately be derived. In addition to Theobald et al. other authors also point out that evidence is an argumentative chain of conclusions based on true statements.

First of all, it should be mentioned that a proof according to Brunner (2014, p. 17 f.) Is a process that has to follow strictly logical rules in order to gain general recognition. The formal criteria stipulate that a mathematical proof must be fully documented and written down. The writing must present the argumentative chain used, which contains a tracing of the true statement back to previously proven sentences or axioms, in such a way that it can be understood and checked by the mathematical specialist community in order to obtain general recognition and validity, as Brunner (2014, P. 17 f.).

Arens et al. (2010, p. 23 f.) Point out the importance of the formal design of a mathematical proof and at the same time indicate a formal framework that, ideally, one should also adhere to in order to meet the formal requirements. Accordingly, a proof should contain the following structural elements,

- the conditions under which the statement is true,
- the claim to be proven,
- the actual evidence and
- an indication that the proof is finished.

The prerequisites are to be understood as a framework in which the assertion to be proven should be valid. The assertion is the statement that is ultimately to be proven, whereby the prerequisite and assertion are often present together in an aggregated form. The actual proof is the sequence of arguments, which is the reduction of the assertion to propositions or axioms that can be understood by other persons. The indication that a proof has been completed, as this is not always immediately apparent, should be in the form of a recognized abbreviation. Mention should be made here of “w. z. b. w. ”and“ q. e. d. ". The former stands for “what was to be proven” and the latter for “quod erat demonstrandum”. The end can also be represented by a box “■”, as in Arens et al. (2010, p. 23 f.) Can be read.

Other authors such as Theobald et al. (2016, p. 7 f.) Point out this structure and recommend using it. A simple proof will now be provided to illustrate this.

Requirement: [Figure not included in this excerpt]

Assertion: 6x + 3> 3x + 6

Proof:

x> 1

⇒ 3x> 3

⇒ 3x + 3> 6

⇒ 6x + 3> 3x + 6

Following this brief example[3] the terminology should be clear and the objectives pursued by mathematical proofs can now be considered, which the following section will deal with.

3.3 What is the goal of a mathematical proof

Now that the basic structure of a mathematical proof has been presented and explained, this section will look at the objectives of mathematical proofs. This has already been dealt with in previous parts of the elaboration, such as the consistent argumentation of sentences that only acquire the status of a sentence through proof. In the following, the goals are presented in a focused manner and some additions are made.

According to Grieser (2013, p. 142), the aim of a proof is the logical and complete justification of a statement in order to give this general validity, because as long as a statement is not proven, it could be wrong, even if it is supported by countless examples. In this statement there is another goal, which is pursued by mathematical proofs, namely consistency. If only one counter-example to a sentence is found, then according to Forster (2016, p. 6) it is wrong. Brunner (2014, p. 17) also describes the goal of mathematical proof in such a way that an assertion made must be logically derived from statements that have already been proven and names consistency as a central criterion.

In addition, a proof has both a convincing and an explanatory function, as Brunner (2014, p. 23) explains. Here, as already mentioned, the convincing function means the argumentative performance of the proof, which demonstrates the general validity of a statement in a compellingly logical and consistent manner. In addition, the explanatory function means the presentation of the comprehensibility of the relationships. Thus, as already mentioned, evidence pursues the goal of carrying and disseminating knowledge, as can also be read in Brunner (2014, p. 14).

In addition, Arens et al. (2010, p. 22), who see a further aim of mathematical proofing in the fact that the proofs provided can be used to derive widely applicable laws of great importance that can be applied in various sciences such as physics.

4 Mathematical Proof Procedures

After mathematics as a science and mathematical proof in general have been dealt with in the first chapters and thus basic aspects such as formal rigor and general structure of a proof have been stated, as well as the aims pursued, this chapter will focus on the individual methods of the Evidence to be received.

The direct and indirect evidence should be briefly discussed here, but this should only be done very briefly, as this is not the subject of this elaboration. It is more for the general understanding of the matter. Then the proof procedure of the complete induction will be explained in a more extensive framework.

4.1 The direct proof

One of the basic proof methods in mathematics is direct proof. The attempt is made to prove an implication of the form “from A follows B” on the basis of already proven sentences or tracing back to axioms, as Meyer (2007, p. 21) explains. Brunner (2014, p. 25) also describes this procedure in this form, and Arens et al. (2010, p. 23) explain this procedure in the form of argumentative inference. Other authors of relevant specialist literature also describe the procedure in the manner described here. In particular, when it comes to transforming a formula in such a way that a true statement is obtained, according to Arens et al. (2010, p. 23) the direct proof is possible and applicable, other authors also point to this field of application.

Let me give you a brief example to illustrate the direct evidence.[4] We have to prove the statement that the square of any even whole number is also an even number.

Prerequisite: [Figure not included in this reading sample] straight (straight means divisible by two)

Figure not included in this excerpt

This provides the proof that any whole number multiplied by two is always divisible by two and is therefore even.

What can be seen is that a statement of the form “A follows B” here in general and using a logical chain of arguments that can be traced back to the axioms and already proven theorems of mathematics was derived without contradiction and thus proved. In addition, the formal requirements were also established followed and thus the statement is true under the stated conditions and the reasoning for it is valid.

Of course, the example chosen is a very simple one, but in the context of this elaboration it is sufficient to clarify the procedure for a direct proof.

At this point it must be mentioned that the direct route is often not possible or only possible with considerable effort. In such a situation it can be helpful to prove a statement indirectly, as for example Arens et al. (2010, p. 23). This path can also be found in other areas of mathematics. As an example of such a change from the direct to the indirect method, a linear optimization problem can be given, for which there are always two ways to solve it. The primal or dual problem can be solved in each case, with less effort required, as for example Ellinger et al. (2003, p. 59 ff.).

4.2 The indirect proof

After the direct proof was described in the previous section of this chapter and it was also mentioned that the direct route is not always the most comfortable route, the indirect proof should be introduced and explained at this point.

Like the direct proof, the indirect proof is also a central mathematical proof technique. Arens et al. (2010, p. 23) even describe it as an almost universally applicable proof technique.

Due to its approach, the indirect proof is almost the opposite, as, for example, Brunner (2014, p. 25) or Christiaans (2016, p. 297 f.) Write. The approach of the indirect proof does not make the implication "from A follows B", but rather "if B does not hold, then A cannot hold either". This approach is equivalent to the direct implication, which results from the law of contraposition about statements. This looks like this,

Figure not included in this excerpt

and says that if A follows from B, this is equivalent to saying that if B does not hold, it follows that A does not hold either.

[...]



[1] see also Arens et al. 2010, p. 22

[2] see also Cristiaans et al. 2016

[3] taken from Posingies 2012, p. 1

[4] taken from Christiaans et al. 2016, p. 298

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