Fractions, Ratios, and Roots: Rediscover the Basics and Learn About Interesting Applications (essentials) 3658325739, 9783658325732

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Renate Motzer

Fractions, Ratios, and Roots Rediscover the Basics and Learn About Interesting Applications

essentials

Springer essentials

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More information about this series at http://www.springer.com/series/16761

Renate Motzer

Fractions, Ratios, and Roots Rediscover the Basics and Learn About Interesting Applications

Renate Motzer Augsburg, Germany

ISSN 2197-6708 ISSN 2197-6716 (electronic) essentials ISSN 2731-3107 ISSN 2731-3115 (electronic) Springer essentials ISBN 978-3-658-32573-2 ISBN 978-3-658-32574-9 (eBook) https://doi.org/10.1007/978-3-658-32574-9 © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Iris Ruhmann This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

What You Can Find in This essential

• • • •

What fractions “are” and how to calculate with them Why in some contexts fractions stand for parts and in others for ratios How fractions and decimal numbers relate to each other Why there are also numbers that cannot be expressed as fractions (so-called irrational numbers) • Why it is so important for percentages to know what the basic value is • Why different types of averages are sometimes needed in different contexts

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

What Are (Common) Fractions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

3

Fractions as Parts of a Whole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Whole as a Circle or Rectangle? . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Addition and Subtraction of Fractions . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Sample Exercises for Skimming . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 8 10

4

Fractions as Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Ratio Aspect of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Miraculous Multiplication of Areas . . . . . . . . . . . . . . . . . . . . . 4.3 The Simpson Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 16 18

5

Ordering Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6

Decimal Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Finite Decimal Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Infinite Decimal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 29

7

Percentage Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

8

Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Root Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Golden Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 37

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Contents

Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Probability Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 44

10 Various Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Introduction

And remember once and for all the most important of all sayings: It’s no secret about numbers, alone a big one in the cracks. Goethe, Urfaust.

The preoccupation with fractions always entails a break with previous ideas about numbers. Whole numbers have a clear representation. Each constellation of numbers stands for exactly one number. This is no longer the case with fractions,  because different fractions can denote the same number 21 = 24 = 36 = . . . . Additionally, every integer (on the number line) has a unique predecessor and a unique successor. Between two integers, only a finite number of other integers fit. But between two fractions there are always an infinite number of fractions. And another difference: When multiplying with integers, the numbers become larger (at least in terms of amount) (except for the unusual multiplications with −1, 0, and 1), when dividing, they became smaller. With fractions, “anything can come out.” So you have to break with all these familiar ideas of numbers when you are engaged in fractions. For this you not only gain a set of numbers where you can do any division (except the one by zero) without remainder. You can discover a lot and find the necessary tools for many applications. Some of them are presented in this book. It is further explained why this new set of numbers is still not enough to solve all mathematical problems and how the world of fractions can be incorporated into an even larger world of numbers.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 R. Motzer, Fractions, Ratios, and Roots, Springer essentials, https://doi.org/10.1007/978-3-658-32574-9_1

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2

What Are (Common) Fractions?

Many would say these are the numbers written with a fractional line, the numerator at the top and the denominator at the bottom. That’s the most important thing. Most people think of positive numbers, that is, the numerator and denominator are positive integers, that is, natural numbers. Whether the zero belongs to the natural numbers is debatable, but as the denominator it makes no sense (see below). If you want to look at negative  numbers  as well, it is sufficient if the numerator is 3 3 negative, because −4 = −3 4 = − 4 . So you could always write the minus in the numerator, but you usually write it before the fraction line. In the following, only positive fractions will be mentioned. If a negative fraction is needed, this will be explicitly mentioned. But what about 43 and 68 ? Are these two different fractions or two spellings for the same (equal) fraction (or even the “selfsame” fraction)? In German lessons the children learn the difference between “the same sweater” and “the selfsame sweater.” If Hans wears the selfsame sweater Max wore yesterday, then he got it from Max. If he was wearing the same sweater, they might have been in the same (selfsame) store (or only in the same store—e.g., in different branches of the same chain) and everyone bought a sweater. Transferred to mathematics, one could ask whether “the fraction,” or more precisely the “fraction number,” 43 exists only once and is always the same number, no matter how you write it, or whether it is only important that the two fractions 3 6 4 and 8 are equal in value. Equal in value or equal means “aequivalent” in Latin. So it can be said that fractions are based on an equivalence relation, so there are different ways of writing the same (selfsame?) fraction. Now how can one determine whether two fractions ab and dc are equal in value? It is often said that if you can transfer one to the other by expanding or reducing. This is the case with 43 and 68 . One has expanded or reduced with 2, depending  3·2 = 68 . on the direction one thinks in 43 = 4·2 © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 R. Motzer, Fractions, Ratios, and Roots, Springer essentials, https://doi.org/10.1007/978-3-658-32574-9_2

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2 What Are (Common) Fractions?

6 3 But what about 21 28 and 8 ? You could reduce both to 4 , so that could be the criteria: Two fractions are equivalent to each other (they describe the same fraction number) if they can be reduced to the same fraction. If all elements that are considered equivalent on the basis of an equivalence relation are combined to form a set, this is called an equivalence class. A fraction is therefore a class of fractions. The completely truncated fraction in this set is usually taken as the “class representative” (representative). There is another way to check the equality of value. This does not require that the numerator and denominator be checked for common divisors that could be used to reduce. What do you do with equations in which fractions occur? How do you get rid of the fractions? Right, you multiply the equations appropriately. For the equation: a c b = d This means to multiply by b and d and you get: ad = bc. a c So in summary: = shall apply precisely, when ad = bc. b d Here, b and d must not be zero. Why can a zero in the denominator not be interpreted meaningfully? 5:0 = a would mean that the other way round a · 0 = 5. But there is no such a. 0:0 = a would mean that the other way round a · 0 = 0. This is true for any number, so there is no unique solution for a. You cannot divide a whole into 0 parts. And if you were to ask yourself how often zero goes into five, that is, how often zero can be taken away from five, then it would go infinitely often. So there is no meaningful interpretation of the content either. On the basis of the previous considerations, a fraction number is an equivalence class of fractions, that is, a set of fractions of equal value. Each of these fractions can represent  the fraction number. The class of 21 is 21 , 24 , 36 , . . . . . If the numerator of a fraction is smaller than the denominator, this is called a true fraction. If the value of the fraction is greater than one then it is a false one. 2 False fractions can also be written as mixed numbers. Example: 12 5 = 25.

3

Fractions as Parts of a Whole

3.1

The Whole as a Circle or Rectangle?

But what should one imagine by a fraction or a fraction number? With ordinary fractions, one usually thinks of fractions of unit sizes or of other sizes. When asked to represent fractions of unit sizes, one could come up with sketches as in Fig. 3.1. (sketches in this booklet are mostly shown as hand sketches, as they are intended to show how  one  can get a clear idea of the fractions). 1 1 1 1 2 , kg, kg or 2 8 3 4 , 3 kg are shares of other sizes (instead of kg you may think of pound, if you want). The conversion into decimal fractions or smaller units is often known from primary school: 21 kg = 0.5 kg = 500 g. For 43 pizza, divide the pizza into 4 parts and take 3 of them. With a 68 pizza you would get the same amount: 6 out of 8 parts (see Fig. 3.2). The expansion of a fraction is thus done by breaking the whole thing into several parts, but also by taking correspondingly more parts. Usually the pizza or cake is round. But there are also rectangular pizzas and cakes from a rectangular baking tray. Which illustration would you prefer: Circle or rectangle? The circle has the advantage that you can see the unity better. All circles of this world are similar, they differ only in the radii. So one circle is always an enlargement or reduction of the other. In addition, a circle can be reconstructed well if you only have one sector (and a fraction of it is actually always represented as a sector—see Fig. 3.3). If you get a part of a rectangular pizza or a sheet cake, you cannot reconstruct how big the whole pizza or cake was. So you have to represent the unit additionally, which is usually done in a dashed line (see Fig. 3.4).

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 R. Motzer, Fractions, Ratios, and Roots, Springer essentials, https://doi.org/10.1007/978-3-658-32574-9_3

5

6

Fig. 3.1 Fractions as a proportion of unit sizes Fig. 3.2 Expanding a fraction with 2

Fig. 3.3 Making the circle visible of any a fraction which is given

3

Fractions as Parts of a Whole

3.2 Addition and Subtraction of Fractions

7

Fig. 3.4 Visualization of the rectangle, of which only a fraction is given

3.2

Addition and Subtraction of Fractions

But rectangles also have advantages. You can give them different side lengths to match the fractions. For example, if you are interested in 25 + 37 , one direction is 5, the other 7 units long. But perhaps you should consider beforehand whether a cake is enough to show the result. Is it doing it here? Well: 25 is smaller than 21 (how can you tell, even if you have not yet painted the cake?). The same applies to 37 . So, all in all, you will surely have less than a whole. It would be more exciting with 25 + 47 . There it could be scarce. Now the drawing solution. You can choose how big you want to draw the cake. The most obvious numbers are the side lengths of 5 cm and 7 cm. Due to the different division from left to right and from top to bottom, a total of 35 parts were obtained. For 2 of the 5 stripes you have 25 dyed, so you get 14 parts. For those, you take 37 three of the 7ths stripes, so you get 15 parts. Probably 6 parts will overlap. Now you may mark them additionally and have marked 29 of the 35 parts (see Fig. 3.5). The result of the addition is thus: 29 35 . The number 35 is thus the common denominator, which is also called the “principal denominator.” If the second summand 47 had been the second summand, 20 parts would have been marked for it. So with 34 marked parts you would have almost got the whole thing. Addition and subtraction can therefore be well represented on rectangular sheet cake or rectangular pizza. Here is another example of a subtraction that you can’t 15 1 really draw: 25 − 37 = 14 35 − 35 = − 35 . The problem is that you would have to take away a 35th of a piece more than there are 35ths of pieces existing. Those who find it difficult to remember how to add or subtract fractions should imagine the corresponding cake tin and/or sing about the whole thing (a text based on the melody of “Eine Seefahrt die ist lustig (A sea voyage that’s fun)” can be found in Paulitsch 1993).

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Fractions as Parts of a Whole

Fig. 3.5 Sum of two fractions shown in a rectangle

3.3

Multiplication and Division

Also for the multiplication of fractions it is helpful to imagine a rectangle that can be divided in two ways. 3 2 4 of 3 means: First divide in one direction and take 2 parts of it and then divide in the other direction and take 3 parts of it (but only of the 23 -part). This way you get 6 out of 12 parts and if you think about it carefully, it is half of the whole (see Fig. 3.6). Besides the size aspect, the operator aspect plays a major role in multiplication: The fraction stands 43 for “ 43 from …” to be considered. This can now be analogous to what we used to do with 43 a pizza or a kilometer. But it is also possible to quarter a half of something again and take 3 parts of it again. The result must always be described as a proportion of the whole. 3 1 3 1 3 4 of 2 = 4 · 2 = 8 . An example where reduction is possible: 2 1 2 1 3 from 2 pizza = 6 pizza = 3 pizza (see Fig. 3.7). On the one hand, the division of fractions can be understood as the inversion of multiplication. How this works exactly is derived here: x = 21 : 23 therefore means 23 · x = 21 . So the question is: What do I have to multiply 23 by to achieve 21 ?

3.3 Multiplication and Division

9

Fig. 3.6 Multiplication of two fractions

Fig. 3.7 Multiplication task, where it is possible to reduce

Since 21 is less than 23 , the number searched for must be less than 1. 23 multiplied by its inverse fraction 23 is first of all 1 and that again halved leads to 21 , so the number searched for is the half of 23 , thus 21 · 23 = 43 . De facto the inverse fraction 23 (or also called reciprocal) of plays a role (to bring the number to 1 in the first step). Another example x = 43 : 9.2 . A related question is: How often does 29 fit in 34 ? More than 1 time? Yes, 29 is smaller than is 43 · 29 . It is even smaller than 28 = 41 . So it should fit more than three times. Written as multiplication task: 29 · x = 43 . The searched number x must first be 3 equalized 29 and then multiplied to the 43 -fold. Thus x = 29 · 43 = 27 8 = 38.

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Fractions as Parts of a Whole

Now some people may find the written down detour via the corresponding multiplication task as too formal. On the other hand, a basic idea for dividing in the case of fractions is splitting, that is, the question of how often the divisor fits into the dividends. How often 23 fits in 21 ? Since 23 is greater than 21 , less than 1 time. You can make both fractions with the same name (i.e., produce an equal denominator) to make them easier to compare. 23 = 46 and 21 = 36 . So the question is, how many times 46 fits in 36 . This question is equivalent to: How often does 4 go into 3? The answer: 43 times. General: ab : dc means: how often does dc fit in ab . Expand delivers:   a c ad cb a d b : d = bd : bd = ad :cb = b · c . De facto one has multiplied by the reverse fraction. Annelies Paulitsch also includes math songs for multiplying and dividing fractions (1993).

3.4

Sample Exercises for Skimming

Sometimes it is worth thinking about whether what you have determined can be true. Example: 1 13 − 56 = 13 . Why should this make you wonder if you can get the result of 13 here? Instead of doing formal 1 13 − 56 = 86 − 56 = 36 = 21 calculations, one could also do 1 13 − 56 = 13 + 1 − 56 = 13 + 16 = 21 calculations. In any case, the result has to be more than 13 , so that it should be obvious that you have made a mistake, for example if you have 36 accidentally reduced to 13 . To conclude Chapter 3, here are a few rough estimates of the various types of calculation. Instead of calculating the result exactly (which the calculator can do more accurately), you should perhaps think more about the order of magnitude of the numbers: For the following exercises, how can you decide whether the result is greater or less than 1 (the result should not be calculated at all). + 41 (Why is 23 < 43 ? What is missing for each 1? Why is the result of the plus task therefore smaller than 1) 1 1 b) 47 + 19 + 10 (Why 10 < 19 < 17 ? Is the result smaller than 1 this time too?) a)

2 3

3.4 Sample Exercises for Skimming

11

+ 56 (Do you think in eighths or sixths? If you think in eighths, how much is missing from the ones to the 38 first? Why is 56 more? If you think in sixths, how much is missing from 56 to 1 and why is 38 more?) d) 23 · 57 You can do math here. c)

3 8

e) f)

7 2 5 · 3 (Compare with d!) 148 91 257 · 43 (Is the first fraction

h)

2 3

larger or smaller than 21 ? How can the second be estimated? Why is it larger than 2?) 34 21 1 g) 234 125 · 13 · 37 (Comparisons with 2 and 2 also help here.) :

3 4

(Which fraction is larger?) i)

148 4 257 : 17

(Which fraction is larger here?)

Do the following exercises result in anything positive or negative? a) b) c) d) e) f)

2 3 1 4 3 8 4 7 4 7 4 7

− − − − −

1 4 2 3 5 6 2 9 2 9 2 9

(Which number is greater?)

−

1 2 2 10 (Why is 9 < 7 and why does this help with the estimation?) 2 − 15 (Here, 15 = 10 can still be considered.) 2 2 − 5 (Now it should be close to zero. 19 + 15 = 14 45 > 7 because 14

− 7 > 45 · 2, so more is deducted than there is given.)

·

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Fractions as Ratios

4.1

The Ratio Aspect of Fractions

Besides the size and the operator aspect (a proportion of something to be considered), the ratio aspect is the third important basic idea for fractions. Here there is no a priori unambiguous whole to which all occurring fractions (including the result of a calculation) are related. Fractions as ratios therefore only play a role later in school, for example in the treatment of proportionality (double price for the double quantity and the like). In this basic idea, fractions are no longer part of a whole, but rather two quantities/numbers/… that are related to each other. These are rarely added up, but rather compared or used to describe a more general relationship. They can be connected with (fractional) operators for multiplication rather than by addition or subtraction. Proportionality plays an important role in everyday life. Often a ratio is given as a ratio of two sizes, for example, e/kg (“per”) or portions of ice-cream/scored a goal (“per”), when ice cream is promised as a reward for a child in the event of success at a football tournament. Later, when dealing with functions, the units are omitted. With linear functions, there is a clear gradient m = y x . Important in this context is usually again the independence from the representatives (no matter which Δx I choose, the quotient of the corresponding Δy to Δx is always the same). If one chooses Δx = 1, the m can be read directly from the graph (see Fig. 4.1). But please note that unlike the pizzas, the counter m is not drawn as part of x, but goes in a different direction. Proportions also play a role when mixing drinks, medicines and the like. There are two ways of indicating the juice content in a juice spritzer, as juice to total liquid or juice to water. A spritzer, which consists of half juice and half water,

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 R. Motzer, Fractions, Ratios, and Roots, Springer essentials, https://doi.org/10.1007/978-3-658-32574-9_4

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Fractions as Ratios

Fig. 4.1 Slope triangles

  has two fractions: As 1:2 = 21 is the proportion of juice to total liquid and 1:1 as the ratio of juice to water. You have to pay attention to which variant is meant. As mentioned before, it is rarely added up under certain circumstances. But what does it mean to pour two spritzers together? Is that also a kind of addition? Or what does it mean when the juice content is halved in a half and half spritzer? The percentage of the total liquid should result in 41 . The ratio is suddenly 1:3, or is it meant that only half as much juice is poured to the same amount of water? Then the ratio would be 1:2 and the juice would make up 13 of the total amount. As you can see, close attention is needed here—and even then, everyday language is not always suitable for being poured into mathematical equations: For example, what does it mean when “the share is now five times less than before?” Probably shrunk to one of the 15 previous percentages, right? Let us assume again as a starting situation that a spritzer was mixed half and half. Earlier it was 50% juice. So “5 times less” could mean it’s just 1 10% = 15 · 21 = 10 juice. The ratio would then be 1:9. Or should the ratio change from 1:1 to 1:5? Or to 1:6 (because of “5 times less”)? And when something has grown 5 times more, you usually think it has grown 5 times. It says it’s increased five times, which would make it six times. “Five times less” would mean four times less. That’s usually not possible. So it can’t be meant that way.

4.1 The Ratio Aspect of Fractions

15

But now back to pouring the spritzers. The amount of spritzers has certainly increased due to the pouring together, but that is not what the conditions are asking for. So there is no break addition. Is it still possible to describe the pouring together mathematically? Let us assume that one mixture was made by adding 2 glasses of juice to 3 glasses of water and the other by adding 1 glass of juice to 2 glasses of water. All glasses are the same size. If these mixtures, which involve a total of 8 glass fillings, are poured together, the new mixture contains 3 glasses of juice to 5 glasses of water. ˇ 21 = 35 (ratio of juice: water) To this fits on the one hand the “calculation” 23 + 2 ˇ 1 3 and on the other hand 5 + 3 = 8 (ratio of juice: total amount). Both times are “added up” in a way that students may erroneously like to do: numerator + numerator by denominator + denominator. The fact that the value of this fraction in results lies between the two “summands” should be clear in terms of content. Therefore, this argument for the fact that numerator plus numerator by denominator plus denominator lies between the two fractions is also called “spritzer proof.” Would it make sense to define such addition as well? Where could there be another catch? Have you noticed that I have completely poured both spritzers together? Not a part of the first spritzer with a part of the second spritzer? But what happens if you take one completely and only a part of the other? Does the “addition” give the same result? Is this addition, therefore “independent of the representative” of the associated fractions. This is the case with the real fractional addition: 2 1 4 3 7 + = + = 3 2 6 6 6 6 2 24 18 42 7 + = + = = 9 4 36 36 36 6 But what happens if we expand the fractions of our spritzer, for example by factors 3 and 2? 2 1 2 3 6 8 ˇ ˇ + = or + = 3 2 5 9 4 13 3 5

8 and 13 are not of equal value. Which of the two fractions is larger? One could convert to a decimal fraction with a pocket calculator (more on this in Chapter 6) and then compare. Or we could bring both to the common

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Fractions as Ratios

denominator (65). Or we leave this out and multiply “crosswise”: 3 · 13 < 8 · 5, 8 therefore 35 < 13 . How is it that in the first blend the juice content is slightly smaller than in the second blend? In the second mixture, three times the amount of the first spritzer (which has a higher juice content) has been taken, and only twice the amount of the second mixture (if we interpret the numbers in the numerator and denominator as equally sized glasses). Therefore, the first spritzer has a higher percentage of the total spritzer and the juice percentage of the total spritzer is a bit higher than in the first mixture. Depending on how much of the two spritzers you take, the mixture has a different value, which of course always lies between the two initial values. And you could certainly tell me now how to choose the mixture if it should be particularly close to one or the other “summand.” But you could also say, who pours different spritzers together and is so interested in the new mixing ratio? On the one hand, I have told you this in such detail because I wanted to show you that independence from the representative (which applies to the usual fractional addition and subtraction as well as to multiplication and division) is nothing to be taken for granted. On the other hand, such a piling up can actually play a role in everyday life (see Sect. 4.3 “Simpson paradox”). Furthermore, there is a geometric interpretation as a “mean slope” if a graph is composed of different distances. Here is an example that invites you to wonder.

4.2

The Miraculous Multiplication of Areas

Compound gradients are to be considered using a somewhat tricky example: A square is divided into four parts; two triangles and two trapeziums. These are combined in a different way and now form a rectangle. However, if you look at the total area, which should have remained the same, a problem arises (see Fig. 4.2 and 4.3). Does (8 · 8 = ) 64 = 65 (= 5 · 13) apply? Or what is the catch here? Almost the same phenomenon occurs with the following larger square: Here the question arises (8 · 21 = ) 168 = 169 (= 13 · 13)? This time the square has the slightly larger area. Let us consider the slopes involved in the first rectangle: 2 3 5 5 (in trapezium) = ( 8 in triangle) = ( 13 in total rectangle)? Which is the greatest, which is the smallest gradient?

4.2 The Miraculous Multiplication of Areas

17

Fig. 4.2 Conversion of a square into a rectangle I

Fig. 4.3 Conversion of a square into a rectangle II 16 3 15 = 40 , 8 = 40 . Such fractions (in their reduced version) are called 1 “minimally adjacent” (i.e., they differ only by 1. Denominator·2. Denominator ). 5 Then where is 13 ? Like the spritzer evidence in between! In the point where the triangle and the trapezoid meet, there is an “invisible” small bend. The trapezoid has a slightly larger gradient. The total gradient, which could be drawn in separately if you were drawing exactly, lies between the two gradients involved. 5 8 In the other example, the slopes 38 , 13 , and 21 play a role. 5 40 3 39 8 lies Here 13 = 104 is larger than 8 = 104 (again minimally adjacent) and 21 in between. 2 5

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4

Fractions as Ratios

Note: Between two minimally adjacent fractions, as can be seen here, there may be a fraction whose denominator (here 21) is smaller than the common denominator of the minimally adjacent fractions (here 104). If you feel like it, you can think about whether this is always the case with minimally adjacent fractions. Perhaps you will also notice that the same numbers have occurred again. Note the numbers involved: 2, 3, 5, 8, 13, 21, … How does this sequence of numbers continue? These numbers are also called Fibonacci numbers. Leonardo of Pisa (1170– 1240), known as “Fibonacci,” examined this sequence of numbers more closely. Normally, the number 1 is preceded twice more: 1, 1, 2, 3, 5, 8, 13, 21, 34, … You’ve probably already recognized the law of education. The next number is always the sum of the two preceding ones. Through it, you can easily create neighboring fractions whose numerator and denominator are still relatively small, but which are still very close together. A supplementary question (which can no longer be answered with fractions): How would one have to divide a square into two triangles and two trapeziums in order to actually contain an equal-area rectangle in the desired shape—and one without kinks? Pythagoras was wrong about “everything is number” in the sense that any ratio can be represented by a (rational) fraction. For this task, one would need the so-called “golden ratio.” This is an irrational ratio. One can describe a lot with fractions, but not everything (see Chapter 8).

4.3

The Simpson Paradox

The Simpson paradox is named after Edward Hugh Simpson (born 1922). It is a paradox from statistics. The evaluation of different groups turns out differently depending on whether the results of the groups are combined or not. Firstly an example from university life. In 1973, accusations of misogyny arose at the University of Berkeley. The reason was that apparently women were discriminated against in the allocation of university places. This could be seen from the admission figures: 44% of 8442 applicants were men and only 35% of 4321 women. This means that 9% more men than women were admitted. A more detailed look at the data showed that in 85 study programs, men were significantly preferred in only 4 areas, and in 6 study programs, significantly more women were admitted. But why did it seem that women were disadvantaged overall? In

4.3 The Simpson Paradox

19

Table 4.1 Notional registration figures divided between women and men Women

Men

Total

Admitted

Total

Admitted

Section 1

900

720

200

180

Section 2

100

20

800

240

Total

1000

740

1000

420

fact, women were more likely to apply for courses of study with lower admission rates (which affected both genders there). Men were more likely to apply for programs with higher overall admission rates (see Bickel et al. 1975). A simplification to two sections could look like in Table 4.1 The situation should with fractions.  4 be represented  9   In Sect. 1, 720 = = 80% men are admitted and 180 900 5 200 = 10 = 90% women   20 3 = 15 = 20% men and 240 are admitted. In Sect. 2 the 100 800 = 10 = 30% women are admitted. Looking only at percentages, women seem to be favored. But the overall figu740 420 res also play a role: 1000 (= 74%) men and 1000 (= 42%) women get a university place. The fractions involved must not simply be reduced in value and then offset. As with spritzers, you have to pay attention to how much of which “spritzer” you take. Normally one thinks: If a < c and e < g, then a + e < c + g. This is true for ordinary addition. In the Simpson paradox, however, it is not the normal addition that plays a role, but the wrong addition, which is not independent of the representative. a+e If ab < dc and fe < gh , the following may apply: b+f > dc+g +h . The prerequisite g c e a must be that d < f or h < b . This can be seen in the following straight lines. Only the two mixtures have to be chosen appropriately so that the mix value of one mixture is close to the smaller initial value and that of the other closer to the larger one. (see Fig. 4.4—The idea of using such straight lines is from Tan 1986). Simpson paradoxes can also play a role in death statistics or causes of death. Let’s take a look at the current cancer statistics and the forecasts for the next few years: “According to the current population forecast, if the number of cases of cancer remains the same, by 2020 the absolute number of cases of cancer would rise by 12% for men and 7% for women compared to 2013. Taking into account the recent decline in incidence trends for some of the more common

20

4

Fractions as Ratios

Fig. 4.4 Sketch of the Simpson paradox

tumors (including stomach and colon cancer), the predicted increases for cancer are somewhat lower overall, at 9% and 6% respectively. However, only just over half of the long-term increase in cancer incidence, with the absolute number doubling since 1970, can be explained by demographic changes, since age-standardized rates have also risen since 1970 until around the turn of the millennium. Depending on the type of cancer, different factors are of importance here” (from the report on cancer incidence (Bericht zum Krebsgeschehen 2016, p. 24)). Thus, an increase in cancer incidence is predicted for the next few years, although for some types of cancer a decrease is even possible. A de facto decrease is also expected in several age groups. The fact that the overall number of cancer cases continues to rise is mainly due to the fact that society is getting older and older and therefore there are more people in the age groups where cancer is more common. In the past, many people did not get old at all because they died of other causes at a younger age.

4.3 The Simpson Paradox

21

Table 4.2 Tuberculosis cases in New York and Richmond in 1910 Population

Fatalities

Mortality rate

New York

Richmond

New York

Richmond

New York

Richmond

White

4,675,174

80,895

8365

131

0.00179

0.00162

Colored

91,709

46,733

513

155

0.00560

0.00332

Total

4,766,883

127,628

8878

286

0.00186

0.00224

So the following Simpson paradox could occur: Cancers are decreasing in every age group, but overall they are increasing. A good 100 years ago there was a Simpson constellation in connection with tuberculosis: In Table 4.2 on mortality due to tuberculosis in New York and Richmond from 1910, we encounter the following Simpson paradox (Székely 1990, pp. 63, 75 and 133). What is to be concluded from this? • “If you are white, go to Richmond. • If you’re colored, go to Richmond, too. • If you’re white or colored, stay in New York.” At least that is how Timm Grams interprets it in “Klüger Irren- Denkfallen mit System (Erring smarter - avoiding systematic pitfalls)” (2016).

5

Ordering Fractions

The ordering of fractions was already mentioned when comparing the gradients. For fractions with positive denominator and numerator, the following applies: a c b < d exactly when ad < bc. This can be recognized by bringing it to the common denominator and then comparing the numerators or by multiplying the inequality by the principal denominator (b · d). When asking which sum or products are greater or smaller than 1, you have certainly already made some comparisons between fractions. That the product of the numerator of one fraction plays a role with the denominator of the other fraction can be seen by the fact that if two fractions should be equal at first, one becomes larger if it receives a larger numerator or the comparison partner becomes smaller by a larger denominator. Since there are an infinite number of fractions in total, and since there are always an infinite number of fractions between two (fractional) numbers (however close together they may be), it is difficult to arrange them in sequence on the number ray. And yet there are aids. First of all, I will pursue the claim that there are infinitely many fractions between two numbers: How do you find a number that lies between two fractions a c b and d ? Let us take 23 and 43 . These are once again minimally adjacent fractions, 8 9 and 43 = 12 . So what can be in between? On the one hand, because 23 = 12 17 the (arithmetic) mean value: 24 (which also corresponds to numerator + numerator by denominator + denominator, after you have brought both to the common 11 7 denominator). Or in the sense of the “spritzer proof” 57 or 11 16 or 15 or 10 or … 2 3 Expand this to 3 or 4 or both and add again numerator + numerator by denominator + denominator. And if you like it, arrange the fractions you receive according to size. Which fractions are closer to 23 , which are closer to 43 ? It always depends on the

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24

5

Ordering Fractions

“weight” you give to the two fractions, so which fraction do you expand with a larger number? Another way to find many fractions between two numbers would be to convert them into decimal numbers: 23 = 0.6666 … and 43 = 0.75 (see also Chap. 6). 67 Between them, therefore, lie 100 = 0.67 or 0.68 or 0.69 or 0.7 or 0.722225678 = 722,225,678 etc. 1,000,000,000 As you can see, there are really many possibilities. The choice is very large (there are infinite fractions between two fractions). We can still start ordering fractions on the number line. Let’s take the numbers between 0 and 1. And let’s start with small denominators (the counters then remain correspondingly small):

You can see that only reduced fractions occur here. In the series written down 16 , 1 1 5 6 7 7 , 8 , 6 , 7 , 8 are missing. Then one would have all fractions whose denominator is 8 at most. One can well imagine when these six fractions will appear in the next three rows. In the next row, the greatest occurring denominator will be number 13, then 21, then 34. We already know these numbers, they are the Fibonacci numbers. The Stern-Brocot tree is structured accordingly (see Bates et al. 2010). Also for numbers >1, the fractions can be generated accordingly. If you take all numbers up to 2, the representation looks very similar.

6

Decimal Fractions

6.1

Finite Decimal Fractions

Many people are more familiar with decimal fractions than ordinary fractions. Decimal numbers are more common in everyday life, but for prices with two decimal places, they are basically constant. The advantage of decimal numbers over ordinary fractions is that they can be represented in the system of digits and can therefore be ordered much more easily in terms of size (see Fig. 6.1). If unlike with monetary values, the number of decimal places is varied, there are several misconceptions, for example, that the decimal point divides the number into two parts that could be compared or edited separately. For example, 1.3 < 1.25 (since 3 < 25). Or 1.3 + 1.25 = 2.28. The entry in the value table can help to see through these errors (see Fig. 6.2). The 3 are tenths, as well as the 2 directly behind the decimal point. The 5 are hundredths and therefore you have to add the 3 and the 2, not the 3 to the 25. 1.3 could also be written as 1.30. Where length is indicated, 1.3 m is the same as 1 m and 30 cm. 1.25 m is the same as 1 m and 25 cm and both units can be added separately. 1.8 + 1.25 Thus Leads to 1 m 80 cm + 1 m 25 cm = 2 m 105 cm = 3 m 5 cm = 3.05 m. Thus Both the Solution 2.33 and 2.105 Would Be Wrong. For adding and subtracting, it is, therefore, useful to set the two numbers to the same number of decimal places. After that, you can also do calculations in writing, if necessary. In the case of subtraction, one may wonder whether it is more advantageous to subtract or add, depending on the figures. As an example of a head calculation strategy:

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26

6

Decimal Fractions

Fig. 6.1 Value table for decimal numbers

Fig. 6.2 Decimal numbers entered in the value table

1.15 – 0.3 = 1.15 – 0.30 = 1.00 – 0.15 (to obtain a smooth intermediate result) = 0.85 (to break down a whole into 100 sub-units and subtract 15 from it). 1.15 – 0.90: From 0.9 to 1, 0.1 is missing; up to 1.15, 0.1 + 0.15 = 0.25. If the two numbers (also called minuend and subtrahend) are close together, it is often advisable to add from the subtrahend to the minuend. This is also recommended if the minuend is a smooth value. Some sellers still calculate in this way and do not just rely on the value of the change displayed by the cash register: 20.00 e – 11.85 e = 5 ct + 10 ct + 3 e + 5 e = 8.15 e. Someone has shopped for 11.85 e and pays with a 20 e note. The shop assistant says “elevenninety, twelve, fifteen, twenty” and accordingly returns 5 ct, 10 ct, 3 e (as a 2 e and a 1 e piece) and a five e note. This could make the customer realize that he has not lost any money in the store. Although the money in his wallet has become less, he has goods worth e11.85 in his pocket and e8.15 in his wallet, which together make up the e20 he came into the shop with. Of course, if he eats and drinks the goods, has his fortune diminished - or has it just been transformed into another form of energy?

6.1 Finite Decimal Fractions

27

In order to understand the multiplication of decimal numbers, one can also often fall back on quantities. Again, lengths are a good choice, since you cannot multiply weights with each other (you can only multiply them, so one factor would have to be without a name). If you multiply two lengths, you get a surface area (the corresponding rectangular area). 1.15 m · 1.3 m specifies the area of the corresponding rectangle in m2 (how many m2 fit into this area). If you imagine the rectangle, you can already see that only a whole square meter fits in it and the rest is only filled by parts of square meters. In any case, it should be noted that not only a rectangle with side lengths of 15 cm and 30 cm is added to the one square meter. If one would get the idea to divide the number into two parts (before the comma, after the comma) and calculate them separately, one would miss two partial rectangles. A pictorial representation is helpful for clarification (see Fig. 6.3). 1.15 m · 1.3 m = 115 cm · 130 cm = 100 cm · 100 cm + 100 cm · 30 cm + 15 cm · 100 cm + 15 cm · 30 cm = 10, 000 cm2 + 3000 cm2 + 1500 cm2 + 450 cm2 = 14, 950 cm2 = 1.4950 m2 . (You can also count with inches.)

Fig. 6.3 Multiplication of decimal numbers interpreted as a rectangular area

28

6

Decimal Fractions

To do this, you must have an idea of how to convert cm2 into m2 (conversion number: 100 · 100 = 10, 000). Or you must have thought before that 1, … m2 must come out. This means that this task requires almost one and a half square meters. It might also be interesting to look at the squares of decimal numbers. Let us take especially the numbers between 1 and 2 with one decimal place: 1.12 = 1.21; 1.22 = 1.44; 1.32 = 1.69; 1.42 = 1.96; 1.52 = 2.25; 1.62 = 2.56; 1.72 = 2.89; 1.82 = 3.24; 1.92 = 3.61. What stands out? For example: All numbers are between 1 and 4, with two decimal places. At 1.42 , you’ve almost reached 2. More than 3 is 1.82 . How many digits does the square of a number with two decimal places have? For example, 1.112 = 1.2321. Is it always four digits? Yes, because in a square the last digit never becomes 0 unless the number ends in 0. So the number of decimal places always doubles. Consequently, there can never be a decimal number with a finite number of decimal places whose square is exactly 2 or exactly 3. If there is no finite decimal fraction that gives a smooth 2 or 3 in the square, can there be an ordinary fraction? Suppose it’s a truncated fraction ab . The square a2 b2

is then also a truncated fraction because no new prime factors come into play through which one could truncate (see Chap. 8). The number of decimal places in multiplication is usually the sum of the number of both factors. As an exception, one could see the results that end at 0 and therefore can be written shortened. Ex. 1.2 · 1.5 = 1.80 = 1.8. When can something like this happen: Only if one factor ends in the digit 5 and the other in an even number. The final digit is always the product of the final digits of the two factors. (Sure, why? Imagine you calculate the result in writing!) All in all, it can be said that neither an ordinary fraction nor a decimal fraction squared can result in a smooth 2 or 3. And what about the division of finite decimal fractions: 1.2:1.5 = ? The interpretation as length specifications results in: 1.2 m:1.5 m = 4 12 dm:15 dm = 12:15 = 12 = = 0.8. 15 5 A conversion to cm would result in: 1.2 m:1.5 m = 120 cm:150 cm = 0.8. When dividing finite decimal fractions, a finite decimal fraction need not 2 necessarily result: 1.2 m:1.8 m = 12:18 = 12 18 = 3 . So we cannot avoid thinking about the connection between ordinary fractions and decimal fractions / decimal numbers.

6.2 Infinite Decimal Numbers

6.2

29

Infinite Decimal Numbers

So far I have used the terms “decimal number” and “decimal fraction” synonymously. For numbers with a finite number of decimal places, the terms are also synonymous. Any decimal number with a finite number of decimal places can be written as a fraction: 34,567 234,567 2.34567 = 2 100,000 = 100,000 . The denominator contains the corresponding power of ten, that is, a 1 followed by as many zeros as there are decimal places. When dividing finite decimal numbers you could formally write a fraction and then expand the fraction so that there are no more commas: 234,567 2.34567:12.456 = 2.34567 12.456 = 1,245,600 (it was expanded with 10,000). But would the result be a finite decimal number? I don’t think so. For this is only the case if, after reducing the fraction, the denominator is a power of ten, or if the fraction can be brought to a power of ten by expanding. Since powers of ten only have the prime factors 2 and 5, this is only the case if the reduced fraction in the denominator only has powers of 2 and 5. 4 16 2 Example: 12 75 = 25 = 100 = 0.6. The denominator after reducing is 5 and can 2 2. therefore be expanded from 2 to the power of ten 10 If the denominator (after truncation) contains other prime factors than 2 and 5, there is no corresponding finite decimal fraction. A periodic decimal fraction can be obtained by written division with the addition of further decimal places:

The number is not always repeated immediately. It is possible that the repetition starts later and consists of a sequence of numbers. Since there can never be more than n – 1 different remainders when dividing by a number n, the period is at most n − 1digits long.

30

6

Decimal Fractions

You can see that the first decimal place (3) does not repeat, the next six will repeat. Since 14 is an even number, if multiples of 14 are subtracted from tens, only even remainders are possible. 3 can therefore no longer appear as a remainder. Even remainders 2, 4, 6, 8, 10, and 12 all appear in this case. Fractions whose denominators (in abbreviated form) do not only contain the prime factors 2 and 5, therefore have an infinite, periodic decimal fraction. If one looks even closer, one could prove that it becomes a pure periodic decimal fraction (the period begins directly after the decimal point) if the denominator does not contain the factors 2 and 5 at all. It becomes mixed periodic if both factors 2 and/or 5 and other prime factors are included (as was the case with 14). Now you could imagine that you also specify infinite decimal numbers whose decimal places are not periodically repeated. e.g., 0.10100100010000001000001 … (always one more 0 until the next 1 comes). No fraction can belong to this decimal number, since fractions always lead to finite or periodic decimal fractions (see Chap. 8). It is therefore not a decimal fraction. Finally, the question of how to convert a periodic decimal fraction into an ordinary fraction will be discussed. Example: 1.23454545 … = 1.23 + 0.01 · 0.454545 … (Split into the nonperiodic and periodic part). Now be x = 0.454545 …, then 100 · x = 45.454545 … Subtracted from each other, this results in 99 · x = 45, therefore x = 45 99 = 5 11 . 123 1 5 1358 + 100 · 11 = 1353+5 That is 1.23454545 . . . = 100 1100 = 1100 . And if you are not quite sure whether it is allowed to calculate with “… numbers” in such a way that you can subtract them from each other? Then you 1353 can check the assumption that 1100 could be the number by calculating 1358:1100 in writing.

7

Percentage Calculations

In addition to ordinary fractions and decimal fractions, one often encounters percentages in everyday life. Discounts are usually indicated in the shop window by large percentage signs—and the subconscious is immediately excited about a possible bargain. Instead of a “20% discount” the shop could also advertise with a “ 15 - discount.” You could probably calculate the reduction even faster, but it would still be an unusual sight for our eyes. Percentages are used because numbers between 0 and 100 are usually easier to handle than 0, …—numbers or ordinary fractions. There is, of course, nothing mysterious about percentages, because % means 1 nothing other than 100 (“percent,” i.e., per hundred). 20% = 20 ·

20 1 1 = = . 100 100 5

If you have even smaller numbers (e.g., the alcohol content in the blood), use 1 . “per thousand” (per thousand) ‰ = 1000 There is a nice formula for calculating percentages: p Percentage value = Basic value · Percentage or P = G · 100 = G · p%. The formula could be solved to G or p, as required. But you can also remember the context in another way. The percentage of the basic value is written as a fraction GP . Multiplied by 1 = 100% is the percentage. Example: A reduction of 34 e at a price of 200 e is a reduction of the share 34 17 200 , i.e., to 100 = 17%. On the other hand, if one knows that there is supposed to be a reduction of 17%, the result is 200 e:

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32

7

Table 7.1 Example of a four-field table

Percentage Calculations

Women

Men

Interested in football

13

12

25

Are not interested in football

7

3

10

20

15

35

200 e · 17% = 200 e ·

17 = 34 e. 100

The fact that the basic value is sought and the percentage value is known is, for example, the case when calculating the net price without VAT. With value-added tax, you pay 119% of the basic value (= net price). So if something costs 238 e with value-added tax, one could use the equation: 119 solve by doing math, 238 e · 100 238 e = G · 100 119 = 200 e. You could also solve by the rule of three: 119% costs 238 e 100% costs … e by first dividing both sides by 119 and then multiplying by 100: 1% costs 2 e 100% costs 200 e. The biggest problem with percentages is often the question of what the associated basic value is. In the previous task, the e238 was not the basic value, but the percentage value. The base value was wanted. If there is a link between (at least) two conditions, you have to look very carefully. As an example, look at a (fictitious) small group of men and women to find out whether they are interested in football. The results of the survey can be written in a four-field table (see Table 7.1). In this group, 13 out of 35 are women (= 13 35 = 37. .. %) and are interested in football. 13 Of the 20 women, 13 (= 20 = 65%) are interested in football. 13 13 (= 25 = 52%) women are among the football fans here. So you have to look carefully. The same 13 women (percentage value) are related once to the basic value of all respondents (35), once to all women (20) and once to all football fans (25). How to calculate from the percentage referring to one basic value to another basic value is discussed in the chapter on conditional probability (Sect. 9.2). Another example: According to UN reports, approximately 41 of all adults worldwide are illiterate. 23 of which are women. So what percentage of all adults

7

Percentage Calculations

33

are female and illiterate? What percentage of women are illiterate? (What would you calculate?). 1 2 1 4 · 3 = 6 is the answer to what? Let us assume that there are about the same number of men and women. 1 1 1 6 : 2 = 3 is then the answer to what?  “

1 of all people" is what percentage of the half of all people? 6



Now a comparison to the use of percentages and absolute numbers: In the first days of 2017, the newspapers reported that the New Year’s Eve fireworks had generated as much particulate matter as 15% of all road traffic in a year. Elsewhere it was said that it was as much as two months of road traffic. Do both figures fit together. Which do you find more frightening? Another interesting phenomenon with fractions and percentages: People often confuse the fraction 41 with 4% or 40%. The same applies to 16 as 6% or 60%. Of these, 60% is more than 40%, while 16 is less than 14 . But sometimes the numbers can be used mixed up: In Austria, about one year’s 16 boys currently complete their AHS (Allgemeinbildende Höhere Schule (general secondary school)) and about one year’s 14 girls. Note on content: At BHS (berufsbildenden höheren Schule (vocational secondary school)) about the same number—even a few more. Mathematically back to the AHS numbers: Since there are about the same number of boys and girls in a year, 60% of AHS graduates are girls and 40% boys. How come the numbers 4 and 6 are swapping? 1 6 1 4 4 = 24 and 6 = 24 . So the ratio of girls to boys is 6:4, that is, out of 100 school-leavers 60 are girls and 40 boys. The same phenomenon of swapped digits would occur if they were 13 and 17 (in which case 70% and 30% would result).

8

Irrational Numbers

8.1

Root Extraction

We have already encountered irrational numbers several times in this booklet, but only in the margins. We have found, for example, that roots of natural numbers that are not square numbers cannot be fractions. If you can no longer reduce a fraction, you cannot shorten its square either (see Sect. 6.1). The task of dividing a square into two triangles and two trapeziums as in Sect. 4.2 in such a way that the assembled figure fills exactly one rectangle, cannot be solved by fractions either. We have also found that there are decimal numbers with an infinite number of decimal places that do not represent a fraction. These are decimal numbers whose decimal places are not periodically repeated. Here is another example: 1.234567891011121314 … There is no mechanism for finding all such numbers. So you can’t put them in order. Therefore the set of irrational numbers is called uncountable. There is also no generation system that generates all irrational numbers from pairs of natural numbers or pairs of integers or from fractions (which could be analogous to the fact that fractions are formed by two numbers, the numerator and the denominator). A generation mechanism for finding a concrete irrational number is the formation of a nested interval. You approach the number from the left and the right as you like so that the distance between the left and right boundary is close to zero. As an example, the Heron procedure for the root of 2 is presented (generally for the root of a natural number a, which is not a square number) It was described around 100 AD by Heron of Alexandria in his first book Metrica, but can already be traced back to the time of Hammurabi I (around 1750 BC).

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8

Irrational Numbers

First, you can see that the value for the square root of 2 (or for the square root of a) must be between 1 and 2 (or a). 1 To get closer to the number, you can take the mean: 1.5 (or a + 2 ). The   1 2 root is slightly smaller than this mean, because 1.52 = 2.25 > 2 (or a + = 2 a 2 + 2a + 1 4

> 2a +4 2a = a, if a > 2). What do we take as an approximate value from   below? The number multiplied

by 1.5 gives 2 (or by the

a+1 2

value a). We get

2 1.5

=

4 3

or

a

a+1 2

=

2a a+1

. This

value is smaller than the root because the product of the two approximate values is exactly 2 (or a). The lower value was thus deliberately chosen. First, we found pairs of numbers with 1 and 2 (or a), then with ( 43 and 23 or 2a a+1 a + 1 and 2 ), between which the root must lie. 2 a (or Mean ) again. We can find the next pair if we take the mean and Mean 2 a Again the mean is the upper and Mean (or Mean ) the lower limit. Here are the next pairs for the root of 2: In again the mean of the previous two borders:  step17we calculate  4 the3next 17 24 24 :2 = + and 2: 3 2 12 12 = 17 (How far apart are the values? 17 = 288 17 289 204 and 12 = 204 . You can see that the fractions are once again minimally adjacent. Those were also 43 and 23 . One could now show that if this approximation procedure is continued for the following steps, it will be possible to specify minimally adjacent fractions as approximations). 577 1 1 It follows 816 577 and 408 (the difference is only 577 · 408 = 235,416 ). A pocket calculator gives the next approximate value only rounded and if you square this rounded value 1.414213562, the pocket calculator display will be exactly 2. 470,832 665,857 The exact values are: 1,331,714 and 470,832 . Let us look at the example a = 5. The root is between 1 and 5. 15 14 7 105 47 In the next step, there is then 53 and 3, 30 14 = 7 and 6 = 3 , 47 and 21 (a 4 difference of 987 ). As you can see here, the fractions are no longer minimally 4935 adjacent, but they are still very close to each other. 2212 and 2212 987 form the next pair of numbers and then the approximate value 2.236073705 (compared to the approximate value 2.236067977, which the calculator gives for the square root of 5). You could also start the procedure with the approximate values 2 and 25 , √ because you know that 5 is between 2 and 3.

8.2 The Golden Ratio

37

9 Then the minimally neighboring fractions would result in 20 9 and 4 . Next 360 161 115.920 51.841 values are then 161 and 72 . Then the result is 51.841 and 23.184 . The smaller value of the two √ corresponds to rounded 2.236067977, that is, the pocket calculator value of 5. The value squared results in 4.999999998. Instead of starting the approximation with 1 and a, you will reach your goal even faster if you start with the integer part of the root. The Heron method also has a geometrical interpretation: The lower and upper approximations are chosen in such a way that they can be regarded as the side lengths of a rectangle with the area a. The product of the two side lengths always gives a. Due to the approximations, the side lengths become more and more similar, the rectangle thus approaches a square. We are looking for exactly the side length of a square whose area is a.

8.2

The Golden Ratio

The next irrational number is also about a rectangle and a square. This time a rectangle of the equal area is to be obtained from the square. Do you remember the task of dividing a square into two triangles and trapeziums in such a way that the rectangle can be assembled from the surface pieces? (see Fig. 8.1 and Sect. 4.2). So the slope 1 −1 x (slope in the triangle) must be exactly the same as the slope x − (1 − x) (slope in the trapezoid above). x This results in the Eq. 1 − x = 2x x− 1 . Multiplied by x: x − x 2 = 2x − 1 and from this x 2 + x − 1 = 0.

Fig. 8.1 Division of a square into an equal area rectangle

38

8

Irrational Numbers

Fig. 8.2 Distribution of a route in the golden ratio

The quadratic solution formula leads to x = √ −1 + 5 2

−1 ±

√

12 − 4(−1) . 2

Since it is a

is not a fraction, but is known as length, the minus solution is omitted. the so-called “golden ratio.” The golden ratio divides a length so that the ratio of the longer part to the total length is exactly the same as the smaller part to the longer part (Fig. 8.2). Thus x1 = 1 −x x , again multiplied by x: x 2 = 1 − x or x 2 + x − 1 = 0. The golden ratio is often found in architecture, also in humans, the navel often divides the body in the golden ratio (and the knees against the lower part and the neck the upper). You can measure at your body. See e.g., the compilation by Dr. Dr. Ruben Stelzner: Der goldene Schnitt – Das Mysterium der Schönheit (The golden ratio—The mystery of beauty at https://www.golden-section.eu/ home.html). The golden ratio can be well approximated by the ratio of consecutive Fibonacci numbers. For this purpose, consider the golden rectangle (see Fig. 8.3), that is, a rectangle whose sides are in the proportion of the golden ratio. If you cut off the square above the smaller side in this rectangle (done on the right side here), a golden rectangle remains (on the left), because the square divides the longer side in the golden ratio, so that the shorter side, that remains in relation to the other side of the remaining rectangle (which corresponds to the longer part), again corresponds Fig. 8.3 Spiral in a golden rectangle

8.2 The Golden Ratio

39

Fig. 8.4 Fibonacci spiral

to the golden ratio by definition. And so one can (theoretically) continue indefinitely. The fact that one can continue indefinitely also speaks for the fact that there is no fraction that determines the ratio exactly. Similar to their rule in “the miraculous multiplication of areas” (see Sect. 4.2) 3 5 8 , 5 8 , and 13 are good approximations for the golden ratio. The Fibonacci spiral in a Fibonacci rectangle is therefore very similar to the golden spiral in a golden rectangle. Only on the very inside is it finite and begins in 2 squares with a semicircle (see Fig. 8.4).

9

Probabilities

9.1

Probability Approaches

“Probability always has something to do with uncertainty” (Büchter and Henn 2004, p. 133). The higher you state a probability, the more certain you seem to be. There are several ways to arrive at such a classification of (un-)certainty. The probability values that are named after the mathematician Laplace (1749– 1827) are the probabilities that arise when a result set with a finite number of possibilities equally probable possibilities is obtained. The proportion f avorable all possibilities then indicates the probability of the associated event. If you roll a normal die and you ask what probability that a number divisible by 3 is rolled, two possibilities are favorable: 3 and 6. In total there are 6 possibilities of how to roll the die. So the probability you are looking for is 26 = 13 . An important question in this approach to probabilities is how to count all possibilities appropriately. Now I had a die someday, which showed the high numbers (4, 5 or 6) remarkably often. The probability of this would be 36 = 21 = 50%. It was 18 out of 30 times the 18 6 = 10 = 60% in this case is the relative frequency of “a high number of case. 30 eyes”. Relative frequencies are often considered as probabilities when no other data are available. However, 30 times rolling the dice is not enough to say anything about the dice’ behavior in the long run. One would have to set up a much longer series of tests. The law of large numbers says: “With increasing numbers of experiments the relative frequency of an observed event stabilizes” (Büchter and Henn 2004,

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Probabilities

p. 145). For this purpose, the corresponding random experiment must be carried out again and again under the same conditions. So you would have to throw the dice several hundred times, or even several thousand times, to see whether the theoretical probabilities do not apply to it, that is, whether it is a so-called “loaded” dice. Finally, there are subjective probabilities (“I am 80% sure that this team will win the game”). Such probabilities do not seem too mathematical at first. However, since one adapts one’s subjective view to “reality” from time to time through experience, such probabilities can also lead to sound judgement. This is the so-called Bayesian concept of probability. According to Bayes, a posteriori probabilities can be calculated from a-priori probabilities through additional information. This brings us closer to suitable estimates (see Büchter and Henn 2004, p. 183.).

9.2

Conditional Probability

In the case of conditional probabilities, there are already preconditions that must be met. Under these preconditions, a statement is then to be made as to how things could go on. It is similar to the question of relative frequency (the percentage) in relation to the basic value. You may remember the four-field board with the football fans. What percentage of women are football fans? Here is the condition that it must be a woman. When you ask what percentage of football fans are women, the precondition is that they must be football fans. Such a question becomes particularly serious when it comes to a disease investigation. Does a positive test result in a virus test already indicate that you really carry the pathogen inside you? How likely is a false alarm? With frequently used tests, the probabilities of false diagnoses are already known. These values are probabilities or relative frequencies for false diagnoses under the precondition that someone has the virus (or the previous knowledge that they do not have it). What is required, however, is the probability under the condition that someone receives a positive test result, that is, precisely the opposite case with regard to prior knowledge. Example of the AIDS test The prevalence indicates which part of the population under investigation is likely to carry the virus. In Germany, it is assumed that about every 1000th person has

9.2 Conditional Probability

43

the virus in his body. If one belongs to a risk group, the prevalence is significantly higher. The sensitivity indicates which part of the infected persons is recognized as such. It is a conditional probability of knowing “the person carries the virus.” The sensitivity should be very high. We assume it is 99%. The specificity indicates which part of the non-infected is recognized as such. This conditional probability should also be very high. We assume 99% again. How likely is it that someone who receives a “positive” test result really carries the virus? We look at 100,000 people who take the test and see what test results can be expected (not that it will happen the same way, but this distribution is the most likely) (see Fig. 9.1). A positive test result with these numbers will therefore be 1098 people. Have you seen what the problem is? Only 99 of these are actually infected, which means 99 = 9.02%. 1098 Therefore, one must not conclude from a positive test result that one really has the virus inside oneself. But you certainly belong to a risk group, namely to the group of those who tested positive, whose risk is about 10%. If you now do a second (independent) test, the prevalence is about 10%. The test must be independent of the first test. It should not make the same mistake again, but there is a certain risk of making another mistake that is just as likely. 1 , extrapolated to 100,000 people from this risk group, the For p = 10 consideration looks like Fig. 9.2.

Fig. 9.1 AIDS testing task tree diagram

44

9

Probabilities

Fig. 9.2 Tree diagram for the second AIDS test

Table 9.1 Four-field table on the AIDS testing task

Infected

Not infected

Positive test

9900

900

10,800

Negative test

100

89,100

89,200

10,000

90,000

100,000

A positive test is expected for 10,800 people. Of these, 9900 are really infected, which means… 9900 = 91.60%. 10800 So if you get a positive test again in the second round, you have to be seriously concerned. One could also present the relationship in a four-field table (the disadvantage being that the prevalence, sensitivity, and specificity can no longer be entered directly). For p = 0.1 and 100,000 people tested, it looks as shown in Table 9.1. Which presentation do you like better? The tree diagram or the four-field table?

9.3

Hypothesis Testing

In a sense, the AIDS test discussed earlier is also a hypothesis test. You test the hypothesis that you have the virus inside you. Usually what is called hypothesis testing in everyday statistics is a little bit different. One does not know about prevalence and, to use the analogy with the AIDS test, one tries to make the

9.3 Hypothesis Testing

45

specificity and sensitivity as high as possible (e.g., 95% or more). However, only conditional probabilities can be given with the hypotheses as preconditions. The conditions cannot be interchanged. Let us remember the die I mentioned in Sect. 9.1. It showed remarkably high numbers. So one hypothesis could be that there is something wrong with the die. The probability of a high number could be greater than 50%. One will test the hypothesis that the probability is equal to 50% against the hypothesis that it is greater. To do this, you have to consider how often you want to roll the dice and where you want to set the limit. Suppose we decide to roll the dice 100 times. At what point do we want to believe that there is something wrong with the dice? When we rolled the die 30 times, 60% of the dice had high scores. Is that unusual? If we rolled the die 100 times, 60 high scores would probably seem unusual. Should we then classify 55 as high—or is that within the normal range of fluctuations? A measure for the usual fluctuations is the so-called dispersion or standard deviation. Here, the so-called square root n law applies: If we roll the die four times as often, the standard deviation, that is, the range around the expected value, which has also quadrupled, should increase. However, not four times as large, but only twice as large. If one rolls the die 100 times as often, the result should not deviate 100 times, but 10 times at most. Take the die example: If you roll 10 times, you would expect 5 times a high number. If that happens 3–7 times, you probably wouldn’t be surprised. Now, if you roll 1000 times, you wouldn’t be surprised about 490 high numbers either, so don’t just consider the 498–502 (i.e., plus/minus 2) range as ordinary. But only 300 times or 700 times would be very unexpected. If you roll the die 100 times as often, you should rather get 10 times larger deviation, that is, 480–520 as the expected range. Are then 470 already a little or 530 already unexpectedly much? √ As a rule of thumb for the standard deviation, there is the formula npq and anything more than twice the standard deviation is very suspicious. “Significant” is what you call such a deviation from the expected value np. The number n indicates the number of attempts (how often the dice are rolled), p is the probability that the desired result will be achieved, q = 1 − p is the opposite probability. In many studies, the probability of a significant result would be less than 5% if the hypothesis that there is no particular case (e.g., the die is perfectly ok) was true. In other words, assuming that the hypothesis is correct, the probability of a significant result is less than 5% for most experimental designs. In the example discussed so far p = q = 0.5. n was first 30 and then 100. √ With n = 30 one expects np = 15 high numbers and the standard deviation is 7.5 = 2.74.

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Probabilities

Thus 18 = 15 + 3 is less than 15 + 2 • 2.74 and my original observation is not significant. √ With 100 dice, you would have to get over 50 + 2 · 25 = 60 high numbers for the result to appear significant. So 60% of 30 is not yet significant, 60% of 100 would be the limit of significance. And if one has as a counter-hypothesis: “The probability of a large number is 60% for this cube,” a test with 100 attempts would not be selective. This is because the expected 60 high numbers are not yet significant for the “normal” die, so they could also happen by chance with a normal die. If you want the limit for significance to lie exactly between the 50% for the normal cube and the 60% for an appropriately prepared cube, you would have to roll the die at least 400 times, four times as often, so that the standard deviation is then twice as large and the scatter ranges no longer overlap. Let’s recalculate again: 50%, that is, 200 high scores, would be expected for a normal die. √ The standard deviation is around 400 · 0.5 · 0.5 =10, 200 + 2 · 10 = 220 (and this corresponds to 55% of 400). For the counter hypothesis (60%) one would expect√400 · 60% = 240 high numbers of eyes with a standard deviation of thus 400 · 0.6 · 0.4 = 9.8 and 240 − 2 · 9.8 = 220.4. 220 would be a really good limit to distinguish cubes with 50% probability from those with 60% probability. Why is the standard deviation at 60% slightly smaller than that at 50%? This is because of one term pq = p(1 − p) under the root. This term yields the largest √ value when p and q are the same size, that is, each is 50%. Thus: npq ≤ √ √ √ n · 0.5 · 0.5 = 0.5 n. Thus, double the standard deviation is always ≤ n, √ that is, a deviation of more than n speaks for a significant result. Note: Why does p(1 − p) give the largest value for p = (1 − p) = 0.5? One could recalculate it for different values of p. Or you could think about looking at the distance from p to 0.5 and name it h. Then p(1 − p) = (0.5 − h)(0.5 + h) = 0.52 +0.5 h −h 2 = 0.52 −h 2 this formula applies regardless of whether p = 0.5 + h or 0.5 − h, that is, regardless of whether p is greater or less than 50%. 1 − p is always the other factor. The value 0.52 – h 2 is greatest for h = 0, that is, when p is really 0.5. A summary of the results for a test where the die is rolled 400 times is shown in Fig. 9.3. Where did the 2.5% come from? 95% is approximately the probability that the √ √ result falls within the interval between np − 2 npq and np + 2 npq (expected

9.3 Hypothesis Testing

47

Fig. 9.3 Hypothesis test for 400 times dice

value − 2 standard deviations and expected value + 2 standard deviations). 5% of the test results do not fall within this interval, 2.5% is below and 2.5% is above it. Another example with a similar consideration: In the first Bundesliga the probability of a draw has been around 41 for many years. This probability does not come from a theoretical consideration but is a relative frequency based on years of experience. After all, there are 306 matches a year. If you currently look at the table of the second Bundesliga (as of March 21, 2017), after the 25th match day (i.e., after 9 · 25 = 225 matches) you can calculate that there have been 66 draws so far. In the first Bundesliga, only 51 are playing (after 225 matches as well). 66 is clearly above the expected value of 41 · 225 = 56.25 and 51 is below. √ The rule of thumb “a deviation of at least one times of n is significant” tells √ us that both deviations do not speak significantly against the usual quarter ( 225 = 15—nicely enough, the root can even be given exactly). √ The more accurate value of 2 npq is about 13 and even such a deviation has not yet been reached. 66 225 = 0.29333 . . . However, the fact that about 30% of the games are played in the second half of the season speaks for itself. Could we say at the end of the season whether 25% or 30% is the better figure for the second Bundesliga (see Fig. 9.4)? How √ do you attain the number 94 or 75? 306 = 17.5 as a first approximation for twice the standard deviation.  √ 1 3 It would be more precise 2· 306 · 4 · 4 ≈ 15.1 or 2 · 306 · 0.3 · 0.7 ≈ 16.

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Probabilities

Fig. 9.4 Hypothesis test for the frequency of draws in the Bundesliga

Fig. 9.5 Draw in the Bundesliga in 4 seasons

Even with the more accurate values, there is such a large overlap range (92 – 16 = 76) and (76.5 + 15.1 = 91.6) that the expected values of one variant are just at the edge of twice the standard deviation of the other variant. A season can therefore not clarify whether there are a 5% more or less draws. To do this, one would have to look at four seasons (see Fig. 9.5). As you can see, this is quite selective (only at 335 or 336 would it still be uncertain which of the two probabilities fits better). As in the dice example before, one situation (100 dice rolls or one season) was such that the second hypothesis was just the limit of twice the standard deviation of the first hypothesis. Each time, the number of attempts had to be quadrupled in order to be able to distinguish the two hypotheses well, because quadrupling the attempts quadruples the expected value, but only doubles the standard deviation. So let’s go back to the current tables (status: March 2, 2017). If we didn’t know anything about the long-term draw rate of 41 and we only had the two numbers: 51 draws in the first Bundesliga and 66 draws in the second Bundesliga, then you can √ see that the difference is exact 15 = n. So it looks like the second Bundesliga will probably have slightly more draws than the first Bundesliga. But it could be that the “truth” is somewhere in the middle (maybe near 117:450 = 0.26—that is, a total draw due to matches in total) and that there were “by chance” a few

9.3 Hypothesis Testing

49

draws less in the first league and a few more in the second league. This thesis cannot be rejected at the significance level of 5%. How good it is that this is not the first Bundesliga season and we can check if there have been such differences in the past. In the last four seasons it was like this: 15/16 there were 71 draws in the first Bundesliga, 86 in the second Bundesliga (so this time there were 15 more in the whole season—which also looks like significance), 14/15 there were 82 in the first league and 88 in the second, this time you can’t speak of a real difference. In 13/14 the difference was even bigger: 64 to 88, and in 12/13 there was also a clear difference (although not quite matching the significance level) 78 to 90. The first Bundesliga, at 295 in these four years, remains below the expected value of 306, but not so much below that one would have to reject 41 the hypothesis of “ 41 ” (twice it was also just above it). The second Bundesliga even reaches a total of 352 draws, which does not quite reach the 30% limit, but as an approximation, 30% is a significantly√ better hypothesis than 25%. And the difference of 57 clearly √ surpasses this n = 4 · 306 ≈ 35, albeit not twice. One can therefore say with a relatively clear conscience: There are more draws in the second Bundesliga than in the first Bundesliga. Another example from everyday life: The 13 in the lottery. The first number drawn in the lottery when it was introduced in 1955 was the number 13, but soon after it was recognized that the number 13 was the least drawn in the lottery and it remained that way throughout the years. So is 13 an unlucky number, at least when it comes to playing the lottery? I found data from 1974 in my math book from my school days (and in the schoolbook at that time) and current data from the internet at “dielottozahlende” (as of March 14, 2017). I have therefore divided it into the two periods 1955–1974 and 1974–2017 (see Table 9.2). Table 9.2 The frequency of 13 for the lottery numbers

Period

1955–1974 1955–2017 1974–2017

Number of draws

996

5572

4576

Expected value for 122 13

682

560

Frequency of 13

96

619

523

Standard deviation 10

24.5

21.4

μ-2σ

633

517

“Significant.”

102

50

9

Probabilities

6 Why do you have to count on p = 49 ? Because 6 out of 49 numbers are drawn 6 every time, so every number has the probability of being 49 in a draw. As you can see, 13 was significantly less likely to be drawn in the first 20 years. Also in terms of the total time, it is significantly less likely to be drawn. But if you look at the second period alone, 13 still occurred less than expected, but this rarity value is no longer significant. So the rare occurrence at the beginning is what makes it so rare in the whole period. You can now decide for yourself whether you would not tick 13 on the lottery ticket for these reasons or whether you would give this “unlucky number” the chance to become a lucky number for you.

Various Mean Values

10

Most of the time, mean values are thought of as the arithmetic mean between two numbers. For 1 and 2, 1.5 would be taken as the mean value. For any number a and b the value is 21 (a + b). Perhaps if a is the smaller number, you would calculate, a + 21 (b − a) that is add half the distance to a and half the distance to b. As you can easily calculate, this is also 21 (a + b). But there are also other mean values. I want to make you aware of two important mean values with two questions. 1. The world population doubled in the last century about every 40 years.

2000

About 6 billion (bln)

1960

3 billion

1920

1.5 billion

How many people were there in 1980? How many in 1940? One is inclined to give the Fig. 4.5 billion for 1980 and 2.25 billion for 1940. But why should the population have increased by 0.75 billion each from 1920 to 1940 and from 1940 to 1960, and by 1.5 billion each in the two periods thereafter (1960–1980 and 1980–2000)?

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Various Mean Values

Let’s take a closer look at the growth again: Starting at 1920

1.5 (billion)

After 40 years

1.5 · 2

After 80 years

1.5 · 2 · 2

After 120 years one would expect: 1.5 · 2 · 2 · 2 (the connection does not apply beyond 2000!). after n · 40 years: 1.5 · 2n . And if you now no longer use a natural number for n, but 0.5 or 1.5 (then you get the data for 1940 or 1980): 1

√ 2 ≈ 2.12

After 20 years

1.5 · 2 2 = 1.5 ·

After 60 years

1.5 · 2 2 ≈ 4.24

After 1 year

1.5 · 2 40 = 1.5 · 1.017 ≈ 1.53 (for verification: 1.01740 ≈ 2) x  x 1 1.5 · 2 40 = 1.5 · 2 40 ≈ 1.5 · 1.017x

After × years

3

1

a = 1.017 also means growth factor. The population grew by about 1.7% every year. Is it possible to calculate the 2.12 as the mean value of 1.5 and 3? And 4.24 from 3 and 6? √ Yes, there is the so-called geometric mean: m geo = a · b. √ √ 1 3 Note: 1.5 · 2 2 = · 1.5 · 1.5 · 2 and 1.5 · 2 2 = 1.5 · 2 · 1.5 · 22 . What does that have to do with geometry? If a and b are the side lengths of a rectangle, then the geometric mean is the side length of an area equal square. Perhaps you remember the Heron method for calculating the roots of a number a. The left and right limits were each chosen so that the product was a. You could draw a rectangle with the area a. Now the mean value was searched for, for which the corresponding rectangle is a square with the same area. So the geometric mean is searched for. The arithmetic mean is used as an approximation. In this context it was also shown that the arithmetic mean is always a little bit larger than the geometric mean. A comment on the world population, if it had always developed as it did in the twentieth century: When would Adam and Eve have lived? Can you get that without a calculator and logarithm? Yes, you can. In the twentieth century we had a doubling time of about 40 years.

10

Various Mean Values

53

Ten doubling times result in 400 years. So in 400 years, two people become 2000 people (not 40, as one might be inclined to say at first, because 210 = 1024, so about 1000). In a further 10 doubling times, this will result in 2 million people (again 1000). We are at 800 years. Another 400 years and we’ll have reached two billion. Double it twice more (so 2 · 40 years more) and we have 8 billion. So it might be about 1300 years since Adam and Eve. This calculation thus shows that population growth in the past could not be as intensive as it was in the twentieth century. 2. You drive a distance of 120 km on the country road. You advance at an average speed of 60 km/h. On the way back you take the motorway (same distance). This time you advance with an average speed of 120 km/h. What is the total average speed? Again, the arithmetic mean is 90 km/h. But why is it wrong? Because the lower speed means that you can travel much longer, twice as long. Specifically, you need 2 h for the forward journey and only 1 h for the return journey. So a total of 3 h for 240 km. The average speed is therefore 240 km/3 h = 80 km/h. This is the so-called harmonic mean: Be the speed for the forward journey a and for the return journey b. Let the distance be s long. So for the forward journey you need: as h, for the return journey bs h (if you are no longer sure which value belongs in the numerator and which in the denominator: The longer the distance s, the more time is needed, so s must be in the numerator. The higher the speed a is, the less time is needed, so a must be in the denominator. Or you check with the unit check: km km = h). h

+ sa Total time: as + bs = sb ab . And this for a distance of 2 s. sb + sa Thus the average speed is: sb 2s = 2s · sb ab + sa = 2s : ab + sa = 2ab b +a.

ab

2sab s(b + a)

=

You can see that the length of the track is not important for the average speed. It shortens out.   1 1 1 1 Instead of mharm = b2ab + a you can also write m = 2 a + b , which may express the mean value property even better. If you calculate the three mean values for some pairs of numbers, you can see that the harmonic mean is even smaller than the geometric mean.

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10

Various Mean Values

Fig. 10.1 Different mean values of a and c

This can also be explained by the remarkable connection that the product of the harmonic with the arithmetic mean is the square of the geometric mean. Or to put it another way: In this book we have already calculated several harmonic averages. This is because the lower limits, which result from the Heron method, are the harmonic means of the two previous limits. This should now be recalculated: √ 2 2ab a+b · = ab = ab b + a 2 And how can you remember that it is called “harmonic mean”? Harmony is present when the relations are right (in German the word “Verhältnisse” can also mean proprotions as well as relations). The ancient Greeks already knew that. That’s why there is a harmony theory in music. The frequency ratios between the tones must be correct. Then it sounds harmonious. You can plot the mean values of two positive numbers as three lengths in a trapezoid (Fig. 10.1). At half height the arithmetic mean m is found. Through the point of intersection of the diagonals, which intersect in the same ratio, namely in the ratio of the side lengths a and c, the line leads to the harmonic mean h. The distance to the geometric mean g is such that the trapezoid is divided into two similar trapeziums, that is, a:g = g:c or g2 = ac.

Conclusion

11

Overall, I hope that you have learned in this book: It may be that you have to deal with fractions in your life again and again (and sometimes it even becomes irrational). But if you look at it, you can put things in order again and maybe find a harmonious centre (mean) in your life.

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What You Learned From This essential

• Fractions can be indicated by various equivalent pairs of numerator and denominator. The reduced version is usually used. • When adding and subtracting, fractions are usually imagined as parts of a whole. Multiplication and division can also be explained in a meaningful way. Multiplication and division can also be interesting for fractions that describe a ratio. • The Simpson paradox is about ratios in data that show opposite tendencies depending on how you read them—separately or together. • Fractions can be converted to decimal numbers by performing the division. Finite and periodic decimal numbers can be converted to fractions. • With percentages, it is very important to know what is seen as a basic value. √ • Test results that deviate at least from the expected value n are suspect. Such a deviation should be “significant”. If one compares two series of mea√ surements, a deviation of 2 n speaks even more for a significant difference between the series of measurements. • When asking for the mean value, one has to look very closely at the specifications. There are different mean values (e.g., the harmonic, the geometric, and the arithmetic mean).

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Literature

Bates, B., Bunder, M., & Tognetti, K. (2010). Locating terms in the Stern-Brocot tree. European Journal of Combinatorics,31(3), 1020–1033. Bickel, P. J., Hammel, E. A., & O’Connell, J. W. (1975). Sex bias in graduate admissions: Data from Berkeley. Science,187(4175), 398. Grams, T. (2016). Klüger Irren – Denkfallen mit System. Heidelberg: Springer. Heigl, F., & Feuerpfeil, J. (1976). Stochastik (2nd ed.). München: Bsv. Paulitsch, A. (1993). Zu Gast bei Brüchen und ganzen Zahlen (3rd ed.). Hallbergmoos: Aulis. Robert Koch-Institut. (2016). Bericht zum Krebsgeschehen in Deutschland 2016. Berlin: Robert Koch-Institut. Székely Gábor, J. (1990). Paradoxa. Klassische und neue Überraschungen aus Wahrscheinlichkeitsrechnung und mathematischer Statistik. Frankfurt a. M.: Harri Deutsch. Tan, A. (1986). A geometric interpretation of Simpsons paradoxon. College Mathematics Journal, 17(4), 340.

Internet source for drawn lotto numbers https://www.dielottozahlende.net/lotto/6aus49/statistiken/haeufigkeit%20der%20lottoza hlen.html. Zugegriffen: 3. Aug. 2017.

Further examples for the appearance of the golden ratio Dr. Dr. Ruben Stelzner: Der goldene Schnitt – Das Mysterium der Schönheit unter https:// www.golden-section.eu/home.html. Zugegriffen: 3. Aug. 2017.

Soccer statistics can be read for example at www.kicker.de

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