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2021 | Book

Introduction Read chapter Read first chapter

Fractions, Ratios, and Roots

Rediscover the Basics and Learn About Interesting Applications

Author: Dr. Renate Motzer

Publisher: Springer Fachmedien Wiesbaden

Book Series : essentials

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Renate Motzer introduces the world of fractions and connects them with decimal numbers. She clearly shows that fractions can be understood as parts of a whole, but also as ratios of two quantities. The author clearly shows why roots cannot be exactly indicated by fractions, how to find good approximations and why an unusual addition of fractions can lead to paradoxical results. Furthermore, she explains the use of fractions in percentage and probability calculations in a practical way and finally discusses different ways of forming mean values.This Springer essential is a translation of the original German 1st edition essentials, Brüche, Verhältnisse und Wurzeln by Renate Motzer, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2018. 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, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
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
Renate Motzer
Chapter 2. What Are (Common) Fractions?
Abstract
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.
Renate Motzer
Chapter 3. Fractions as Parts of a Whole
Abstract
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 make sketches as in Fig. 3.1.
Renate Motzer
Chapter 4. Fractions as Ratios
Abstract
Besides the size and 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 so on).
Renate Motzer
Chapter 5. Ordering Fractions
Abstract
The ordering of fractions was already mentioned when comparing the gradients. For fractions with positive denominator and numerator, the following applies: \(\frac{a}{b} < \frac{c}{d}\) 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).
Renate Motzer
Chapter 6. Decimal Fractions
Abstract
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).
Renate Motzer
Chapter 7. Percentage Calculations
Abstract
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.
Renate Motzer
Chapter 8. Irrational Numbers
Abstract
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 your square either (see Sect. 6.1).
Renate Motzer
Chapter 9. Probabilities
Abstract
“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.
Renate Motzer
Chapter 10. Various Mean Values
Abstract
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 \(\frac{1}{2}\left( {a + b} \right)\).
Renate Motzer
Chapter 11. Conclusion
Abstract
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.
Renate Motzer
Backmatter
Metadata
Title
Fractions, Ratios, and Roots
Author
Dr. Renate Motzer
Copyright Year
2021
Electronic ISBN
978-3-658-32574-9
Print ISBN
978-3-658-32573-2
DOI
https://doi.org/10.1007/978-3-658-32574-9

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