The Little Book of Numbers

2025
Buch

Verfasst von: Chris Brink
Verlag: Springer Nature Switzerland
Enthalten in: Springer Professional "Wirtschaft+Technik"; Springer Professional "Technik"; Springer Professional "Wirtschaft"

Über dieses Buch

Dieses Buch ist eine konzeptionelle Erforschung von Zahlen und bietet eine einzigartige Perspektive, die mathematische Erkenntnisse mit philosophischen Überlegungen verbindet. Es enthält Kapitel zu bestimmten Zahlen, um allgemeinere theoretische Fragen zu veranschaulichen, wobei das Konzept der Unendlichkeit besonders hervorgehoben wird. Der Inhalt ist so konzipiert, dass er zugänglich und unterhaltsam ist, wobei schwere mathematische Terminologie und Notation vermieden werden, um die Lesbarkeit zu erhalten. Das Buch enthält auch historische und biografische Auszüge sowie gelegentlich respektlose Kommentare und bietet eine zugängliche Einführung in die Welt der Zahlen. Dieses Buch richtet sich an alle, die mehr über die Philosophie der Zahlen wissen wollen, auch an diejenigen, die vielleicht keinen mathematischen Hintergrund haben, aber neugierig auf das Thema sind.

Mit KI übersetzt

Inhaltsverzeichnis

Frontmatter

1. 0 (‘Zero’)

Abstract

How can the absence of something be something? That, exactly, is the paradox of zero. If you go out and buy half a dozen eggs, then the number of eggs you have is six. If there are no eggs in the house, and you have already borrowed half a dozen eggs from your neighbour, then your egg count is minus six. But it seems a bit of a stretch, from there, to conclude that if you neither have eggs nor owe any eggs then, somehow, you do have a number of eggs, which is zero.

Chris Brink

2. 1 (‘One’) and Other Natural Numbers

Abstract

One is the simplest number there is. As much as zero is subtle, one is straightforward. It only does two things, but it does those two things so effectively that it is the ancestor of all numbers.

Chris Brink

3. √2 (‘The Square Root of 2’)

Abstract

Here is a little experiment. Take an A4 sheet of paper, and measure its sides, then use your calculator to find the ratio of its length to its breadth. Now find the middle of the long side, and then fold the A4 sheet into two equal halves at that point. Both of these halves are then sheets of size A5, as you probably know. Now repeat the experiment: measure the sides of an A5 sheet, and then calculate the ratio of its length to its breadth. You will find that this ratio is the same for an A5 sheet as for an A4 sheet. You can repeat the experiment with an A3 sheet, and you will find the same again: the ratio of length to breadth is the same for all these sheets.

Chris Brink

4. φ (‘Phi’ or ‘The Golden Ratio’)

Abstract

Here is a little geometric construction. Begin with a square, and divide it into two equal rectangles. Then find the diagonal of one of these rectangles, starting at a corner of the square. Use this diagonal as the radius of a circle, with its centre at the other end of the diagonal, and draw a circle-arc on one side of the square. Then extend the square to a rectangle, the breadth of which is just the side of the square, and the length of which is its base extended until it meets the arc you have drawn. The whole process looks like this:

Chris Brink

5. π (‘Pi’)

Abstract

A circle is one of those really simple mathematical objects which unlock a treasure trove of results. All you need to define a circle is a fixed point and a given length: if you collect together all the points which are that given length away from the fixed point, you have a circle. The fixed point is called the center, and the fixed length is called the radius of the circle. If you double the radius, you have the diameter of the circle, and the length of the circular path going once around the circle is its circumference. The diameter, then, is the longest straight line you can draw between two points on the circumference of the circle, and it goes through the center.

Chris Brink

6. e (‘Euler’s number’)

Abstract

Imagine that you have a $1000 to invest for a year, and Bank A is so keen to gain your custom that they offer to pay you 100% interest at the end of the year. However, when Bank B gets to hear of this, they offer to pay you 50% interest 6-monthly. Which would you choose?

Chris Brink

7. i (‘The Square Root of −1’)

Abstract

When it comes to public relations, the mathematicians can really shoot themselves in the foot. Not only do they call some numbers odd, and some numbers irrational, but then, to make matters worse, they absent-mindedly acquiesce in some numbers being called imaginary. This is quite awful terminology, because it reinforces a stereotype that mathematicians spend their time doing wonderfully complicated work on things that don’t really exist—and hence, by implication, don’t really matter.

Chris Brink

8. 10 (‘Ten’) and Other Numbers Famous for Being Famous

Abstract

Ten is quite an unremarkable number, except for one thing: we use it a lot. Every time you buy a liter of milk or a kilogram of flour, or walk a kilometer, or worry about the percentage interest on your mortgage, what you do is based on the number ten. For this ubiquitous manifestation of ten we can praise or blame the French Revolution of 1789.

Chris Brink

9. Prime Numbers

Abstract

The prime numbers are absolutely maddening. It is easy to understand what a prime number is, and given any number we know how to determine whether it is prime or not, but we have never been able to find a method which will generate the next prime number if you know all the previous ones.

Chris Brink

10. Large Numbers

Abstract

Here is a question: how long do you think it would take you to count to one million?

Chris Brink

11. Infinity

Abstract

I am sorry if this comes as a shock to you, but in mathematics the symbol ‘∞’ does not actually denote anything. There is no mathematical entity called ^‘ ∞ ^’. In particular, ‘∞’ does not denote a specific entity called ‘infinity’.

Chris Brink

12. What Is a Number?

Abstract

There are five fingers on your hand, five points to a pentagram, and five circles in the Olympic logo. But is there such a thing as a number called five? And if so, what kind of a thing is it? As in our Chap. 2 discussion of one: is five an entity, or an activity, or both? We can say five is a number, but that is just to put the question in more general terms. What is a number? More generally, do numbers really exist, or are they just a convenient fiction we have invented?

Chris Brink

Titel: The Little Book of Numbers
Verfasst von: Chris Brink
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-032-04863-9
Print ISBN: 978-3-032-04862-2
DOI: https://doi.org/10.1007/978-3-032-04863-9

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The Little Book of Numbers

Über dieses Buch

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. 0 (‘Zero’)

2. 1 (‘One’) and Other Natural Numbers

3. √2 (‘The Square Root of 2’)

4. φ (‘Phi’ or ‘The Golden Ratio’)

5. π (‘Pi’)

6. e (‘Euler’s number’)

7. i (‘The Square Root of −1’)

8. 10 (‘Ten’) and Other Numbers Famous for Being Famous

9. Prime Numbers

10. Large Numbers

11. Infinity

12. What Is a Number?

Backmatter

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