What to do without numbers? Communication should be impossible. Numbers play an indispensable role in human communication and economics. Do you want to know something about numbers?
In 'Talking about numbers' the author takes you on a mathematical voyage of discovery in number theory.
Talking about numbers, A mathematical voyage of discovery / Over getallen gesproken, Een wiskundige ontdekkingsreis
Base and sequel
The motivation to write this book mainly arose from a long mathematical study and interest. Because much information is available in both Dutch and English and because this book is a collection of work from a lot of mathematicians, information is provided in both languages. It makes the book accessible for a wide audience. This book is the 3rd, revised edition.
It contains improvements, but above all it is a substantial extension of the 2nd , revised edition. This extension includes 90 new sets of numbers and many special numbers.
The objective is to open a fascinating and informative world of numbers in number theory. The book is divided into four parts. Part I is a personal focus of the author. Part II pays attention to early and recent phrases done by wise men, to basicnumbers and to numeral systems. An overview of 337 mathematicians from 38 countries and a presentation of mathematical terminology is given.
They are part of the descriptions of the number sets in Part III. This third part is the lion’s share of the book and includes 619 descriptions of number sets. The description of each number set has been clarified with examples, the name or names of concerned mathematicians together with personal information. Of course, the collection of the number sets includes the well-known Natural numbers and Prime numbers. But moreover ABA numbers, Brilliant numbers, Fibonacci numbers, Lazy caterer numbers, Twin prime numbers, Quaternions, Weird numbers, Schizophrenic numbersand many others which are usually unknown. Some elaborated examples are:
It concerns solutions of the Frobenius problem or Coin problem. Suppose there exist only coins with a value of 5 and 9 cents. Then the largest monetary amount that cannot be obtained using only 5 and 9 cents is 31. This number is called the Frobenius number. It can be calculated by a general formula. Applied to the numbers 5 and 9 it shows: (5 – 1) ∙(9 – 1) – 1 = 31.
The International Bank Account Number (IBAN) belongs to these numbers To check if the IBAN is valid, the 11-proof will be applied. Is the number valid then the control number, that follows the country code, is computed.
It concerns integers that are based on the Collatz conjecture. Take any positive integer n. If nis even, divide nby 2. If nis odd, multiply nby 3 and add 1. Repeat this process and the final number will be 1. For example:
- n= 7, then the sequence according to the two rules is: 22, 11, 34, 17, 52, 52, …, 4, 2, 1.
It concerns an integer sequence in which each number is the product of the previous numbers plus 1. The sequence starts with 2, 3, 7, 43, 1807, 3263443, …
The extension in this 3rd , revised edition includes 90 new number sets, including:
- - Braille numbers: The digits 0, …, 9 are represented by blocks of 3 x 2 positions.
- - Brazilian numbers: The numbers are represented by one and the same digit.
- - Chinese remainder numbers: Numbers that leave a given rest after division.
- - Descartes numbers: Odd numbers which are odd perfect numbers.
- - Factories: Numbers that are equal to the sum of the factorials of the digits.
- - Obstinate numbers: Positive odd numbers which cannot be expressed as sum of a power of 2 and a prime number.
- - Maya numbers: Numbers that belong to the vigesimal (base-20) positional notation.
- - Ramsey numbers: Numbers that are the solution of the party problem.
- - Clock numbers: The 12 clock numbers are represented by arithmetic expressions.
- - Black hole numbers: Numbers which are the final result of operations on natural numbers.
Following the 619 number sets, special numbers are given with amazing properties, followed by numbers that appear in the Bible, with surprising mathematical relationships.
Some examples of special numbers
- - Münchhausen number: 3435 = 33+ 44+ 33+ 55
- - pi-Approximation Day: 22/7= 3.1428… or July 22
- - 221859= 223+ 183+ 593
- - 1666= MDCLXVI, the year with all Roman numbers in descending order.
Some examples of Biblical numbers
- 61 the 18th Prime number. 2 Peter is Bible book 61 and contains 3 chapters.
The chapters contain successively 21, 23 and 18 verses. These numbers added together results in 61.
- 153 = 13 + 53 + 33is a Narcissistic number
- 666 = 1 + 2 + 3 + 4 + 567 + 89 = 6 + 63 + 6 + 63 + 6+ 63
- 6 and 28 are the only Perfect numbers
The Biblical numbers in this 3rd , revised edition are filled up with numbers that indicate money (e.g. dram, talent), weights (e.g. gerah, shekel), lengths (e.g. day’s journey, stadia,), time (e.g. third hour, watch), liquid measures (e.g. hin, firkin), dry measures (e.g. cab, cor).
Finally part III ends with two problems. The first problem (Is P = NP?) is waiting for solution and the second problem (Fermat Point) has already been solved a long time ago.
Part IV includes references to literature and websites. It finishes with two bulky indices of number sets in alphabetical order both in Dutch and English. Both indexes also contain a sub index with number sets that have more than one name.
Lecturers and students in mathematics and technique and everyone who is interested in or curious about surprising characteristics of numbers. Because of the book is written in Dutch and English, besides not a 1:1 translation, the book is not only suitable for the Dutch reader but also for non-Dutch.
The book focuses on 619 number sets part from number theory. The chapters about special numbers and Biblical numbers present an additional view on numbers. Many special numbers are presented like the well-known and e, but also numbers in relation to temperature, universe and constants. The chapter about Biblical numbers highlights on numbers that have an explicit mathematical relationship. It may be considered as an unique approach.
Title: Talking about numbers, A mathematical voyage of discovery 3rd, revised edition / Over getallen gesproken, Een wiskundige ontdekkingsreis 3de, herziene druk
Author: Prof. Dr. Ir. Maarten Looijen
ISBN: Hard copy 978 94 018 0028 0 / eBook 978 94 018 0601 5
Price: € 30,79 (ex VAT € 28,25)
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