Book I. Construction of numbers, length and area
Publisher: Naumov I.
Published: 2018
Place of publication: Finland
Language: eng
Format: eBook
Number of pages: 171
Description: The 1-st book is done to provide a complete construction of numbers, length and area. It is obviously the most reasonable way to start a course of mathematics. Almost everyone knows that when we operate with numbers, we do it according to the certain rules. Unfortunately, there is never any explanation of where these rules came from, despite the fact that
it is a very important question, to say the least.
Really, almost anyone knows the “sign rules” [A] or a commutative law [B], or an associative law [C]. But it’s not typical of people to ask: “Why are these laws true?”. If someone gets a “math education”, he will be told that the real numbers R form a “field” and even a “complete field”, and that’s where all the properties like [A],[B],[C] came from. But it explains
nothing, because it immediately leads to the question: ”Why do the real number R form a field?”.
And the better person knows math, the easier he can understand that there must be some good
explanation of all these properties, as they aren’t obvious at all. So, there are “infinitely many” real
numbers. Why do all of them without any exceptions obey to the certain rules? Why can we take
some line and compare exactly one (appropriate) real number to every point of that line, such that
every real number will be compared to some point? I’ve never seen any students curriculum where
any of these fundamental questions are normally explained.

Is it so because it is something obvious or something insignificant? No, in reality, not at all.
Eventually, there exist only one reasonable way to answer all these questions - to provide a
complete and clear construction of the field R . We will start from the simpliest basic objects and
then, step by step, we will be building real numbers, and during the process we will check
straightly all their properties, without accepting any of them for granted. As far as I’m concerned,
the numbers construction is one of the most amazing and the most important part of mathematics. There exist several known approaches to such construction, and some of them have serious flaws, because if we follow them, most of the properties of numbers will stay unexplained. The main goal of this book is to provide the most convenient construction, where definitions are clear and there is a strict logical structure, there are no “gaps ” and “uncertainties”,
and there are no inconvenient and protracted proofs at all. My main goal was to give only a good quality information and to make the narrative consecutive and clear.
All The Best, Naumov Ivan
Table of contents: title: Natural numbers .............................................................................. 5; title: Groups, Rings, Fields.......................................................................... 33; title: Matrixes........................................................................................... 46; title: Integer numbers................................................................................ 53; title: Rational numbers.............................................................................. 73; title: Sequences and limits.......................................................................... 84; title: Real Numbers.................................................................................... 93; title: Construction of length........................................................................ 130; title: Construction of area........................................................................... 149; title: Angles.............................................................................................. 165; title: Literature.......................................................................................... 171