Exploring the History and Operations of Numbers: A Comprehensive Guide

[jd12345]

I want to study the history of numbers - the way we assigned symbols to numbers and then invented place values to make addition,subtraction easier. It should also deal with binary numbers, octals, decimals etc. And then how multiplication , division is defined.

Basically i need a book about numbers and how various operations are defined. It should explain it in the most rigorous and deepest way.

 

[chiro]

I want to study the history of numbers - the way we assigned symbols to numbers and then invented place values to make addition,subtraction easier. It should also deal with binary numbers, octals, decimals etc. And then how multiplication , division is defined.

Basically i need a book about numbers and how various operations are defined. It should explain it in the most rigorous and deepest way.

Hey jd12345.

In terms of arithmetic, normal algebra, and the different bases systems, the history is spread across a variety of times and places.

For example the sexigesimal system was used in the Babylonian times for a variety of reasons and is also used in angular and other periodic measures for things like geometry and time.

With regards to zero even becoming a number you can thank the Middle East for this crucial yet long due discovery.

With regards to binary, you can go back before even modern computers (before even things like the ENIAC) to some of the machine devices used to do calculations (one that comes to mind is Babbages device who was programmed by Ada consider by some to be the first programmer).

With regards to arithmetic, the idea of exponentiation was greatly influenced by Rene Descartes as he was the first to even think about talking about xⁿ instead of thinking about say x*x or x*x*x or some other thing that reduces purely down to multiplication.

As for the number systems, if you look at the history of the written language, you will see how the alphabets of various languages have transformed and evolved from the past to the present day. You will see that things have become more standardized and clearer with the development of language and various languages that have come and gone have had influence on what we currently use today across the world not only in structure and syntax, but in the graphical description.

You say want a single book to talk about this, but the truth is that things have developed in a non-local way where bits and pieces have been added in different ways.

If you want to define division and multiplication in terms of a basewise description, you just need to look at the high school formulas used and use a particular base.

The actual specifics for all of base 2 are defined in any introductory digital circuits book that cover addition, subtraction, multiplication and division for base 2. The exact same ideas translate into any base if you modify the rules for what to carry (just like you did for base 10 in high school).

Division is defined using the normal long division which uses a DIV/MOD algorithm that is used in programming all the time. The high school algorithm and the DIV/MOD algorithm that a computer uses are the same thing. Review this for a proper understanding of definining division in any base.

In terms of the symbols used, the base 10 system that we take for granted today was invented somewhere in middle east/india area.

Take a look at things 'like' this (to your own satisfaction):

http://www.endlesssearch.co.uk/philo_enneagram_dec.htm

For the rigorous definition in terms of some base, the above will tell you everything you need in terms of that. Most of the formal stuff in mathematics doesn't deal with this, but deals with defining the language, structure, and properties of things like sets, functions, and so on (and not bases and descriptions of arithmetic in terms of these bases), but if you get a copy of a digital electronics book for 1st year electrical/computer/telecommunications engineering, you'll get a list of all the base 2 algorithms that can be extended to any base.