Number System: Auotomatic Algebraic Solutions is an innovative text that introduces high quality methods for easily learning various concepts of number system. It is the outcome of deep analysis on the number system. The book is very useful for students, scientists, engineers, etc.

Chapter 1 covers many high speed methods for division by 9, 99, 999, etc, formula to create fast methods for any division. The division methods are also useful for computer programs.

Chapter 2 covers Raghavendra’s analysis (R-analysis) and its applications. R-analysis is the unique key for solving several problems. It is the easiest and best way for algebraically solving various problems of the number system.

There is only one method is there to algebraically derive formulae for finding the sums of first ‘n’ natural numbers, squares, cubes, etc. That method is R-analysis. Other than R-analysis there are several ways to derive formulae for finding the sums of these series, but none is merely an automatic algebraic derivation where we can just turn the crank and get the answer with out any insight.

For example in one previously used method, it is used a neat trick to derive formula for finding the sum of first ‘n’ natural numbers [that trick is: S_{n} = 1 + 2 + 3 + … + n and S_{n} = n + … + 3 + 2 + 1 2S_{n} = (n + 1) + (n + 1) + … up to n times (n + 1) 2S_{n} = n (n + 1) S_{n }= n (n + 1) / 2] but the same method (trick) completely fails to derive formula for finding the sum of other series;

similarly, in another method, {which is completely depend on the result of the [(n + 1)^{2} – n]} based on the result of [(n + 1)^{2} – n], remaining steps are created to derive formula. But same derivation cannot be possible if we give any value for ‘n’ in the starting steps, i.e., in the first step if we give any value for n, say for example 4, then we cannot able to do the same derivation from the result of [(4 + 1)^{2} – 4].

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The number system conversion methods developed using R-analysis is very useful for rapidly converting any base number to decimal equivalent and vice versa, useful for binary, octal etc. arithmetic operations and hetero base arithmetic operations, useful to invent many useful concepts of the number system and we study the applications of these conversion methods in the Chapter 3.

Chapter 3 covers Decimal Coded Binary (DCB) number system and detailed study of Quotient, Quotient Difference, Difference, Add and Step methods (which are created using R-analysis).

Using the Quotient method we can convert base x (when x ≥ 8) number to decimal number more rapidly than any other usual methods and using the same method we can perform decimal number to binary, octal, etc. conversion, Multi base conversions, binary, octal, etc. arithmetic operations, etc.

The Quotient method is very good for computer program, since in one program (formula) we can rapidly convert base x number to decimal equivalent and vice versa; it convert negative numbers also. But if we use usual method in the programs, it fails to convert negative numbers and it requires more calculations.

The Difference method is useful for rapidly converting any base number to decimal equivalent and Quotient Difference method is useful for rapidly converting base x (x ≥ 8) number to decimal equivalent.

Using the Add method, we can add up to nine binary numbers at a time and using same method we can very easily perform binary multiplication, Hetero base addition/multiplication, converting sum or products of binary numbers to DCB number, converting DCB number to binary equivalent, etc.

The DCB number system and Hetero base arithmetic operations are new to mathematics. A formula is developed for converting decimal number to DCB by studying the Add method. Converting decimal number to DCB number is very useful for rapidly converting decimal number to binary equivalent; since, we can rapidly convert any DCB number to binary equivalent very easily.

Using the Add method, we can easily perform decimal number to binary conversion, binary and Hetero base arithmetic operations in a non-calculator (for example calculators in the mobile phone). But we cannot do this in normal calculator without learning the Add method; even in widely used scientific calculators we can perform only some limited calculations, for example we cannot do binary arithmetic operations for the numbers with fractions, and for binary division it gives only integer part of the answer, etc.

Using the Step method, we can convert decimal number to binary, octal etc. It is also useful for converting all terms of sequence to binary, octal etc. equivalent and to find out various new concepts of the number system. We can also perform Step method calculations in calculator.

The R-analysis, new formulae, conversion methods, rapid calculations methods, various new concepts in this text are very much useful for solving various mathematical problems easily, extremely useful for several research and technological applications.

I would like to thank Sri Satish Vaman Naik Sir, for his encouragement, giving expert guidelines to improve the text and for writing foreword to this book. I would also like say thank for all who contributed to the success of this book in one way or another.

I welcome any comments concerning the text. Comments may be forwarded to the following email address: numbersystem@ymail.com or rahaven@gmail.com.

**Raghavendra Lingayya**_{ }

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