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# The Number System of Algebra: Number Systems from the Egyptians to the Greeks to the Europeans to…

## byHenry B. Fine, Ph.D.

### Kobo ebook | June 22, 2013

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I THEORETICAL

1 THE POSITIVE INTEGER,

AND THE LAWS WHICH REGULATE THE ADDITION AND

MULTIPLICATION OF POSITIVE INTEGERS

The number concept

Numerical equality

Numeral symbols

The numerical equation

Counting

Addition and its laws

Multiplication and its laws

2 SUBTRACTION AND THE NEGATIVE INTEGER

Numerical subtraction

Determinateness of numerical subtraction

Formal rules of subtraction

Limitations of numerical subtraction

Symbolic equations

Principle of permanence. Symbolic subtraction

Zero

The negative

Recapitulation of the argument of the chapter

3 DIVISION AND THE FRACTION

Numerical division

Determinateness of numerical division

Formal rules of division

Limitations of numerical division

Symbolic division. The fraction

Negative fractions

General test of the equality or inequality of fractions

Indeterminateness of division by zero

Determinateness of symbolic division

The vanishing of a product

The system of rational numbers

4 THE IRRATIONAL

Inadequateness of the system of rational numbers

Numbers defined by regular sequences. The irrational

Generalized definitions of zero, positive, negative

Of the four fundamental operations

Of equality and greater and lesser inequality

The number defined by a regular sequence its limiting value

Division by zero

The number-system defined by regular sequences of rationals a closed and continuous system

5 THE IMAGINARY COMPLEX NUMBERS

The pure imaginary

Complex numbers

The fundamental operations on complex numbers

Numerical comparison of complex numbers

Adequateness of the system of complex number

Fundamental characteristics of the algebra of number

6 GRAPHICAL REPRESENTATION OF NUMBERS THE VARIABLE.

Correspondence between the real number-system and the points of a line

The continuous variable

Correspondence between the complex number-system and the points of a plane

The complex variable

Definitions of modulus and argument of a complex number and of sine,

cosine, and circular measure of an angle

Construction of the points which represent the sum, difference, product,

and quotient of two complex numbers

7 THE FUNDAMENTAL THEOREM OF ALGEBRA

Definitions of the algebraic equation and its roots

Demonstration that an algebraic equation of the nth degree has n roots

8 INFINITE SERIES

8.1 REAL SERIES

Definitions of sum, convergence, and divergence

General test of convergence

Absolute and conditional convergence

Special tests of convergence

Limits of convergence

The fundamental operations on infinite series

8.2 COMPLEX SERIES

General test of convergence

Absolute and conditional convergence

The region of convergence

A theorem respecting complex series

The fundamental operations on complex series

9 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS UNDETERMINED COEFFICIENTS. INVOLUTION AND EVOLUTION. THE BINOMIAL THEOREM

Definition of function

Functional equation of the exponential function

Undetermined coefficients

The exponential function

The functions sine and cosine

Periodicity of these functions

The logarithmic function

Indeterminateness of logarithms

Permanence of the laws of exponents

Permanence of the laws of logarithms

Involution and evolution

The binomial theorem for complex exponents

II HISTORICAL

10 PRIMITIVE NUMERALS

Gesture symbols

Spoken symbols

Written symbols

11 HISTORIC SYSTEMS OF NOTATION

Egyptian and Phoenician

Greek

Roman

Indo-Arabic

12 THE FRACTION

Primitive fractions

Roman fractions

Egyptian (the Book of Ahmes)

Babylonian or sexagesimal

Greek

13 ORIGIN OF THE IRRATIONAL

Discovery of irrational lines. Pythagoras

Consequences of this discovery in Greek mathematics

Greek approximate values of irrationals

14 ORIGIN OF THE NEGATIVE AND THE IMAGINARY.

15 ACCEPTANCE OF THE NEGATIVE

16 RECOGNITION OF THE PURELY SYMBOLIC CHARACTERS OF ALGEBRA

### Details & Specs

The following ISBNs are associated with this title:

ISBN:9990006052671