CHAPTER I 
INTRODUCTION, DEFINITIONS OF ALGEBRAS, 
ILLUSTRATIONS 
The co-ordinates of the numbers of an algebra may 
be ordinary complex numbers, real numbers, rational 
numbers, or numbers of any held. By employing a general 
field of reference, we shall be able to treat together 
complex algebras, real algebras, rational algebras, etc., 
which were discussed separately in the early literature. 
We shall give a brief introduction to matrices, partly 
to provide an excellent example of algebras, but mainly 
because matrices play a specially important role in the 
theory of algebras. 
i. Fields of complex numbers. If a and h are real 
numbers and if i denotes V — i, then a+bi is called a 
complex number. 
A set of complex numbers will be called a field if the 
sum, difference, product, and quotient (the divisor 
not being zero) of any two equal or distinct numbers 
of the set are themselves numbers belonging to the set. 
For example, all complex numbers form a field C. 
Again, all real numbers form a field 9?. Likewise, the 
set of all rational numbers is a field R. But the set of all 
integers (i.e., positive and negative whole numbers and 
zero) is not a field, since the quotient of two integers is 
not always an integer. 
Next, let a be an algebraic number, i.e., a root of 
an algebraic equation whose coefficients are all rational 
numbers. Then the set of all rational functions of a