Abel's theorem in problems and solutions - School of Mathematics

52 Chapter 2
From this definition it is clear that in the complex numbers there is
nothing of the ‘supernatural’: the complex numbers are nothing but pairs
of real numbers. However, a question may arise: is it correct to call such
objects numbers? We will answer this question at the end of this section.
Another question, which perhaps the reader may put, is the reason way
the operations of addition and multiplication of complex numbers are
defined exactly in this manner and, in particular, way the operation of
multiplication is so strange. We will answer this question in §2.3.
First, we clarify the remarkable properties of the set of complex numbers
which we had defined.
202. Prove that the complex numbers form a commutative group
under addition. Which complex number is the unit element (zero) of this
group?
In the sequel it will be convenient to denote the complex numbers by
a single letter, for example by (or
203. Prove that the operation of multiplication of complex numbers
is commutative and associative, i.e., that and
It is easy to verify that
for every complex number The complex number (1, 0) is therefore
the unit element in the set of complex numbers under multiplication.
204. Let be an arbitrary complex number and suppose that
Prove that there exists a complex number such that
The results of Problems 203 and 204 show that complex numbers
form a commutative group under multiplication.
205. Prove that the operations of addition and multiplication of complex
numbers possess the distributive property, i.e., that
for any complex numbers
From the results of Problems 202–205 it follows that the complex
numbers with the operations of addition and multiplication defined by
(2.2) and (2.3) form a field. This field is the field of complex numbers.