1.3. The Complex Number System as a Number Field
We have seen that in the field of real numbers, the linear equation ax + b = 0 is always solvable.
It is now time to investigate the solvability of the quadratic equations.
Consider the equation x
2
= 9. This can also be written as x
2
– 9 = 0. We can rewrite the left side
to come up with the equation (x – 3)(x + 3) = 0. From Theorem 1.10, we know that the solutions
of the given quadratic equation are x = 3 and x = 3.
Similarly, the equation x
2
= 3 can be rewritten as x
2
– 3 = 0 or (x 
3 )(x +
3 ) = 0. This latter
equation implies that the solutions are x =
3 and x = 
3 .
Suppose, now, that we are given the equation x
2
+ 1 = 0. We easily see that in R, this equation
has no solution, since the equation can be written as x
2
= 1 and there is no real number whose
square is 1 (or any negative number, for that matter!). This situation behooves us to find another
system that includes the set of real numbers and allows the solvability of the equation of the form
x
2
= p, where p is any positive real number.
Let us invent a special symbol for a “number” whose square is 1. Let us call it
i
. Thus, what we
are willing to accept the fact that
i
2
= 1. Clearly, this “number” is not a real number. We are
now ready to define a
complex number
.
Definition 1.22
. Complex number
A
complex number
is of the form x + y
i
, x and y are real numbers and i
2
= 1. The real
number x is called the
real part
, while the real number y is called the
imaginary part
.
For example, in the complex number 2 + 3
i,
the real part is 2 while the imaginary part is 3. The
number 1 is seen as a complex number with imaginary part equal to 0. We can easily conclude
that every real number is a complex number with the imaginary part equal to 0. A complex
number of the form y
i
, y ≠ 0, is sometimes called a
pure imaginary
.
Definition 1.23
. Conjugate of a complex number
The
conjugate
of a complex number x + y
i
is the complex number x – y
i
.
Time to think
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 Spring '11
 dikopaalam
 Math, Real Numbers, Equations, Complex number, Study of Structures Section

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