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Complex numbers are covered, involving i. The fundamental theorem of algebra is referenced. The absolute value of a complex number is defined. The triangle inequality is stated.

It is useful to build from the real numbers another number system called the complex numbers. Although the real numbers R have many of the properties we expect, i.e., every positive number has a positive square root,every number has a cube root, and so on, there are somewhat less prominent properties that R fails to possess. For instance, negative numbers do not have square roots.This is actually a property that is missing in any ordered field, since every square is positive in an ordered field.See part (e) of [link] . One way of describing this shortcoming on the part of the real numbers is tonote that the equation 1 + x 2 = 0 has no solution in the real numbers. Any solution would have to be a number whose square is - 1 , and no real number has that property. As an initial extension of the set of real numbers,why not build a number system in which this equation has a solution?

We faced a similar kind of problem earlier on. In the set N there is no element j such that j + n = n for all n N . That is, there was no element like 0 in the natural numbers. The solution to the problem in that case was simply to “create” something called zero, and just adjoin it to our set N . The same kind of solution exists for us now. Let us invent an additional number, this time denoted by i , which has the property that its square i 2 is - 1 . Because the square of any nonzero real number is positive, this new number i was traditionally referred to as an “imaginary” number. We simply adjoin this number to the set R , and we will then have a number whose square is negative, i.e., - 1 . Of course, we will require that our new number system should still be a field; we don't want to give up ourbasic algebraic operations. There are several implications of this requirement:First of all, if y is any real number, then we must also adjoin to R the number y × i y i , for our new number system should be closed under multiplication. Of course the square of i y will equal i 2 y 2 = - y 2 , and therefore this new number i y must also be imaginary, i.e., not a real number. Secondly, if x and y are any two real numbers, we must have in our new system a number called x + y i , because our new system should be closed under addition.

Let i denote an object whose square i 2 = - 1 . Let C be the set of all objects that can be represented in the form z = x + y i , where both x and y are real numbers.

Define two operations + and × on C as follows:

( x + y i ) + ( x ' + y ' i ) = x + x ' + ( y + y ' ) i ,

and

( x + i y ) ( x ' + i y ' ) = x x ' + x i y ' + i y x ' + i y i y ' = x x ' - y y ' + ( x y ' + y x ' ) i .
  1. The two operations + and × defined above are commutative and associative, and multiplication is distributive over addition.
  2. Each operation has an identity: ( 0 + 0 i ) is the identity for addition, and ( 1 + 0 i ) is the identity for multiplication.
  3. The set C with these operations is a field.

We leave the proofs of Parts (1) and (2) to the following exercise. To see that C is a field, we need to verify one final condition, and that is to show that if z = x + y i 0 = 0 + 0 i , then there exists a w = u + v i such that z × w = 1 = 1 + 0 i . Thus, suppose z = x + y i 0 . Then at least one of the two real numbers x and y must be nonzero, so that x 2 + y 2 > 0 . Define a complex number w by

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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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