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# 4.2 Complex Conjugate.

4.21   Definition (Complex conjugate.) Let be a field in which is not a square. Let be an element of . We define

is called the conjugate of .

The following remark will be needed somewhere in the proof of the next exercise.

4.22   Remark. If is a field in which is not a square, then in , since

4.23   Exercise. A Let be a field in which is not a square. Let . Show that
a)
.
b)
.
c)
if .
d)
.
e)
If , then . If then .
f)
.
g)
.

4.24   Example. The results of the previous exercise provide a way to write expressions of the form in the form . Write

and calculate away. For example, in , we have

4.25   Exercise. A Write each of the following elements of in the form where .
a)
b)

4.26   Note. The first appearance of complex numbers is in Ars Magna (1545) by Girolamo Cardano (1501-1576).

If it should be said, Divide 10 into two parts the product of which is 30 or 40, it is clear that this case is impossible. Nevertheless, we will work thus: [16, page 219].
He then proceeds to calculate that the parts are and , and says
Putting aside the mental tortures involved, multiply by making which is . Hence this product is 40 . So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless [16, page 219-220].

Around 1770, Euler wrote

144. All such expressions as are consequently impossible, or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing nor greater than nothing, nor less than nothing; which necessarily constitutes them imaginary, or impossible.

145. But notwithstanding this, these numbers present themselves to the mind; they exist in our imagination, and we still have a sufficient idea of them; since we know that by is meant a number which, multiplied by itself, produces ; for this reason also, nothing prevents us from making use of these imaginary numbers, and employing them in calculation. [20, p 43]

The use of the letter to represent was introduced by Euler in 1777.[15, vol 2, p 128] Both Maple and Mathematica use I to denote .

The first attempts to justify'' the complex numbers appear around 1800. The early descriptions were geometrical rather than algebraic. The algebraic construction of used in these notes follows the ideas described by William Hamilton circa 1835 [25, page 83].

You will often find the complex conjugate of denoted by instead of . The notion of complex conjugate seems to be due to Cauchy[45, page 26], who called and conjugates of each other.

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