Recall that a real number is a point on the real number line. Just as a real number should be considered as a point on the line, a complex number is considered a point in the plane which can be identified in the usual way using the Cartesian coordinates of the point. Thus

and

You should prove the following theorem.
Theorem 1.3.1 The complex numbers with multiplication and addition defined as above form a field satisfying all the field axioms. These are the following list of properties.
Something which satisfies these axioms is called a field. In this book, the field of most interest will be the field of complex numbers or the field of real numbers. You have seen in earlier courses that the set of real numbers with the usual operations also satisfies the above axioms. The field of complex numbers is denoted as ℂ and the field of real numbers is denoted as ℝ. An important construction regarding complex numbers is the complex conjugate denoted by a horizontal line above the number. It is defined as follows.

What it does is reflect a given complex number across the x axis. Algebraically, the following formula is easy to obtain.
Observation 1.3.2 The conjugate of a sum of complex numbers equals the sum of the complex conjugates and the conjugate of a product of complex numbers equals the product of the conjugates. To illustrate, consider the claim about the product.

Now by induction, the first product in the above can be split up into the product of the conjugates. Similar observations hold for sums.
Definition 1.3.3 Define the absolute value of a complex number as follows.

Thus, denoting by z the complex number z = a + ib,

Also from the definition, if z = x + iy and w = u + iv are two complex numbers, then
Notation 1.3.4 Recall the following notation.

There is also a notation which is used to denote a product.

The triangle inequality holds for the absolute value for complex numbers just as it does for the ordinary absolute value.
Proof: Let z = x + iy and w = u + iv. First note that

and so



so this shows the first version of the triangle inequality. To get the second,

and so by the first form of the inequality

and so both
With this definition, it is important to note the following. Be sure to verify this. It is not too hard but you need to do it.
Remark 1.3.6 : Let z = a + ib and w = c + id. Then
For example, consider the distance between