Complex Numbers

(1)

Department of Mathematics and Statistics, Villanova University, Villanova, PA, USA

There is no real number a for which $\,a^{2} + 1 = 0\,$ . In order to develop an expanded number system that includes solutions to this simple quadratic equation, we define the “imaginary number” $\,i = \sqrt{-1}\,$ . That is, this new number i is defined by the condition that $\,i^{2} + 1 = 0\,$ .

Since $\,i^{2} = -1$ , it follows that $\,i^{3} = i^{2} \cdot i = -1 \cdot i = -i\,$ . Similarly, $\,i^{4} = i^{3} \cdot i = -i \cdot i = -i^{2} = -(-1) = 1\,$ , and $\,i^{5} = i^{4} \cdot i = 1 \cdot i = i\,$ .

As a quick observation, notice that the equation a ⁴ = 1 now has not only the familiar solutions a = ±1 but also two “imaginary” solutions a = ±i . Thus, there are four fourth roots of unity, namely, ± 1 and ± i . The inclusion of the number i in our number system provides us with new solutions to many simple equations.

4.1 The complex number system

The complex number system, denoted by ℂ, is defined to be the set

$\displaystyle{\mathbb{C} = \left \{a + b \cdot i\,:\, a\ \mathrm{and}\ b\ \mathrm{are\ real\ numbers}\right \}.}$

To carry out arithmetic operations in ℂ, use the usual rules of commutativity, associativity, and distributivity, along with the definition $i^{2} = -1$ . Thus, $$(a + bi) + (c + di) = (a + c) + (b + d)i$$

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

and $(a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i$ . Also, $$-(a + bi) = -a + (-b)i$$

. Momentarily, we will look at division of one complex number by another.

A geometric view of ℂ . Each complex number $\,z = a + bi\,$ is determined by two real numbers. The number a is called the real part of z and is denoted by $\,\mathfrak{R}z = a\,$ , while the number b is called the imaginary part and is denoted by $\,\mathfrak{I}z = b\,$ . (It is important to keep in mind that the imaginary part of a complex number is actually a real number, which is the coefficient of i in the complex number.) In this sense, the complex number system is “two-dimensional,” and so can be represented geometrically in the xy-plane, where we plot the real part of a complex number as the x-coordinate and the imaginary part of the complex number as the y-coordinate.

A real number a can also be written as $\,a = a + 0 \cdot i\,$ , and so corresponds to the point (a, 0) on the x-axis, which is known therefore as the real axis. Similarly, a purely imaginary number $\,b \cdot i = 0 + b \cdot i\,$ corresponds to the point (0, b) on the y-axis, which is called the imaginary axis.

The distance to the origin from the point (a, b) , corresponding to the complex number a + bi, is equal to $\sqrt{a^{2 } + b^{2}}$ . Accordingly, we make the following definition.

Definition 4.1.

The modulus of the complex number a + bi, denoted by | a + bi | , is defined by

$\displaystyle{ \vert a + bi\vert = \sqrt{a^{2 } + b^{2}}. }$

(4.1)

The modulus of a complex number is analogous to the absolute value of a real number. Indeed, for a real number $\,a = a + 0 \cdot i\,$ , we get that $\,\vert a + 0 \cdot i\vert = \sqrt{a^{2 } + 0^{2}} = \vert a\vert \,$ in the usual sense. Note also that $\,\vert a + bi\vert = 0\,$ if, and only if, $\,a = b = 0\,$ . A central observation is that

$\displaystyle{(a + bi) \cdot (a - bi) = a^{2} + b^{2} = \vert a + bi\vert ^{2}\,.}$

With this in mind, we make another definition.

Definition 4.2.

The conjugate of a complex number a + bi , denoted by $\,\overline{a + bi}\,$ , is defined by

$\displaystyle{ \overline{a + bi} = a - bi\,. }$

(4.2)

The central property, to repeat, is that

$\displaystyle{ (a + bi) \cdot \overline{(a + bi)} = (a + bi) \cdot (a - bi) = a^{2} + b^{2} = \vert a + bi\vert ^{2}\,. }$

(4.3)

The conjugate of a complex number is the key ingredient when it comes to the arithmetic operation of division. For example, notice that

$\displaystyle{(5 + 12i)(5 - 12i) = 5^{2} + 12^{2} = 13^{2} = 169.}$

It follows from this that

$\displaystyle{ \frac{1} {5 + 12i} = \frac{5 - 12i} {169} = \frac{5} {169} - \frac{12} {169}\,i\,}$

whence

$\displaystyle{ \frac{3 + 4i} {5 + 12i} = (3 + 4i) \cdot \frac{1} {5 + 12i} = (3 + 4i)\left ( \frac{5} {169} - \frac{12} {169}\,i\right ) = \frac{63} {169} - \frac{16} {169}i.}$

In general, we get that

$\displaystyle{ \frac{1} {a + bi} = \frac{1} {a^{2} + b^{2}}(a - bi)\,.}$

This enables us to divide any complex number by (a + bi) , provided that a and b are not both 0 . The act of dividing by the nonzero complex number (a + bi) is re-expressed as multiplication by (a − bi) and division by the nonzero real number (a ² + b ²) .

4.2 The complex exponential function

Consider these well-known Taylor series:

$\displaystyle{\cos (x) =\sum _{ n=0}^{\infty }(-1)^{n} \frac{x^{2n}} {(2n)!} = 1 -\frac{x^{2}} {2!} + \frac{x^{4}} {4!} -\frac{x^{6}} {6!} + \cdots \,,}$

$\displaystyle{\sin (x) =\sum _{ n=0}^{\infty }(-1)^{n} \frac{x^{2n+1}} {(2n + 1)!} = x -\frac{x^{3}} {3!} + \frac{x^{5}} {5!} -\frac{x^{7}} {7!} + \cdots \,,}$

and

$\displaystyle{\exp (x) = e^{x} =\sum _{ n=0}^{\infty }\frac{x^{n}} {n!} = 1 + x + \frac{x^{2}} {2!} + \frac{x^{3}} {3!} + \frac{x^{4}} {4!} + \cdots \,.}$