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1.1: Motivation

Last updated

May 3, 2023
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- 1: Complex Algebra and the Complex Plane
- 1.2: Fundamental Theorem of Algebra

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The equation $x^2 = -1$ has no real solutions, yet we know that this equation arises naturally and we want to use its roots. So we make up a new symbol for the roots and call it a complex number.

Definition: complex numbers

The symbols $\pm i$ will stand for the solutions to the equation $x^2 = -1$ . We will call these new numbers complex numbers. We will also write

$\sqrt{-1} = \pm i \nonumber$

Note: Engineers typically use $j$ while mathematicians and physicists use $i$ . We’ll follow the mathematical custom in 18.04.

The number $i$ is called an imaginary number. This is a historical term. These are perfectly valid numbers that don’t happen to lie on the real number line. (Our motivation for using complex numbers is not the same as the historical motivation. Historically, mathematicians were willing to say $x^2 = -1$ had no solutions. The issue that pushed them to accept complex numbers had to do with the formula for the roots of cubics. Cubics always have at least one real root, and when square roots of negative numbers appeared in this formula, even for the real roots, mathematicians were forced to take a closer look at these (seemingly) exotic objects.) We’re going to look at the algebra, geometry and, most important for us, the exponentiation of complex numbers.

Before starting a systematic exposition of complex numbers, we’ll work a simple example.

Example $\PageIndex{1}$

Solve the equation $z^2 +z + 1 = 0$ .

Solution

We can apply the quadratic formula to get

$z = \dfrac{-1 \pm \sqrt{1-4}}{2} = \dfrac{-1 \pm \sqrt{-3}}{2} = \dfrac{-1 \pm \sqrt{3} \sqrt{-1}}{2} = \dfrac{-1 \pm \sqrt{3} i}{2}$

Think: Do you know how to solve quadratic equations by completing the square? This is how the quadratic formula is derived and is well worth knowing!

Definition: complex numbers

Example 1.1.1\PageIndex{1}

Solution

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Example $\PageIndex{1}$