Everyday usage and use in mathematics

The word complex in everyday English is often used to describe something that is complicated, intricate, or consisting of many different and connected parts. For example, one might describe a machine with many interrelated parts as complex or a situation involving multiple factors as a complex situation.

The word imaginary in everyday English means "not real" or "fanciful."

In mathematics, the term complex is specifically related to complex numbers, which are numbers that include a real part and an imaginary part. A complex number is expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit defined as \( \sqrt{-1} \).

The number $b$ is called the imaginary part to contrast with the real part.

The complex plane is a two-dimensional plane where each point represents a complex number. The horizontal axis (real axis) represents the real part of the complex number, and the vertical axis (imaginary axis) represents the imaginary part. For example, the complex number \( 3 + 4i \) would be plotted at the point \( (3, 4) \) on the complex plane, where 3 is the distance along the real axis and 4 is the distance along the imaginary axis.

The everyday word complex is used to describe complex numbers because they have two parts -- the imaginary part and the real part.

Importance in mathematics

The roots of a polynomial with real coefficients might or might not be real numbers. For example, consider the equation \( x^2 + 4 = 0 \). To solve this equation, we rewrite it as \( x^2 = -4 \) and then take the square root of both sides to obtain \( x = \pm \sqrt{-4} \). Since the square root of a negative number is not real, we express it as \( x = \pm 2i \), where \( i = \sqrt{-1} \). Here \( 2i \) is a complex number with a real part of 0 and an imaginary part of \( 2i \).

In fact, any non-constant polynomial with real coefficients will have a complex root. This is the fundamental theorem of algebra.

Exercises