# Imaginary Numbers?

/Extension 2 Mathematics students often ask " why do we learn complex numbers and what use are they". In my last blog I mentioned that i = √-1 is used in the theory of relativity to form a fourth dimension using time and the speed of light. Complex numbers occur in many applications of Physics so I feel it is important to increase the awareness of complex numbers to try to allay the suspicion that students have of them.

## What are complex numbers?

If we walk 3 m to the east from a sign post our position is given as x = 3, y=0. If we then walk 4 m north the final position is given as x = 3, y = 4. This point, represented by two real numbers, is written as the ordered pair (3,4). Wouldn't it be great if we could treat this pair of numbers just like a single real number, such as 0.5 or -2, when we carry out algebraic manipulations such as addition and multiplication? Yes, we can do this by forming a **complex number** z written as z = 3+4*i*. In this number 3 is called the **real part**, 4 is called the **imaginary part** and *i *is called the **imaginary unit** or imaginary operator. When *i* is chosen to be √-1 the laws of algebra involving the manipulation of complex numbers take a convenient form. Think of a complex number as a parcel containing two bits of information. By using the rules of algebra we can manipulate these two bits of information simultaneously as a single complex number z. An excellent explanation of complex numbers is given in the book *An Introduction to Ordinary Differential Equations* by Earl A Coddington, Dover Publications 1989, on pages 1 to 6.

## Why is the second part called the imaginary part?

The great Mathematician Leonhard Euler in 1777 used the term imaginary to describe the second part of a complex number. The use of the word imaginary is an endless source of confusion to students as it creates the wrong mindset. Imaginary in this context **does not mean that it does not exist**. The imaginary part of a complex number is a **real number corresponding to the Y value** **of the point** when it is plotted in the complex plane. The real part corresponds to X value of the point. Remember that complex numbers represent points in the X-Y plane.

## How were complex numbers discovered?

Many students think that complex numbers were developed to overcome the problem of the discriminant being negative when the quadratic formula is used to solve a quadratic equation. This false impression has been promulgated by a number of school textbooks. Complex numbers do allow us to obtain **two solutions** to any quadratic equation but historically complex numbers were developed to overcome difficulties in the solution of the **cubic equation**. A cubic equation has **three solutions** at least one of which must be a real number. The formula for the solution of a cubic equation is more involved than the quadratic formula and has a term in it involving *i*. An excellent description of the history and uses of complex numbers is given in Paul J Nahin's book, *An Imaginary Tale the Story of **√-1*, Princeton University Press 1998.

## What are some applications of complex numbers?

Firstly, complex numbers allow us to solve polynomial equations. A polynomial of degree n with real coefficients has n roots in the complex plane. The polynomial equation

z ^{5 } - 5 z ^{ 4 } + 7 z ^{ 3 } - z^{ 2 } - 8 z + 6 = 0

has the solutions (roots in the complex plane) z = -1, z = 1, z = 3, z = 1 - *i* and z = 1 + *i.*

One equation that is not included in the NSW Extension 2 Mathematics Course is Euler's famous equation

*e*^{ i 𝜽 } = cos( 𝜽 ) + *i* sin( 𝜽 )

As this equation has terms involving cos( 𝜽 ) and sin( 𝜽 ), Euler's equation has many uses in Physics problems that involve repetitive movements such as wave motion.

Another application of complex numbers is in fluid mechanics where the force exerted by a fluid flowing over a structure, such as an aircraft wing, can be calculated using an integral in the complex plane known as Blasius' theorem.

## What are some interesting formulae involving *i *?

*e*^{i𝜋}= -1*i*^{i}= e^{-𝜋/2}≅ 0.207879576351..- ln( -1 ) =
*i*𝜋 - ln(
*i*) = 𝜋*i*/2 - sin(
*i*) =*i*(*e*-*e*^{-1})/2 - cos(
*i*) = (*e*+*e*^{-1})/2 ≅ 1.54308063482..