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+4i. 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 ?
- ei𝜋 = -1
- ii = e-𝜋/2 ≅ 0.207879576351..
- ln( -1 ) = i𝜋
- ln( i ) = 𝜋 i/2
- sin( i ) = i (e - e-1)/2
- cos( i ) = (e + e-1)/2 ≅ 1.54308063482..