Do Complex Numbers Scare You?

Beniel Seka

Some years ago, I was holding a book called ‘Complex Variables’. My friend who had come to visit me remarked, “You are studying complex variables! I have problems with simple variables and you have the guts of welcoming complex variables!”

My friend was studying law. He had dropped Mathematics since he was in Form Four. He was thinking of x and y which gave him a lot of troubles in algebra. He could not even solve a simple linear equation leave alone quadratic equations. He could not imagine that some people would embrace a subject that had so many complications.  “I cannot even solve the simple variables and here you are dealing with complex variables! Let me keep aloof,” he told me as he stepped aside.

I once did a research where I used a questionnaire to collect information about topics taught in Advanced Mathematics for forms five and six. One of the items in the questionnaire asked pupils to state topics which were difficult. Some pupils mentioned Complex numbers. A follow-up research revealed that the topic had not been taught in that school. The pupils were carried away by the words in the said topic.

May be some of you have the same thoughts. I can assure you that complex variables are not as frightening as they look. This could have been caused by the word ‘complex’ but this is what the first mathematicians who introduced the new system of numbers called them. They used the term to differentiate the new system from ordinary numbers which were later called real numbers.

However an item analysis of the topic in the Advanced Certificate of Secondary Education Examinations (ACSEE) mathematics results for a number of years has shown good performance. Also many candidates choose questions involving complex numbers. This indicates that the topic is not as difficult as some people think.

The ordinary numbers could be squared to obtain square numbers, for example 3 squared are 9 while 5 squared are 25. It is possible to perform the reverse operation known as square root, for example, the positive square root of 9 is 3 while the positive square root of 25 is 5.

When negative numbers were introduced, they could also be squared to give square numbers. This is possible because negative number times a negative number is a positive number. For example, (-1) times (-1) is (+1). So the square root of 1 is (+1) or (-1). This is a solution of the equation x squared=1. Problems start when the right-hand side of this equation is negative. Is there a number which multiplies by itself gives a negative number? Rene’ Descartes who developed coordinate geometry said it was not possible.

Some physical problems had been observed in that form and had remained unsolved. But they needed solution! Sooner or later, mathematicians introduced the concept of ‘imaginary’ numbers. They defined i= square root of (-1) where ‘i’ meant ‘imaginary’. This has widened the scope of the number system. We have a new number system called complex numbers. It consists of the ordinary numbers which we will now refer to as real numbers and the imaginary numbers.

A complex number, usually denoted by Z, is written as Z= a+bi where a is known as the real part and bi is the imaginary part. When b=0 we have no imaginary part and the real part is Z=a. this is the form we are so much used to. It forms the real number system. When a=0, we remain with the imaginary part only. Perhaps complex numbers will not bother you if your limit is Basic Mathematics but if you are doing Additional or Advanced Mathematics you will certainly study the topic.

Sometimes I wonder what the world would look like if complex numbers had not been invented.  Some of the modern technologies awe a lot to the complex numbers. Even renowned mathematicians such as Carl Frederick Gauss and Cauchy were able to make their discoveries after using complex numbers.

Those who will study mathematical Analysis at higher levels will definitely need complex numbers. Theory of a complex variable is a basic part of Mathematical analysis. Its influence can be seen in almost every field of mathematics. In addition to its prominence in Pure Mathematics and its elegant logical structure, the theory represents one of the most powerful mathematical instruments of Applied Mathematicians, engineers and physicists. So do not fear it.