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Complex Numbers

by Dexter Booth 10th September 2019

Dexter Booth brings his eight-part blog series to a close with a concluding essay on complex numbers

When we multiplied or divided two imaginary numbers we were thrown back amongst the real numbers as indeed we were when we raised the imaginary unit to the power of the imaginary unit. As a consequence we need to consider the possibility of real numbers and imaginary numbers being mixed together. Let’s look at one of the simplest possible arrangement of symbols, namely: Which represents the sum of the Type 1 and Type 2 units by which I mean the addition of the real unit to the imaginary unit. This mixture of a real number and an imaginary number is called a complex number. We could easily have called it a mixed number because that is what a complex is, a mixture and, as we shall see, it is a consistent mixture because it retains its form throughout the entire arithmetic of complex numbers. Some typical complex numbers are: Which, as you see, contain positive and negative rational and irrational real numbers as well as imaginary numbers. Every complex number is represented as a sum (or difference) of a real part; and an imaginary part as we have just seen exemplified.

The arithmetic of complex numbers holds few surprises; they add and subtract as you would expect. For example: And: Complex numbers are also closed under multiplication and division. For example, it can be shown that: And: Finally, complex numbers are also closed under raising to a power. For example, we can show that:
So the problem of defining what is meant by raising a number to a complex power reduces to the problem of defining what meant by raising a number to the power i, and in this particular case it can be shown that So there we have it, quite a journey. Starting from a consideration of how to record a simple count we have constructed our number system from the whole numbers through natural numbers, integers, rational numbers, irrational numbers, imaginary numbers to complex numbers. The ultimate number is the complex number and all the others can be thought of as special forms of complex number. For instance, the real numbers are complex numbers with zero imaginary part and the imaginary numbers are complex numbers with zero real part.
dexter-diagram-8

We have also seen how we can apply the five arithmetic operations to the complex numbers; we can add, subtract, multiply and divide complex numbers and we can raise one complex number to the power of another complex number and so our journey is complete. The only problem outstanding is that of division by zero which we cannot define. So Edward, who started the ball rolling when he was 18 months old is now 12 and able to read and understand this blog. Ironically, his brother Henry, who is only 10 years old, has an obsession with balls - of the ovoid variety.

For humanity to travel this far has taken thousands of years but, with 20-20 hindsight, we have been able to traverse the route in the space of eight short essays. On reflection we have discovered something quite remarkable and very precious. Imagine the primitive soul who could only count by association; by crude utterances; by notches on a bone or a stick; by equivalent numbers of pebbles. From that early beginning humanity has created a complete and consistent number system that stands at the very foundations of our art, our science and our engineering. There is, of course, still one outstanding question. Did humanity create the number system or was it there all along just waiting to be discovered? After all, the behaviour of the universe as we perceive it is explainable by us through the number system and the universe was here long before we were.
The details of how these arithmetic operations are performed are sufficiently involved that to display them would detract from the ultimate aim of simply demonstrating the existence of the complex numbers.

This is the last of Dexter Booth's 'The Invention of Numbers' series. The rest can be found below:

  1. A Case for Mathematics
  2. Inventing the Idea of Numbers
  3. Numerals And Numbers
  4. Whole Numbers, Natural Numbers And Integers
  5. Rational Numbers
  6. Irrational Numbers
  7. Imaginary Numbers
  8. Complex Numbers (the one you're reading now!)
Featured image credit: Mandelbrot Set created by Woflgang Beyer. Available on Wikimedia Commons via CC BY-SA 3.0.