Complex Numbers: Why √(-1) = i Is A Misleading Starting Point For Understanding Complex Numbers
Understanding The Number Line
The way complex number concept is introduced misleads students.
Most people are familiar with the number line and most ancient tribes had an idea of counting. The base-ten comes as a natural counting process for most tribes. The reason can be as simple as the fact with ten digits (fingers). Some animals have a way of knowing that one or two members are missing.
The straight number line that stretches from minus-inifinity to plus-plus infinity is well understood to most by second grade. Before that most people can count from to infinity through stages from ten, hundred, thousand depending on thier abilities. Negative numbers concept may a little bit difficult to grasp for many before first grade. It does not take long to understand that the number line starts before zero. Teachers can use the concept of borrowing and lending as a pedagogy.
"If you borrow Susan's orange today, you have minus one orange, when you bring two oranges tomorow you will only be able to eat one orange because you need to give back Susan's orange from the two." Something like this is a good teaching aid.
What is not easy to conceptualize is the concept of a complex plane. The situation is not made easier when people are introduced to this concept through the assertion: √(-1) = i .
While √(-1) = i is a true statement, it is only an instant of the whole concept. A better introduction to the complex plane is preferable if science and engineering students will have a lasting appreciation.
This article discusses complex number fundamentals and what they mean to engineers.
Operator Concept: √(-1) = i
The strength of mathematics comes from its abstract nature, and therein comes its weakness. This is a strong statement.
The introduction of complex numbers as √(-1) = i is adequate from a pure mathematics point of view. Science and engineering students are better served if complex numbers are introduced with i , ( the √(-1) ), as an operator that shifts the phase of a number by 90-degrees. A second operation would result in a 180-degree shift. This explains why (i)2 = -1 . This explanation is not only practical, but is even a better abstraction than √(-1) = i.
Algebraically the two statements are the equivalent.
To say , √(-1) = i is a correct statement but it does not explain what is taking place, it is more like explaining a concept through the most obvious example. Or worse by giving the consequence as an explanation of the cause.
A better understanding of the complex number plane will help understanding natural phenomenon like echos, colour spectrum, sound quality and image processing, to name but just a few.
The Complex Number Plane
Real numbers are a subset of complex numbers. Human beings (and other animals) were using complex numbers before they understood the mathematics. A bat using echo-location applies the concept of phase shift to calcuate distances.
The very concept of interest rates, that money today is not the same as money tomorrow carries a deeper understanding of complex numbers than people to accept. There is a time difference on the arrival of the money which can be expressed as a phase shift.
Animals have abilities to sense minute differences in sound, color, distances, speeds and acceleration than any other mechanical or electronic sensor ever manufactured. Our ears, eyes and fingers can pick up differences in time of arrivals of signals to make sense of where a sound is coming from or how far one object is from another. Or to be able to track and chase a target.
The above paragraph explains why humans are still the best machines out there. Machines are getting advanced partly due to development of algorithms . These algorithms must be able to mimic how we sense and calculate natural occurrences.
To understand complex numbers, a scientist must abstract a concept of numbers spread out on a plane rather than just on a straight line. Numbers on a complex plane have both magnitude and phase differences.