# Re-defining the complex unit for teaching purposes

I often come across students who are confused by the idea that the complex unit, $i$, is defined as $i^2 = -1$. Since we are using the complex numbers in an engineering course, we use the complex numbers to encode information about a sinusoid of a given frequency, where the argument is the phase of the sinusoid with respect to a reference and the magnitude is the amplitude. I've started to explain the need for the complex numbers by re-defining the complex unit as: $i^5 = i$, which helps to explain that the complex numbers inherently encode periodic information, hence why they were so useful for encoding sinusoids.

My question is this: conceptually this helps the students to better understand why we use the complex numbers (they tend to hate the fact that we use the term "imaginary number"). However, this isn't the original definition of $i$. Is there anything seriously wrong with approaching the concepts with my definition? Is there anything special about the fact that $i^2 = -1$ that is not also present in my definition?

• $i^4=1$, $i^5=i$ – danimal Jul 13 '16 at 14:33
• Edited, sorry for the mis-type! – Michael Stachowsky Jul 13 '16 at 14:35
• I think you should just walk them thru the sequence $i,i^2,i^3,i^4,i^5,...$ and show that it's $i,i^2,i^3,i^4,i,...$. You can also show the simple tank circuit (AC sinusoidal voltage source, series resistor, followed by an inductor and capacitor in parallel.) Using complex methods gives a complex (time-dependent) formula for the current. The imaginary part of it corresponds to current circulating in the sub-loop containing the inductor and the cap. – DanielWainfleet Jul 13 '16 at 18:20

An alternative to the use of the embarrassing $i=\sqrt{-1}$ is to work with coordinate couples $(a,b)$ and introduce the product rule

$$(a,b)\cdot(c,d):=(ac-bd,ad+bc).$$

Then the constant $i$ is the innocuous $(0,1)$.

An important issue is to relate the complex product to rotations. It is easy to make the connection with

$$(\cos\theta,\sin\theta)\cdot(\cos\phi,\sin\phi)=(\cos\theta\cos\phi-\sin\theta\sin\phi,\sin\theta\cos\phi+\cos\theta\sin\phi)\\ =(\cos(\theta+\phi),\sin(\theta+\phi)).$$

• That is much better, thanks! – Michael Stachowsky Jul 13 '16 at 14:40

Well, for one thing any one of $x=1,-1,i,-i$ satisfies $x^5 = x$, so $x^5 = x$ cannot be the definition of $i$ unless you append the condition that $k = 5$ is the smallest positive integer power greater than $1$ such that $x^k = x$.