I was discussing integer powers of i in my summer Algebra 2 last month and started with the “standard” modulus-4 pattern I learned as a student and have always taught. While not particularly insightful, my students and I considered another approach that might prove simpler for some.
I began with the obvious and before invoking the definition of i to get . From these three you can see every time the power of i increases by 1, you multiply the result by i and simplify the result if possible using these first 3 terms. The result of is simple, taking the known results to
But , cycling back to the value initially found with . Continuing this procedure creates a modulus-4 pattern:
They noticed that i to any multiple of 4 was 1, and other powers were i, -1, or –i, depending on how far removed they were from a multiple of 4. For an algorithm to compute a simplified form of i to an integer power, divide the power by 4, and raise i to the remainder (0, 1, 2, or 3) from that division.
They got the pattern and were ready to move on when one student who had glimpsed this in a math competition at some point noted he could “do it”, but it seemed to him that memorizing the list of 4 base powers was a necessary requirement to invoking the pattern.
Then recalled a comment I made on the first day of class. I value memorizing as little mathematics as possible and using the mathematics we do know as widely as possible. His challenge was clear: Wasn’t asking students to use this 4-cycle approach just a memorization task in disguise? If I believed in my non-memorization claim, shouldn’t there be another way to achieve our results using nothing more the definition of i?
A POTENTIAL IMPROVEMENT:
By definition, , so it’s a very small logical stretch with inverse operations to claim .
Even Powers: After trying some different examples, one student had an easy way to handle even powers. For example, if n=148, she invoked an exponent rule “in reverse” to extract an term which she turned into a -1. Because -1 to any integer power is either 1 or -1, she used the properties of negative numbers to odd and even powers to determine the sign of her answer.
Because any even power can always be written as the product of 2 and another number, this gave an easy way to handle half of all cases using nothing more than the definition of i and exponents of -1.
A third student pointed out another efficiency. Because the final result depended only on whether the integer multiplied by 2 was even or odd, only the last two digits of n were even relevant. That pattern also exists in the 4-cycle approach, but it felt more natural here.
Odd Powers: Even powers were so simple, they were initially frustrated that odd powers didn’t seem to be, too. Then the student who’d issued the memorization challenge said that any odd power of i was just the product of i and an even power of i. Invoking the efficiency in the last paragraph for n=567, he found
In the end, powers of i had become nothing more complicated than exponent properties and powers of -1. The students seemed to have greater comfort with finding powers of complex numbers, but I have begun to question why algebra courses have placed so much emphasis on powers of i.
From one perspective, a surprising property of complex numbers for many students is that any operation on complex numbers creates another complex number. While they are told that complex numbers are a closed set, to see complex numbers simplify so conveniently surprises many.
Another cool aspect of complex number operations is the stretch-and-rotate graphical property of complex number multiplication. This is the basis of DeMoivre’s Theorem and explains why there are exactly 4 results when you repeatedly multiply any complex number by i–equivalent to stretching by a factor of 1 and rotating . Multiplying by 1 doesn’t change the magnitude of a number, and after 4 rotations of , you are back at the original number.
So, depending on the future goals or needs of your students, there is certainly a reason to explore the 4-cycle nature of repeated multiplication by i. If the point is just to compute a result, perhaps the 4-cycle approach is unnecessarily “complex”, and the odd/even powers of -1 is less computationally intense. In the end, maybe it’s all about number sense.
My students discovered a more basic algorithm, but I’m more uncomfortable. Just because we can ask our students a question doesn’t mean we should. I can see connections from my longer studies, but do they see or care? In this case, should they?