@swamidass, one of the main reasons why an old timer like me is prone to expression mathematical equations in such a crude format is that I wrote a lot of software in the 1970’s in FORTRAN and Pascal.
Among my sidelines in those days was helping physics professors and nuclear medicine specialists at the university’s cyclotron tweak their software. There were lots of “i” constants appearing in my programs. (I used to make a lot of bad puns in those days when I consulted for the cyclotron physicists, usually along the lines of: “It is settled then. The i’s have it!” )
When I said that “negative imaginary number” doesn’t mean anything, I was mostly commenting on the “negative” part. I am not questioning the importance of imaginary numbers.
There are two square roots of -1. Which one of those is negative? How do we distinguish between them?
I recognized that and so I don’t think I misunderstood you. I thought your intended meaning was clear but perhaps others read it differently. I don’t know.
To my knowledge, algebra textbooks have always shown them as (+i) and (-i). I assume that the topic is still taught that way----but I admit to being retired and well outside the loop of how mathematics education is conducted nowadays.
I think I see what you are saying—and it reminds me of the teaching approach used by a mathematics professor whose office happened to be adjacent to mine back when I taught at a modest-sized university where all of the faculty of the various science departments’ were housed in one building. The math prof and I used to have lunch in a shared faculty lounge and talk about his teaching methodologies, especially because he found himself assigned to teaching a course in “Methodologies for Elementary & Middle School Teachers of Mathematics” for the School of Education. He was really big on number lines and helping young people (and education majors who struggled with even basic mathematics) visualize numbers more effectively.
So he would take a number like (-5) and tell the future teachers to always factor it into (-1) and (5), as in (-1)*5= -5. He said that once one factors out the (-1), one can immediately recognize it as “code” for "go in the opposite direction----which on a traditional number line means “take N steps to the left”.
He extended these concepts to things like rays on a graph, where -90degrees meant a motion akin to the minute hand on a clock going counter-clockwise 90 degrees (or 3hours on a twelve-hour clock.)
Anyway, I know his School of Education students who struggled with math gave him rave reviews.
Your comment also brings to mind the long history of controversies in mathematics over how best to notate mathematical concepts (e.g., factorials using an exclamation point, the integral sign, the square root sign, summation Sigma, etc.)
no there is only one square root of any number real or imaginary. The square root of -1 is given the symbol i by mathematicians and the symbol j by the electrical engineers who use the symbol i for current.
i*i = -1 That is all you need to know. All of complex algebra builds on that logical premise.
@Patrick, I’m curious about the background to why mathematicians use “i” but electrical engineers use “j”. Is it because engineers were already using “i” for some other purpose? (Of course, I’m aware that an upper-case “I” has a “standard meaning” among electrical engineers. But what about the lower-case “i”?)