> When I tried to calculate the square root of -1 on my calculator, it gave me an error.
This was a red flag to me. This person takes the answer from a calculator as the ultimate truth, whereas it (obviously) is not. I wonder how he would have reacted had he had an advanced calculator (HP50g or some TI) and got "i" as the answer instead of an error.
> To this day I do not understand imaginary numbers. It makes no sense to me at all.
Coming from a mixed EE/CS background, I had a lot of math and math-heavy classes in the first 2.5 years (linear algebra, real, vector and complex analysis, discrete math, systems theory)..
What I observed during the 5-yr master's degree was that the people who were seeking "meaning" in math courses were the same people who 1) had the most problems with math exams, 2) had the least ability to apply the math to other domains in order to do something useful.
What worked for me was to accept that math is manipulation of abstract symbols according to some rules, combined with some ingenuity (i.e., when you have to consider external information in addition to the rules).
In the case of complex numbers, you introduce a new abstract symbol called "i" with the property that i X i=-1. Other rules that you (hopefully) learned earlier still apply. So computing (x+y)X(x+z) or (2+3i)X(2-6i) is in essence the same process except that you replace i^2 with -1 when you encounter it.
I actually enjoyed doing math on that level: as a manipulation of abstract rules and invention of new rules (theorems).
In simple EE, inducntances and capacitances are modeled as pure imaginary "resistances"). However, I somehow figured out how to make use of math (not only complex numbers) in other domains. I made use of math in electronics for drawing Bode plots, information theory, systems theory, and even economics! (In economics, I was reading a lengthy chunk of text with some summations, only to realize that, in actuality, integration was described and everything could be summarized to few sentences explaining the meaning of the involved integral. On the exam, I gave MY explanation -- based on the integral -- instead of the lecturer's, and I passed!)
To me the problem seemed to be that lecturers in other subjects resorted to awkward, special-cased explanations instead of showing us how to model the problems at hand with the math we have already learned.
> When I tried to calculate the square root of -1 on my calculator, it gave me an error.
This was a red flag to me. This person takes the answer from a calculator as the ultimate truth, whereas it (obviously) is not. I wonder how he would have reacted had he had an advanced calculator (HP50g or some TI) and got "i" as the answer instead of an error.
> To this day I do not understand imaginary numbers. It makes no sense to me at all.
Coming from a mixed EE/CS background, I had a lot of math and math-heavy classes in the first 2.5 years (linear algebra, real, vector and complex analysis, discrete math, systems theory)..
What I observed during the 5-yr master's degree was that the people who were seeking "meaning" in math courses were the same people who 1) had the most problems with math exams, 2) had the least ability to apply the math to other domains in order to do something useful.
What worked for me was to accept that math is manipulation of abstract symbols according to some rules, combined with some ingenuity (i.e., when you have to consider external information in addition to the rules).
In the case of complex numbers, you introduce a new abstract symbol called "i" with the property that i X i=-1. Other rules that you (hopefully) learned earlier still apply. So computing (x+y)X(x+z) or (2+3i)X(2-6i) is in essence the same process except that you replace i^2 with -1 when you encounter it.
I actually enjoyed doing math on that level: as a manipulation of abstract rules and invention of new rules (theorems).
In simple EE, inducntances and capacitances are modeled as pure imaginary "resistances"). However, I somehow figured out how to make use of math (not only complex numbers) in other domains. I made use of math in electronics for drawing Bode plots, information theory, systems theory, and even economics! (In economics, I was reading a lengthy chunk of text with some summations, only to realize that, in actuality, integration was described and everything could be summarized to few sentences explaining the meaning of the involved integral. On the exam, I gave MY explanation -- based on the integral -- instead of the lecturer's, and I passed!)
To me the problem seemed to be that lecturers in other subjects resorted to awkward, special-cased explanations instead of showing us how to model the problems at hand with the math we have already learned.