It is really surprising how always the authorized official or professional opinions would come first as an obvious answer for so many questions in mathematics, where also people would much appreciate them as they confine to their own learning, even though they aren’t of anything new, but a repletion of some materials that were long time well established and so available everywhere
If you simply Google the words (contradiction in complex numbers), then you would certainly find so many issues in this regard, and if you go after them carefully, you would be surprised to find this not better than many other huge fallacy in mathematics, consider at least this recent Wikipedia reference below:
https://webcache.googleusercontent.com/search?q=cache:0mK7ybi4DY4J:https://en.wikipedia.org/wiki/Mathematical_fallacy+&cd=10&hl=en&ct=clnk&gl=sa
Which most likely trying to outline the public concern, even without referring to the many actual public opinions or discussions, but considering some official reference as the general behaviors of wiki writers or researchers
And if this is really a fallacy, then there must be something that went unquestionable or unnoticeable and had been quite wrong from the early beginning, even though it appears as useful concept in little matters that had been so designed for
With strict rules on how to properly use it, as you must do this step first, then the second step should be arranged as this, or else, you wouldn’t get it correctly, and there are other cases where you have to choose the correct answer yourself, as if you must need a catalogs on how to use the imaginary or generally the complex numbers, so unlike the natural or the normal beautiful mathematics!
After so much comparison and investigations in this matter, I was convinced more than ever, that was only another huge fallacy in mathematics, despite its apparent usefulness that had been designed for, which also could had been done without all those extra magical tools that was never of any great discovery but rather a convention or better word was only an agreement, just to convey the unnecessarily talents and extra baseless volume of nonsense mathematics!
To show this fallacy to a layperson, remember that one day there weren’t the negative integers or zero concepts, but zero was introduced as an integer to facilitate the calculations mainly, even it is really meaningless to be called as real object as any other positive integer on the real line number, but zero was the bridge to further create the set of negative integers which are actually nothing but a mirror image of the natural integers on the number line (by the definition),
Where also the set behavior of negative integers under division or multiplication operations had been defined wrongly for a purpose as this:
Negative times negative is equal to positive (unbelievable huge mistake), then how? I would like to explain it to a layperson:
Just consider the simple example of multiplication (3*5 = 15), on a number line (say, positive X-axis), then mark the three numbers on the positive X-axis, (3, 5, and 15), now, consider Y-axis acting like a mirror (being the artificial symmetry we do create deliberately for our own narrow purpose only),
Now, observe the mirror image of the multiplication operation marked on the positive X-axis, in the mirror, where naturally the image would be seen as this (-3)*(-5) = (-15)
Since we do accept that (-3) is the mirror image of 3, and similarly (-5) is the mirror image of (-5),
But oddly we deny the product image of (-3)*(-5) = (-15) as the mirror image of the actual product (3)*(5) = (15), where this was another huge fallacy in mathematics for so many reasons!
Or more precisely (-1)*(-1) = (-1), so Sqrt(-1) = -1, and not even the legendary fake imaginary number (i)?
How come people can be fooled to this limit and for so many centuries!?
Now, I think I do understand why the ancient mathematicians as (Babel yon, Egyptians, Arabs, Indians, Chinese Greek, Muslims,…etc), all had missed that genius idea of negative integer concept basically,
Otherwise they would certainly discover most of the mathematics we know today within few years as an immediate subsequent results, (if they allowed themselves to establish so easily such concepts)
But I think they were even much wiser, even not to accept the negative integers, because they knew from the definition of the line, that is the shortest distance joining two points and extend endlessly in both opposite directions, nothing was being defined as negative in the elementary definition of a line or generally a number line, but only two opposite direction,
In other words, the Y-axis is only made to show an artificial symmetry (+, -) that is never a natural symmetry, which would create later the fake “fundamental theorem of algebra” and many more, which is easily refutable if you go back in time and question this type of fake symmetries, or genius discoveries of this form,
What is the solution of (x + 1 = 0)?, so easy discovery, let there be negative integers, but truly speaking, I can’t show you the negative sheep
What is the solution of (x^2 + 1 = 0)?, much easier, let there be imaginary numbers, but truly speaking, this isn’t interesting imagination, but boring magic nations
What is the Solution of (x^3 + 2 = 0), So silly problem, let there be a cube root of (2), but truly speaking at the paradise of fools, you would infinitely find infinitely many more of this type of numbers (without a single proof and unlike the case of Sqrt(2) with rigorous proof)
I also do appreciate other answers provided here which authors had no responsibilities or any guilt, because this was being so as global education adopted
Hopping also a tolerance for my own point of view in this regard!
Thanking You Sincerely and Best Regards
Bassam King Karzeddin
8TH, Oct, 2016