Imaginary numbers... (last one, I promise)
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Mathematicians are special. They cannot calculate sqrt(-1) and instead of giving up math, they simply state that we have discovered a new number. I swear, you cannot win against these people...
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One of my math professors had said that "all numbers are imaginary". For example, I can show you two oranges, or two cats, but I cannot show just "two". So, all the numbers we deal with, are in our head, mind. We write it, of course, as 2, but that is just a representation of "two" in my written language. The same 2 is written as ೨ in my language Kannada, and as २ in Devanagari script. But none of them are the concept called "two"; the concept called "two" is imagined in my mind, it is imaginary. This was the logic of my professor. Stated otherwise, I cannot see, hear, touch, smell or taste "two" or "three". None of the numbers is tangible in that sense. The concept of "imaginary number i" takes that imagination abstraction to the next level.
You missed the joke.....
ed
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I can further the development: see (-1)^(1/4) :-\ Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Kenneth Haugland wrote:
Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Actually, it was to solve the cubic. The only way to solve a real cubic that has 3 real solutions is to have an intermediate quadratic solution of complex conjugates. The quintic cannot be solved, and that was proven by Abel, who refined the proof that Ruffini had done (and which was a mess). Later on, Galois used Abel's work to come up with the idea of permutation groups, from which spring the discipline of Group Theory. Arnold came up a with proof that does not rely on Group Theory, but instead relies on the complex plane and functions that wind around the plane.
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Kenneth Haugland wrote:
Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Actually, it was to solve the cubic. The only way to solve a real cubic that has 3 real solutions is to have an intermediate quadratic solution of complex conjugates. The quintic cannot be solved, and that was proven by Abel, who refined the proof that Ruffini had done (and which was a mess). Later on, Galois used Abel's work to come up with the idea of permutation groups, from which spring the discipline of Group Theory. Arnold came up a with proof that does not rely on Group Theory, but instead relies on the complex plane and functions that wind around the plane.
My comment got autotuned/autocorrected on me, but yes. Between the proofs and the Renaissance Italians, there was also the paper from Lagrange that showed the similarities in how to get the solutions to these two equations, cubic and quartic, which is basically the start of group theory. I wanted to study complex numbers in detail but never got around to it. I mean, I know the basics, but there are a lot of neat and cool theorems you can use from them to solve real-world problems.