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Formulation of the Complex Plane

by IxRxPhysicist
Tags: complex, formulation, plane
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Feb23-13, 11:59 AM
P: 26
Hey all,
More of a fundamental question, could possibly be a chicken-egg question. I understand mathematical constructs and minuta if I know from where it hailed. So my question is this, what was the motivation behind developing the complex plane? Was it theoretically developed or was it found necessary due to equations such as x^2 = -1?

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Feb23-13, 12:36 PM
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Imaginary numbers were already being used, although not really understood, in the 1600s when formulas for solving cubic equations were discovered. Even for an cubic polynomial with all real roots, the formula for the solution sometimes necessitates taking square roots of negative numbers. These occur in pairs and end up canceling each other out when all the roots are real, but the formula doesn't work without this feature.

Euler (1700s) is the one who took ##\sqrt{-1}##, gave it a name (##i##), and developed a good deal of the theory of complex numbers and functions.

One thing he would have recognized at an early point is that if you append ##i## to the real numbers, then no matter how you add, subtract, multiply, or divide, the result will always be a number of the form ##a + bi## with ##a,b \in \mathbb{R}##, so from that it's already clear that the complex numbers are 2-dimensional over the reals.

Moreover, he worked out the famous identity ##e^{ix} = \cos(x) + i\sin(x)##. Putting ##x = \pi/2##, this becomes ##e^{i\pi/2} = i##, which makes it clear that the angle between ##i## and ##1## is ##\pi/2##, a right angle, so it makes sense to have the real numbers on the "x" axis and the imaginary numbers on the "y" axis.

Apparently, Argand (early 1800s) gets the credit for first publishing the geometric interpretation (complex plane), but I find it hard to believe that this wasn't already fully understood by Euler.
Feb23-13, 12:37 PM
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I think it's fair to say that the origins were fairly theoretical.

In fact, complex numbers first showed up with the work of Cardano. Cardano tried to solve polynomials of the third degree. He (and others) eventually found a nice method to solve those equations.

Now, if you solve a quadratic equation [itex]ax^2 + bx + c=0[/itex], then you usually define the discriminant [itex]D= b^2 - 4ac[/itex]. We know now that if D is negative, then there are complex solutions. But back in Cardano's day, they did not know complex numbers. So they simply said that if D is negative, then the quadratic has no solutions.

Now, Cardano's method also involved some discriminant. And this discriminant can be used to find the solutions to the cubic equation. And in the solution, you will have to take the square root of the discriminant (as with the quadratic equation). So you can think that the same happens with the quadratic: if the discriminant is negative, then the square root doesn't exist, so there are no solutions. But they discovered that this is not at all the case. The found out that even if the discriminant is negative, that there are still real solutions. And it was even more weird, those real solutions could be found by exactly the same formula where you take the square root of a negative number.

So they introduced expressions such as [itex]\sqrt{-1}[/itex] as tools to help them solve equations. They were not called numbers though, they were treated as mere tricks that happened to work. It was close to mathematical heresy to call the square roots of negative numbers actual numbers. Remember, this is a time where even negative numbers were treated suspicious.

Of course, if there is anything that mathematicians have learned by now, then that is that there are no coincidences in math. In fact, coincidences are usually symptoms of a deeper underlying structure that is yet to be discovered. After a long time, the mathematicians in history also discovered this and started to look at complex numbers as actual numbers. But it took until (just before) 1800 that mathematicians started to do actual work with complex numbers. This was the birth of complex analysis.

Feb24-13, 12:06 PM
P: 26
Formulation of the Complex Plane

Thanks guys, this is a start and it is going to take some time to understand that structure and benefits of using the complex plane, for instance I know why you preferably execute quantum phenomena in k-space.


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