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What is really the need for complex numbers? Is there any physical meaning associated with it?

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- Thread starter koolraj09
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What is really the need for complex numbers? Is there any physical meaning associated with it?

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HallsofIvy

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Historically, the motivation for defining complex numbers was Cardano's formula for solving cubic equations. That involves taking a square root, followed by other calculations. There exist cubic equations which

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lavinia

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What is really the need for complex numbers? Is there any physical meaning associated with it?

There are many physical processes that are described in terms of complex numbers. In quantum mechanics, the wave function is a complex distribution that evolves according to a complex version of the heat equation. Quantum mechanical things have phases which are complex numbers.

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Pythagorean

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lavinia

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Complex numbers can be multiplied as well as added. Adding them is the same as 2D vector addition.

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lavinia

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The modeling of quantum mechanical systems as complex distributions gives a picture of the world that is fundamentally composed of complex number values.

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Historically they came about as a way of solving cubic equations. At the time there was a solution by radicals to cubic equations and sometimes a square root of a negative number would arise, but go away by the time the answer was simplified. It was quite mysterious at the time as you could 'allow' these weird items in order to get real answers.

Then I believe (and I forget who did it first) there was a movement to see complex numbers as vectors. Because, as noted above, you could add them and multiply them (and they follow the parallelogram law for force addition). It was from this point that Hamilton tried finding a 3D version of the complex numbers, after all that's what you'd want. The complex numbers just don't cut in a 3D world. However he discovered the quaternions, which are 4D and have units of 1, i, j, and k. Those eventually were hijacked by Gibbs and Heaviside to form what we think of as vector analysis today.

I'm not fully certain of the question so I'll posit another possibility by posing a question myself. What is the physical significance of the real numbers? When I teach mathematics that involves complex numbers (I dislike the term 'imaginary') I always spend a few minutes on this question.

As for 'need' that depends. For example, do we need real numbers? I'm not sure, we could probably work out all of our results on the rationals but it would be ridiculuous, supposing its even possible. The real numbers are a construction that make analysis much nicer. We can take limits (and the limit points exist) and so voila: calculus. In the same way the complex plane makes many things very nice and very convenient. Could we work without them? Maybe, but its probably not a good idea.

Then I believe (and I forget who did it first) there was a movement to see complex numbers as vectors. Because, as noted above, you could add them and multiply them (and they follow the parallelogram law for force addition). It was from this point that Hamilton tried finding a 3D version of the complex numbers, after all that's what you'd want. The complex numbers just don't cut in a 3D world. However he discovered the quaternions, which are 4D and have units of 1, i, j, and k. Those eventually were hijacked by Gibbs and Heaviside to form what we think of as vector analysis today.

I'm not fully certain of the question so I'll posit another possibility by posing a question myself. What is the physical significance of the real numbers? When I teach mathematics that involves complex numbers (I dislike the term 'imaginary') I always spend a few minutes on this question.

As for 'need' that depends. For example, do we need real numbers? I'm not sure, we could probably work out all of our results on the rationals but it would be ridiculuous, supposing its even possible. The real numbers are a construction that make analysis much nicer. We can take limits (and the limit points exist) and so voila: calculus. In the same way the complex plane makes many things very nice and very convenient. Could we work without them? Maybe, but its probably not a good idea.

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Alot of applications, and interests.

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