Complex Algebra - Miami

Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense not real. Later, when probably one of the students of Pythagoras discovered that numbers such as 2 are irrational and cannot be written as a quotient of integers, legends have it that the discoverer suffered dire consequences. Now both negatives and irrationals are taken for granted as ordinary numbers of no special consequence. Why should 1 be any different? Yet it was not until the middle 1800's that Complex numbers were accepted as fully legitimate. Even then, it took the prestige of Gauss to persuade some. How can this be, because the general solution of a quadratic equation had been known for a long time? When it gave Complex roots, the response was that those are meaningless and you can discard them. Complex Numbers As soon as you learn to solve a quadratic equation, you are confronted with Complex numbers, but what is a Complex number?

The magnitude or absolute value of a complex number z= x+ iyis r= p x2 +y2. Combine this with the complex exponential and you have another way to represent complex numbers. rsin rcos x r rei y z= x+iy= rcos +ir sin = r(cos i ) = rei (3:6) This is the polar form of a complex number and x+ iyis the rectangular form of the same number. The ...

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  Complex, Algebra, Exponential, Complex algebra, Complex exponential

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