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Unformatted text preview: 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. 3.1 Complex Numbers As soon as you learn to solve a quadratic equation, you are confronted with complex numbers, but what is a complex number? If the answer involves √- 1 then an appropriate response might be “What is that ?” Yes, we can manipulate objects such as- 1 + 2 i and get consistent results with them. We just have to follow certain rules, such as i 2 =- 1 . But is that an answer to the question? You can go through the entire subject of complex algebra and even complex calculus without learning a better answer, but it’s nice to have a more complete answer once, if then only to relax* and forget it. An answer to this question is to define complex numbers as pairs of real numbers, ( a,b ) . These pairs are made subject to rules of addition and multiplication: ( a,b ) + ( c,d ) = ( a + c,b + d ) and ( a,b )( c,d ) = ( ac- bd,ad + bc ) An algebraic system has to have something called zero, so that it plus any number leaves that number alone. Here that role is taken by (0 , 0) (0 , 0) + ( a,b ) = ( a + 0 ,b + 0) = ( a,b ) for all values of ( a,b ) What is the identity, the number such that it times any number leaves that number alone? (1 , 0)( c,d ) = (1 . c- . d, 1 . d + 0 . c ) = ( c,d ) so (1 , 0) has this role. Finally, where does √- 1 fit in? (0 , 1)(0 , 1) = (0 .- 1 . 1 , . 1 + 1 . 0) = (- 1 , 0) and the sum (- 1 , 0) + (1 , 0) = (0 , 0) so (0 , 1) is the representation of i = √- 1 , that is i 2 + 1 = 0 . (0 , 1) 2 + (1 , 0) = (0 , 0) . This set of pairs of real numbers satisfies all the desired properties that you want for complex numbers, so having shown that it is possible to express complex numbers in a precise way, I’ll feel free to ignore this more cumbersome notation and to use the more conventional representation with the symbol i : ( a,b ) ←→ a + ib That complex number will in turn usually be represented by a single letter, such as z = x + iy ....
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This note was uploaded on 01/30/2012 for the course PHYS 315 taught by Professor Nearing during the Fall '08 term at University of Miami.
- Fall '08