ME1302 - Part 2 - COMPLEX NUMBERS The simple algebraic...

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COMPLEX NUMBERS The simple algebraic equation 2 10 x + = has no solution in terms of real numbers. To overcome this limitation, we introduce the unit imaginary element i , defined as 1 i = In terms of this imaginary element, the solution of the above equation becomes 2 11 x or x i =− =± − =± Similarly, the solution of the equation 2 16 0 x + = is given by: 2 16 16 4 x or x i =± − The above equation has two different solutions, given by x = 4 i and x = –4 i . Numbers like i , 6 i , –4 i are called purely imaginary numbers, and they represent special cases of the general complex number a + ib , in which a and b are real numbers. The real numbers a and b are called the real and imaginary parts of the complex number z = a + ib , and denoted by R e () Im azbz = = Basic Algebraic Rules for Complex Numbers Sum and Difference If z 1 = a + ib and z 2 = c + id are any two complex numbers, then their sum is 12 ( zz aci bd ) + =++ + and their difference is ( ) =−+ −
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Product If z 1 = a + ib and z 2 = c + id are any two complex numbers, then their product is 2 12 () ( ) z z a ib c id ac i ad bc i bd =+×+ =+ + + Taking into account that i 2 = –1 gives ( z z ac bd i ad bc ) = −+ + The product of a real number k and a complex number z is given by kz k a ib ka ikb = +=+ Complex conjugate To any complex number a + ib there corresponds a complex number a ib , obtained by changing the sign of the imaginary part. The complex number a ib is called the complex conjugate of a + ib . The usual notation for complex conjugate is z . It follows that the product z z is a real number, given by 22 ( ) zz a ib a ib a i ab ab i b =+×− =+ − − 2 and since i 2 = –1, the result is a b = + Division If z 1 = a + ib and z 2 = c + id are any two complex numbers, then their quotient is given by 1 2 za c b db c a i zcd cd d + =+ ++ The above result is easy to prove as follows: ( ) 2 11 2 ai bci d z z z ac bd i bc ad c i d c i d cd z +− ++ − =×= = + Equality of complex numbers If two complex numbers z 1 = a + ib and z 2 = c + id are equal, then a = c and b = d .
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Zero complex number The zero complex number is the number 0 + 0 i , that is, both its real and imaginary parts are zero. Exercises : 1. If z 1 = 2 – 3 i and z 2 = – 1 + i , find: a) z 1 z 2 b) z 1 (2 z 2 +1) 2. If z = 2 – i , find: a) z b) z z 3. If z 1 = 3 + 2 i and z 2 = 1 – 3 i , find: a) z 1 / z 2 b) Re( z 1 / z 2 ) and Im( z 1 / z 2 ) 4. If z 1 = 1 – 2 i and z 2 = 2 + i , find: a) z 1 3 b) ( z 1 z 2 ) 2 Modulus-Argument Form of a Complex Number An alternative and important representation of a complex number involves the use of polar coordinates. This allows the complex number zxi y = + to be written in the form
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( ) cos sin zr i θ =+ where the modulus r is given by ( ) 1/2 22 rz x y == + and the argument θ is such that () cos sin xx yy and rr xy θθ == ++ The convention for the argument θ is that it always lies in the interval π < θ π .
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This note was uploaded on 04/20/2011 for the course ME 1331 taught by Professor Zeshan during the Spring '11 term at American College of Computer & Information Sciences.

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ME1302 - Part 2 - COMPLEX NUMBERS The simple algebraic...

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