Chapter2-ComplexNumbers - 2 2.1 Complex Numbers The...

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2 Complex Numbers 2.1 The imaginary number i [see Riley 3.1, 3.3] Complex numbers are a generalisation of real numbers. they occur in many branches of mathematics and have numerous applications in physics. The imaginary number is i = - 1 i 2 = - 1 The obvious place to see where we have already needed this is in the solution to quadratic equation. Eg. finds the roots of z 2 + 4 z + 5 = 0 ( z + 2) 2 + 1 = 0 ( z + 2) 2 = - 1 and z 1 , 2 = - 2 ± - 1 . So in this case we use the imaginary number and write the solutions as z 1 , 2 = - 2 ± i . which is called a complex number. The general form of a complex number is z = x + iy where z is the conventional representation and is the sum of the real part x and i times the imaginary part y : these are denoted as Re ( z ) = x Im ( z ) = y , respectively. The imaginary or real part can be zero, so if the imaginary part is, the number is real and hence real numbers are just a subset of complex numbers. Also when using the quadratic solutions formula, we had situations where there were no (real) roots as b 2 - 4 ac < 0. we could have solved the above quadratic to get the same results: z 1 , 2 = - 4 ± 16 - 20 2 = - 4 ± - 4 2 = - 4 ± 2 - 1 2 = - 2 ± i . A complex number may also be written more compactly as z = ( x, y ) where x and y are two real numbers which define the complex number and may be thought of as Cartesian coordinates. 1
z=x+iy q r Re(z ) Im(z) Argand diagram Recall that in Cartesian coordinates x = r cos θ y = r sin θ Therefore we can represent z in polar coordinates as z = x + iy = r (cos θ + i sin θ ) . The number r is called the modulus of z , written as | z | or mod( z ). This can be written in terms of x and y as | z | = p x 2 + y 2 . The angle θ is called the argument of z , written as arg( z ) (or arg z ) and is defined as arg( z ) = tan - 1 y x . so arg( z ) is the angle that the line joining the origin to z on an Argand diagram makes with the positive x - axis. The anti-clockwise direction is taken to be positive by convention. However, θ is not unique since θ + 2 ( n is zero or any integer) are also arguments for the same complex number. We therefore define a principal value of a complex number as that value of θ which satisfies - π < θ π . (it could also be 0 < θ 2 π ). Also, account must be taken of the signs of

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