Complex Numbers(4) by pptfiles

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									Complex Numbers All complex numbers consist of a real and imaginary part. The imaginary part is a multiple of i (where i = 1 ). We often use the letter ‘z’ to represent a complex number eg. z = 3 +5i The conjugate of z is written as z* or z If z1 = a + bi then the conjugate of z (z* ) = a – bi

Similarly if z2 = x – yi then the conjugate z2* = x + yi z z* will always be real (as i2 = -1)

For two expressions containing complex numbers to be equal, both the real parts must be equal and the imaginary parts must also be equal. If z1 = a + bi , z2 = x + yi and 2z1 = z2 + 3 then hence 2( a + bi) = x + yi + 3 2a + 2bi = x + 3 + yi so 2a = x + 3 (real parts are equal) and 2b = y (imaginary parts are equal) When adding/subtracting complex numbers deal with the real parts and the imaginary parts separately eg. z1 + z2 = a + bi + x + yi = a + x + (b + y)i When multiplying just treat as an algebraic expression in brackets eg. z1 z2 = (a + bi)(x + yi) = ax + ayi + bxi + byi2 = ax - by + (ay + bx)i

(as i2 = -1)

Division by a complex number is a very similar process to ‘rationalising’ surds – we call it ‘realising’

z1 a  bi  z2 x  yi z1 z2*   * z2 z2   (a  bi )( x  yi ) ( x  yi )( x  yi )  z 2*   *  1  z2 

ax  ayi  bxi  i 2by x 2  xyi  xyi  i 2 y 2 ax  by  (bx  ay )i  x2  y 2
Argand Diagrams We can represent complex numbers on an Argand diagram. This similar to a normal set of x and y axes except that the x axis represents the real part of the number and the y axis represents the imaginary part of the number. imaginary 4+4i 4

-3 +2i

2

-4

-2

0

2

4

real

-2 -4 – 3i 2-3i

-4

The argand diagrams allow complex numbers to be expressed in terms of an angle (the argument) and the length of the line joining the point z to the origin (the modulus of z). Hence the complex number can be expressed in a polar form. The argument is measured from the real axis and ranges from –п to п.

so for z=4+4i
4  4i  z arg( z ) 
2


2

z  4  42 4 2 When in this form some expressions for complex numbers can be drawn as loci. z  z1  r

This means that the distance between the fixed point z1 and the loci z is a constant value r, thus z is a circle of radius r about.
arg( z  z1 )  

This means that the argument of the line between the loci z and the point z1 has an argument of  . Thus the loci z is the line from z1 at an argument of  .

z  z1  z  z2
This means that the line joining the point z1 to the loci z is equal in length to the line joining z 2 to the loci z. therefore the loci is the perpendicular bisector of the line joining the two points.

z  z1  k z  z2 The same as above but rather than the locus being equidistant from both points it is k times further away from z1 than z 2 .
From an argand diagram complex numbers can be express using a modulus and an argument, the component real and imaginary parts of these numbers can then be expressed in a similar way to a resolved vector.
z R arg( z )  

 z  R (cos   i sin  )


								
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