VIEWS: 26 PAGES: 3 POSTED ON: 1/21/2010 Public Domain
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 )