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First year: (I) Complex numbers Definition, addition and multiplication A complex number is a number of the form a ib , where a and b are real numbers and where i is an imaginary number such that i 1 . Note that since i 1 , we have i 2 1 . Let z a ib . - The real part of z is a . We write Re( z ) a . The imaginary part of z is b . We write Im( z ) b . (1) Adding complex numbers: Say z1 a ib and z2 c id . Then z1 z2 a c i(b d ). Example: z1 1 i3, z2 2 i5 z1 z2 (1 2) i(3 5) 1 i2. (2) Multiplying complex numbers: Say z1 a ib and z2 c id . Then z1 z2 ac bd i(bc ad ) . Example: z1 1 i2, z2 2 i3 z1 z2 (1 i2)(2 i3), z1 z 2 2 i3 i 2 2 (i 2) (i3), z1 z 2 2 i3 i 4 i 2 2 3, z1 z 2 2 6 i (4 3) 4 i 7. (3) Exercises (i) Find z1 z2 and z1 z2 , where z1 1 i3, z2 2 i4 , z1 1 i3, z2 2 i 4, z1 1 i, z2 1 i 2. (ii) Find the real and imaginary part of z1 , where z1 (1 i3)(2 i7) i. (II) Modulus and complex conjugate The modulus of a complex number z a ib is z a 2 b 2 . For example if z 1 i3 , then z 12 32 10 . The complex conjugate of z a ib is z a ib . For example if z 1 i3 , then z 1 i3 . Note that z z z, 2 2 for, if z a ib , then z z (a ib )( a ib ) a 2 iab iba (ib )( ib ) a 2 b 2 z . Complex fractions: You will often be asked to find the real and imaginary parts of complex numbers of the form z 1 . 1 i2 To do so, you need to write z in the form a ib : - Multiply the numerator and the denominator by the complex conjugate of the denominator z - Make use of the equality z z z : 2 z - (1 i 2) (1 i 2)(1 i 2) 1 i2 1 i2 5 11 2 2 Finally split the fraction into real and imaginary parts: 1 2 z i 5 5 and conclude: Re( z ) 1 2 , Im( z ) . 5 5 (III) Polar form and De Moivre’s formula z a ib is called the Cartesian form of the complex number z . If the real numbers a are represented on a horizontal axis and the complex numbers ib are represented on a vertical axis, then you can locate the a complex number z a ib the way you would locate a vector in R 2 . b Another way of representing a complex number is to give its modulus z together with the angle between the line [0, z ) and the positive semi-axis [0,) . Note: the angle is called the argument of the complex number z and is measured anti-clockwise. Once you have the modulus and the argument of a complex number z a ib , you can write it in polar form: z re i , with r z a 2 b 2 , tan 1 (b / a) and ei cos( ) i sin( ). Example: Say z 4 i3 . Then r 42 32 5 and tan 1 (3 / 5) 0.54 rad 30.96o , so that z 5e i 0.54 . The angle is usually expressed in radian. De Moivre’s formula z r cos( ) i sin( ) z n r n cos(n ) i sin( n ) r n ein . Example: z 3cos( / 3) i sin( / 3) z 3 27cos( ) i sin( ) 27.