1.6 Complex Numbers

Consider `z = cos theta + i sin theta " where " z in CC, z != 1`

Show that `Re ((1+z)/(1-z)) = 0`

`(1 + z) / (1 - z) = (1 + cos theta + i sin theta) / (1 - cos theta - i sin theta)`

attempt to use the complex conjugate of their denominator

`= ((1 + cos theta + i sin theta) (1 - cos theta + i sin theta)) / ((1 - cos theta - i sin theta) (1 - cos theta + i sin theta))`

`Re ((1+z)/(1-z)) = (1 - cos^2 theta - sin^2 theta) / ((1 - cos theta)^2 + sin^2 theta) (= (1 - cos^2 theta - sin^2 theta) / (2 - 2cos theta))`

using `g cos^2 theta + sin^2 theta = 1`to simplify the numerator

`Re ((1+z)/(1-z)) = 0`

Consider the complex numbers `z = 2 (cos (pi/5) + i sin (pi/5)) and w = 8 (cos ((2k pi)/5) - i sin ((2k pi)/5)), where k in ZZ^+`

(a) Find the modulus of zw.

(b) Find the argument of zw in terms of k. Suppose that `zw in ZZ`

(c) In what year does the number of units sold first exceed 5000 ? Between 1990 and 1992, the total number of units sold is 760 .

(i) Find the minimum value of k.
(ii) For the value of k found in part (i), find the value of zw.

(a)

|zw| = 16

(b)

`"attempt to find " arg(z) + arg(w)`

`"attempt to find " arg(z) + arg(w)`

`= pi/5 - (2k pi)/5 (= ((1 - 2k)pi)/5)` 

(c) 

(i)

`zw in ZZ` 

 `arg(zw)` is a multiple of  `pi`

`1 - 2k` is a multiple of 5

`k = 3`

(ii)

`zw = 16 (cos(-pi) + i sin(-pi)) - 16`

Consider `z = cos ((11pi)/18) + i sin ((11pi)/18)`

(a) Find the smallest value of n that satisfies `z^n = -i, "where" n in ZZ^+`

(b) Hence or otherwise, describe a single geometric transformation applied to z on the Argand diagram that results in z¹⁰

(a)

attempt to use De Moivre's theorem

`(cos ((11pi)/18) + i sin ((11pi)/18))^n = cos ((11n pi)/18) + i sin ((11n pi)/18) (= e^(((11n pi)/18)i))`

EITHER attempt to consider imaginary part in `sin ((11n pi)/18) = -1`

THEN `(11n pi)/18 = (3pi)/2, (7pi)/2, (11pi)/2 ... (= (3pi)/2 + 2k pi, k in ZZ)`

n = 9

(b)

EITHER `z^10 = e^((110pi)/18 1) (= e^((55pi)/9 1) = e^((pi)/9 1))`

recognising that the difference between `arg(z^10) " and " arg(z)` is needed

`arg(z^10) - arg(z) = pi/9 - (11pi)/18 = -pi/2`

THEN a rotation  `(3pi)/2`OR `-pi/2` OR equivalent angle about the origin.

`"Let" z_1 = a (cos (pi/4) + i sin (pi/4)) " and " z_2 = b (cos (pi/3) + i sin (pi/3))`

`"Express" (z_1/z_2)^3 " in the form " z = x + yi.`

`x = (sqrt(2a^3)) / (2b^3) , y = (-sqrt(2a^3)) / (2b^3) " or " x = a^3 / (sqrt(2b^3)) , y = -a^3 / (sqrt(2b^3))`

If `z_1 = a + a sqrt(3) i " and " z_2 = 1 - i` where  is a real constant, express z₁ and z₂ in the form `r cis theta`, and hence find an expression for `(z_1 / z_2)^6` in terms of a and i.

`z_1 = 2a cis (pi/3), z_2 = sqrt(2) cis (-pi/4)`

EITHER

`(z_1/z_2)^6 = (2^6 a^6 cis(0)) / (sqrt(2)^6 cis(pi/2)) (= 8a^6 cis(-pi/2))`

OR

`(z_1/z_2)^6 = ((2a) / sqrt(2) cis ((7pi)/12))^6 = 8a^6 cis (-pi/2)`

THEN

= `-8a^6i`

`(z + 2i)` is a factor of `2z^3 - 3z^2 +8z - 12`

Find the other two factors.

If `(z+2i)`is a factor then `(z - 2i)`is also a factor.

`(z + 2i)(z - 2i) = (z^2 + 4)`

The other factor is `(2z^3 - 3z^2 + 8z - 12) / (z^2 + 4) = (2z - 3)`

The other two factors are `(z - 2i) and (2z - 3)`

Consider the equation `2(p + iq) = q - ip - 2(1 - i)`, where p and q are both real numbers. Find p and q.

`2(p + iq) = q - ip - 2(1 - i)`

`2p = q - 2`

`2q = -p +2`

`p = -0.4 and q = 1.2`

(a) Find the roots of the equation `w^3 = 8i, w in CC`. Give your answers in Cartesian form. One of the roots w₁ satisfies the condition `Re(w_1) = 0`

`b) Given that `w_1 = z / (z - i)` , express z in the form  `a + bi`, where `a, b in Q`

(a)

`w^3 + (2i)^3 = 0`

`(w + 2i)(w^2 - 2wi - 4) = 0`

`w = (2i +- sqrt(12)) / 2`

`w = sqrt(3) + i, -sqrt(3) + i, - 2i`

(b)

`w_1 = - 2i`

`z/(z-i) = -2i`

`z = - 2i (z - i)`

`z (2i + 1) = -2`

`z = -2/ (1 + 2i)`

`z = -2/5 + 4/5i`

Two distinct roots for the equation `z^4 - 10z^3 + az^2 + bz + 50 = 0`are `c + i`and `2 + id, where `a, b, c, d in R, d > 0`

(a) Write down the other two roots in terms of c and d.

(b) Find the value of c and the value of d.

(a)

other two roots are `c - i and 2 - id`

(b)

use of sum of roots

`2c + 4 = 10 => c =3`

use of product of roots

product is

`(c + i)(c - i)(2 + id)(2 - id)`

`(c^2 + 1)(4 + d^2)[= 10(4 + d^2)] = 50`

`d = 1`

`z_1 = 2 sqrt(3) cis ((3pi)/2) " and " z_2 = -1 + sqrt(3) i`

(a) 

(i) Write down `z_1` in Cartesian form.

(ii) Hence determine `(z_1 + z_2)`* in Cartesian form.

(b) 

(i) Write `z_2` in modulus-argument form.

(ii) Hence solve the equation `z^3 = z_2`

(c) `"Let" z = r cis theta ", where " r in RR^+ " and " 0 <= theta < 2pi.`Find all possible values of `r` and `theta` ,

(i) if `z^2 = (1 + z_2)^2`

(ii) if `z = - 1/z_2`

(d) Find the smallest positive value of  for which `(z_1/z_2)^n in R^+`

(a)

(i)

`z_1 = 2 sqrt(3) cis ((3pi)/2) => z_1 = -2 sqrt(3) i`

(ii)

`z_1 + z_2 = - 2 sqrt(3) i - 1 + sqrt(3) i = - 1 - sqrt(3) i`

`(z_1 + z_2)`* = `-1 + sqrt(3)i`

(b) 

(i)

`|z_2| = 2`

`tantheta = - sqrt(3)`

`z_2`lies on the second quadrant

`theta = arg`

`z_2 = (2pi)/3`

`z_2 = 2cis(2pi)/3`

(ii)

attempt to use De Moivre's theorem

`z = root(3)(2) cis (((2pi)/3 + 2k pi) / 3), k = 0, 1 " and " 2`

`z = root(3)(2) cis ((2pi)/9), root(3)(2) cis ((8pi)/9), root(3)(2) cis ((14pi)/9) (= root(3)(2) cis ((-4pi)/9))`

c)

(i)

`z^2 = (1-1 + sqrt(3)i)^2 = -3`

`=> z = +- sqrt(3)i`

`z = sqrt(3)cispi/2`

or `z_1 = sqrt(3) cis ((3pi)/2) (= sqrt(3) cis (-pi/2))`

`"so " r = sqrt(3) " and " theta = pi/2 " or " theta = (3pi)/2 (= -pi/2)`

(ii)

`z = -1/(2cis(2pi)/3)`

`=> z = (cispi)/ (2cis(2pi)/3)`

`=> z=1/2cispi/3`

`r = 1/2 and theta = pi/3`

(d)

`z_1 / z_2 = sqrt(3) cis ((5pi)/6) => (z_1 / z_2)^n = sqrt(3)^n cis ((5n pi)/6)`

equating imaginary part to zero and attempting to solve obtain n = 12