(i) Express the complex number z = 4i/(1-i√3)in Cartesian form.

(ii) Determine the modulus and argument of z, and hence express z in
polar form.
(b) Solve the equation z4 − z = 0, expressing your solutions in Cartesian form.

4i/(1-√3 i)

= 4i(1+√3 i)/(1+3) = -√3 + i
Now, tanθ = -1/√3, so z = (2,-π/6)

z^4-z = 0
z(z^3-1) = 0
z(z-1)(z^2+z+1) = 0
and so on

b) form a) z = -√3 + i

which is in quadrant II if graphed in the Argand plane
so if tanθ = -1/√3 , θ = 5π/6
modulus of z = √(3 + 1) = 2
argument of z = 5π/5
z = 2cos 5π/6 + 2sin 5π/6

check: 2cos 5π/6 + 2sin 5π/6 i = -1.73205.. + (1)i or -√3 + i

(i) To express the complex number z = 4i/(1 - i√3) in Cartesian form, we start by rationalizing the denominator.

Multiply the numerator and denominator by the conjugate of the denominator:
z = (4i/(1 - i√3)) * ((1 + i√3)/(1 + i√3))

Next, simplify the numerator:
z = (4i * (1 + i√3)) / (1 + i√3)

Now, multiply the numerator out:
z = (4i + 4i^2√3) / (1 + i√3)

Simplify the denominator using the formula (a + bi)(a - bi) = a^2 - b^2i^2:
z = (4i + 4i^2√3) / (1 - 3i^2)

Since i^2 = -1, we can substitute it in:
z = (4i - 4√3) / (1 + 3)

Simplify further:
z = (4i - 4√3) / 4
z = i - √3

Therefore, the complex number z in Cartesian form is z = i - √3.

(ii) To determine the modulus and argument of z, we use the Cartesian form z = i - √3.

The modulus or absolute value of z is the distance from the origin in the complex plane:
|z| = √(Re(z)^2 + Im(z)^2)
= √((0)^2 + (-√3)^2)
= √3

The argument of z is the angle it makes with the positive x-axis:
arg(z) = arctan(Im(z)/Re(z))
= arctan((-√3)/(0))
= π/2 (or 90 degrees) in the principal range

So, the modulus of z is √3 and the argument of z is π/2.

In polar form, z can be expressed as z = √3 * cis(π/2), where cis(θ) = cos(θ) + isin(θ).

(b) To solve the equation z^4 - z = 0, we can factor out z:
z(z^3 - 1) = 0

From this, we have two possibilities:

1. z = 0
2. z^3 - 1 = 0

Let's solve each case separately:

1. z = 0

When z = 0, the equation z^4 - z = 0 is satisfied.

2. z^3 - 1 = 0

To solve z^3 - 1 = 0, we can rewrite it as:
(z - 1)(z^2 + z + 1) = 0

From the first factor, z - 1 = 0, we get:
z = 1

From the second factor, z^2 + z + 1 = 0, we solve the quadratic equation by using the quadratic formula:
z = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 1, and c = 1:
z = (-1 ± √(1 - 4(1)(1))) / (2(1))
z = (-1 ± √(-3)) / 2

We have a complex square root because the discriminant is negative.

Using the imaginary unit i, we can express the solutions as:
z = (-1 ± i√3) / 2

Therefore, the solutions to the equation z^4 - z = 0 in Cartesian form are z = 0, z = 1, and z = (-1 ± i√3) / 2.

(i) To express the complex number z = 4i/(1-i√3) in Cartesian form, we need to rationalize the denominator.

First, let's multiply the numerator and denominator by the conjugate of the denominator, which is (1+i√3).
So, z = (4i * (1+i√3))/((1-i√3) * (1+i√3))

Expanding, we have:
z = (4i + 4i√3)/(1 - i^2√3^2)

Simplifying further:
z = (4i + 4i√3)/(1 + 3)
z = (4i + 4i√3)/4
z = i + i√3

Therefore, z in Cartesian form is z = i + i√3.

(ii) To determine the modulus and argument of z and express z in polar form, let's consider the Cartesian form of z, which is z = i + i√3.

Modulus (magnitude):
The modulus of a complex number can be found using the formula: |z| = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number.
In this case, a = 0 and b = 1√3. Therefore, |z| = √(0^2 + (1√3)^2) = √(0 + 3) = √3.

Argument (angle):
The argument of a complex number can be found using the formula: Arg(z) = atan(b/a), where a and b are the real and imaginary parts of the complex number.
In this case, a = 0 and b = 1√3. Therefore, Arg(z) = atan((1√3)/0) = π/2 (since we have a vertical line).

Polar form:
Using the modulus and argument, we can express z in polar form as z = √3 * e^(iπ/2).

(b) To solve the equation z^4 - z = 0, let's express it in Cartesian form and find its solutions.
Let z = a + bi, where a and b are real numbers.

Substituting this into the equation, we get:
(a + bi)^4 - (a + bi) = 0

Expanding and simplifying:
a^4 + 4a^3bi + 6a^2b^2 + 4ab^3i^2 + b^4i^4 - a - bi = 0

Simplifying further:
(a^4 - 6a^2b^2 + b^4) + (4a^3b - 4ab^3)i - (a + bi) = 0

Comparing real and imaginary parts, we have:
a^4 - 6a^2b^2 + b^4 - a = 0 (1)
4a^3b - 4ab^3 - b = 0 (2)

We have a system of two equations (1) and (2) that needs to be solved simultaneously to find the solutions. The solutions to this equation will be in Cartesian form.