Find All Complex Number Solutions z=-6

Problem

z=−6

Solution

1. Identify the complex number in standard form a+b*i Here, z=−6+0*i

2. Determine the modulus r of the complex number.

r=√(,(−6)2+0)

r=6

3. Determine the argument θ of the complex number. Since the number lies on the negative real axis, the angle is π radians (or 180.

θ=π

4. Express the complex number in polar form z=r*(cos(θ)+i*sin(θ))

z=6*(cos(π)+i*sin(π))

5. Generalize the solution for all complex representations by adding multiples of 2*π to the argument.

z=6*(cos(π+2*k*π)+i*sin(π+2*k*π))

6. Simplify using Euler's formula e(i*θ)=cos(θ)+i*sin(θ)

z=6*e(i*(π+2*k*π))

Final Answer

z=6*e(i*(π+2*k*π)),k∈ℤ

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