Find All Complex Number Solutions z=2-2i

Problem

z=2−2*i

Solution

1. Identify the complex number z=a+b*i where a=2 and b=−2

2. Calculate the modulus r using the formula r=√(,a2+b2)

r=√(,2+(−2)2)

r=√(,4+4)

r=√(,8)

r=2√(,2)

3. Determine the argument θ using tan(θ)=b/a

tan(θ)=(−2)/2

tan(θ)=−1

4. Locate the quadrant. Since a>0 and b<0 the complex number is in the fourth quadrant.

θ=(7*π)/4

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

z=2√(,2)*(cos((7*π)/4)+i*sin((7*π)/4))

6. Generalize the solution for all complex representations by adding 2*k*π to the argument, where k is any integer.

θ=(7*π)/4+2*k*π

Final Answer

z=2√(,2)*(cos((7*π)/4+2*k*π)+i*sin((7*π)/4+2*k*π))

Want more problems? Check here!