Solve z=i-1/1-i

$$z = i - \frac { 1 } { 1 - i }$$

Solve for z

$$z=-\frac{1}{2}+\frac{1}{2}i=-0.5+0.5i$$

Steps for Solving Linear Equation

Multiply both numerator and denominator of \frac{1}{1-i} by the complex conjugate of the denominator, 1+i.

$$z=i-\frac{1\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$z=i-\frac{1\left(1+i\right)}{1^{2}-i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$z=i-\frac{1\left(1+i\right)}{2}$$

Multiply 1 and 1+i to get 1+i.

$$z=i-\frac{1+i}{2}$$

Divide 1+i by 2 to get \frac{1}{2}+\frac{1}{2}i.

$$z=i+\left(-\frac{1}{2}-\frac{1}{2}i\right)$$

Combine the real and imaginary parts in numbers i and -\frac{1}{2}-\frac{1}{2}i.

$$z=-\frac{1}{2}+\left(1-\frac{1}{2}\right)i$$

Add 1 to -\frac{1}{2}.

$$z=-\frac{1}{2}+\frac{1}{2}i$$

Assign z

$$z≔-\frac{1}{2}+\frac{1}{2}i$$