Solve 1/1-i+2i/1-i

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

Evaluate

$$-\frac{1}{2}+\frac{3}{2}i=-0.5+1.5i$$

Solution Steps

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

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

Do the multiplications in \frac{1\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

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

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

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

Multiply both numerator and denominator of \frac{2i}{1-i} by the complex conjugate of the denominator, 1+i.

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

Do the multiplications in \frac{2i\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

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

Divide -2+2i by 2 to get -1+i.

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

Add \frac{1}{2}+\frac{1}{2}i and -1+i to get -\frac{1}{2}+\frac{3}{2}i.

$$-\frac{1}{2}+\frac{3}{2}i$$

Real Part

$$-\frac{1}{2} = -0.5$$

Solution Steps

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

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

Do the multiplications in \frac{1\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$Re(\frac{1+i}{2}+\frac{2i}{1-i})$$

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

$$Re(\frac{1}{2}+\frac{1}{2}i+\frac{2i}{1-i})$$

Multiply both numerator and denominator of \frac{2i}{1-i} by the complex conjugate of the denominator, 1+i.

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

Do the multiplications in \frac{2i\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$Re(\frac{1}{2}+\frac{1}{2}i+\frac{-2+2i}{2})$$

Divide -2+2i by 2 to get -1+i.

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

Add \frac{1}{2}+\frac{1}{2}i and -1+i to get -\frac{1}{2}+\frac{3}{2}i.

$$Re(-\frac{1}{2}+\frac{3}{2}i)$$

The real part of -\frac{1}{2}+\frac{3}{2}i is -\frac{1}{2}.

$$-\frac{1}{2}$$