Solve 2/3i+1/1-i

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

Evaluate

$$\frac{1}{2}-\frac{1}{6}i\approx 0.5-0.166666667i$$

Solution Steps

Multiply both numerator and denominator of \frac{2}{3i} by imaginary unit i.

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

Divide 2i by -3 to get -\frac{2}{3}i.

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

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

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

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

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

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

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

Add -\frac{2}{3}i and \frac{1}{2}+\frac{1}{2}i to get \frac{1}{2}-\frac{1}{6}i.

$$\frac{1}{2}-\frac{1}{6}i$$

Real Part

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

Solution Steps

Multiply both numerator and denominator of \frac{2}{3i} by imaginary unit i.

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

Divide 2i by -3 to get -\frac{2}{3}i.

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

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

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

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

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

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

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

Add -\frac{2}{3}i and \frac{1}{2}+\frac{1}{2}i to get \frac{1}{2}-\frac{1}{6}i.

$$Re(\frac{1}{2}-\frac{1}{6}i)$$

The real part of \frac{1}{2}-\frac{1}{6}i is \frac{1}{2}.

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