Solve 3-2i/4-i+i/2-i

$$\frac { 3 - 2 i } { 4 - i } + \frac { i } { 2 - i }$$

Evaluate

$$\frac{53}{85}+\frac{9}{85}i\approx 0.623529412+0.105882353i$$

Solution Steps

Multiply both numerator and denominator of \frac{3-2i}{4-i} by the complex conjugate of the denominator, 4+i.

$$\frac{\left(3-2i\right)\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}+\frac{i}{2-i}$$

Do the multiplications in \frac{\left(3-2i\right)\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}.

$$\frac{14-5i}{17}+\frac{i}{2-i}$$

Divide 14-5i by 17 to get \frac{14}{17}-\frac{5}{17}i.

$$\frac{14}{17}-\frac{5}{17}i+\frac{i}{2-i}$$

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

$$\frac{14}{17}-\frac{5}{17}i+\frac{i\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}$$

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

$$\frac{14}{17}-\frac{5}{17}i+\frac{-1+2i}{5}$$

Divide -1+2i by 5 to get -\frac{1}{5}+\frac{2}{5}i.

$$\frac{14}{17}-\frac{5}{17}i+\left(-\frac{1}{5}+\frac{2}{5}i\right)$$

Add \frac{14}{17}-\frac{5}{17}i and -\frac{1}{5}+\frac{2}{5}i to get \frac{53}{85}+\frac{9}{85}i.

$$\frac{53}{85}+\frac{9}{85}i$$

Real Part

$$\frac{53}{85} = 0.6235294117647059$$

Solution Steps

Multiply both numerator and denominator of \frac{3-2i}{4-i} by the complex conjugate of the denominator, 4+i.

$$Re(\frac{\left(3-2i\right)\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}+\frac{i}{2-i})$$

Do the multiplications in \frac{\left(3-2i\right)\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}.

$$Re(\frac{14-5i}{17}+\frac{i}{2-i})$$

Divide 14-5i by 17 to get \frac{14}{17}-\frac{5}{17}i.

$$Re(\frac{14}{17}-\frac{5}{17}i+\frac{i}{2-i})$$

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

$$Re(\frac{14}{17}-\frac{5}{17}i+\frac{i\left(2+i\right)}{\left(2-i\right)\left(2+i\right)})$$

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

$$Re(\frac{14}{17}-\frac{5}{17}i+\frac{-1+2i}{5})$$

Divide -1+2i by 5 to get -\frac{1}{5}+\frac{2}{5}i.

$$Re(\frac{14}{17}-\frac{5}{17}i+\left(-\frac{1}{5}+\frac{2}{5}i\right))$$

Add \frac{14}{17}-\frac{5}{17}i and -\frac{1}{5}+\frac{2}{5}i to get \frac{53}{85}+\frac{9}{85}i.

$$Re(\frac{53}{85}+\frac{9}{85}i)$$

The real part of \frac{53}{85}+\frac{9}{85}i is \frac{53}{85}.

$$\frac{53}{85}$$