Solve 1-2i/2-i+3-i/3i

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

Evaluate

$$\frac{7}{15}-\frac{8}{5}i\approx 0.466666667-1.6i$$

Solution Steps

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

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

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

$$\frac{4-3i}{5}+\frac{3-i}{3i}$$

Divide 4-3i by 5 to get \frac{4}{5}-\frac{3}{5}i.

$$\frac{4}{5}-\frac{3}{5}i+\frac{3-i}{3i}$$

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

$$\frac{4}{5}-\frac{3}{5}i+\frac{1+3i}{-3}$$

Divide 1+3i by -3 to get -\frac{1}{3}-i.

$$\frac{4}{5}-\frac{3}{5}i+\left(-\frac{1}{3}-i\right)$$

Add \frac{4}{5}-\frac{3}{5}i and -\frac{1}{3}-i to get \frac{7}{15}-\frac{8}{5}i.

$$\frac{7}{15}-\frac{8}{5}i$$

Real Part

$$\frac{7}{15} = 0.4666666666666667$$

Solution Steps

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

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

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

$$Re(\frac{4-3i}{5}+\frac{3-i}{3i})$$

Divide 4-3i by 5 to get \frac{4}{5}-\frac{3}{5}i.

$$Re(\frac{4}{5}-\frac{3}{5}i+\frac{3-i}{3i})$$

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

$$Re(\frac{4}{5}-\frac{3}{5}i+\frac{1+3i}{-3})$$

Divide 1+3i by -3 to get -\frac{1}{3}-i.

$$Re(\frac{4}{5}-\frac{3}{5}i+\left(-\frac{1}{3}-i\right))$$

Add \frac{4}{5}-\frac{3}{5}i and -\frac{1}{3}-i to get \frac{7}{15}-\frac{8}{5}i.

$$Re(\frac{7}{15}-\frac{8}{5}i)$$

The real part of \frac{7}{15}-\frac{8}{5}i is \frac{7}{15}.

$$\frac{7}{15}$$