Solve 1/2+3i-6+7i/3i

$$\frac { 1 } { 2 + 3 i } - \frac { 6 + 7 i } { 3 i }$$

Evaluate

$$-\frac{85}{39}+\frac{23}{13}i\approx -2.179487179+1.769230769i$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$\frac{1\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}-\frac{6+7i}{3i}$$

Do the multiplications in \frac{1\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}.

$$\frac{2-3i}{13}-\frac{6+7i}{3i}$$

Divide 2-3i by 13 to get \frac{2}{13}-\frac{3}{13}i.

$$\frac{2}{13}-\frac{3}{13}i-\frac{6+7i}{3i}$$

Multiply both numerator and denominator of \frac{6+7i}{3i} by imaginary unit i.

$$\frac{2}{13}-\frac{3}{13}i-\frac{-7+6i}{-3}$$

Divide -7+6i by -3 to get \frac{7}{3}-2i.

$$\frac{2}{13}-\frac{3}{13}i+\left(-\frac{7}{3}+2i\right)$$

Add \frac{2}{13}-\frac{3}{13}i and -\frac{7}{3}+2i to get -\frac{85}{39}+\frac{23}{13}i.

$$-\frac{85}{39}+\frac{23}{13}i$$

Real Part

$$-\frac{85}{39} = -2\frac{7}{39} = -2.1794871794871793$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$Re(\frac{1\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}-\frac{6+7i}{3i})$$

Do the multiplications in \frac{1\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}.

$$Re(\frac{2-3i}{13}-\frac{6+7i}{3i})$$

Divide 2-3i by 13 to get \frac{2}{13}-\frac{3}{13}i.

$$Re(\frac{2}{13}-\frac{3}{13}i-\frac{6+7i}{3i})$$

Multiply both numerator and denominator of \frac{6+7i}{3i} by imaginary unit i.

$$Re(\frac{2}{13}-\frac{3}{13}i-\frac{-7+6i}{-3})$$

Divide -7+6i by -3 to get \frac{7}{3}-2i.

$$Re(\frac{2}{13}-\frac{3}{13}i+\left(-\frac{7}{3}+2i\right))$$

Add \frac{2}{13}-\frac{3}{13}i and -\frac{7}{3}+2i to get -\frac{85}{39}+\frac{23}{13}i.

$$Re(-\frac{85}{39}+\frac{23}{13}i)$$

The real part of -\frac{85}{39}+\frac{23}{13}i is -\frac{85}{39}.

$$-\frac{85}{39}$$