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

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

Evaluate

$$\frac{17}{26}+\frac{7}{26}i\approx 0.653846154+0.269230769i$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$\frac{1}{2}+\frac{1}{2}i+\frac{2-3i}{13}$$

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

$$\frac{1}{2}+\frac{1}{2}i+\left(\frac{2}{13}-\frac{3}{13}i\right)$$

Add \frac{1}{2}+\frac{1}{2}i and \frac{2}{13}-\frac{3}{13}i to get \frac{17}{26}+\frac{7}{26}i.

$$\frac{17}{26}+\frac{7}{26}i$$

Real Part

$$\frac{17}{26} = 0.6538461538461539$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$Re(\frac{1}{2}+\frac{1}{2}i+\frac{2-3i}{13})$$

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

$$Re(\frac{1}{2}+\frac{1}{2}i+\left(\frac{2}{13}-\frac{3}{13}i\right))$$

Add \frac{1}{2}+\frac{1}{2}i and \frac{2}{13}-\frac{3}{13}i to get \frac{17}{26}+\frac{7}{26}i.

$$Re(\frac{17}{26}+\frac{7}{26}i)$$

The real part of \frac{17}{26}+\frac{7}{26}i is \frac{17}{26}.

$$\frac{17}{26}$$