Solve 1/2+3i

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

Evaluate

$$\frac{2}{13}-\frac{3}{13}i\approx 0.153846154-0.230769231i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2-3i.

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

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

By definition, i^{2} is -1. Calculate the denominator.

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

Multiply 1 and 2-3i to get 2-3i.

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

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

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

Real Part

$$\frac{2}{13} = 0.15384615384615385$$

Solution Steps

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

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

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

By definition, i^{2} is -1. Calculate the denominator.

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

Multiply 1 and 2-3i to get 2-3i.

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

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

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

The real part of \frac{2}{13}-\frac{3}{13}i is \frac{2}{13}.

$$\frac{2}{13}$$