Solve 1/(3+2i)

$$1 / ( 3 + 2 i )$$

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

Real Part

$$\frac{3}{13} = 0.23076923076923078$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{3}{13}$$