Solve 3-2i/3+2i

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

Evaluate

$$\frac{5}{13}-\frac{12}{13}i\approx 0.384615385-0.923076923i$$

Solution Steps

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

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

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

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

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

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

Multiply complex numbers 3-2i and 3-2i like you multiply binomials.

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

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

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

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

$$\frac{9-6i-6i-4}{13}$$

Combine the real and imaginary parts in 9-6i-6i-4.

$$\frac{9-4+\left(-6-6\right)i}{13}$$

Do the additions in 9-4+\left(-6-6\right)i.

$$\frac{5-12i}{13}$$

Divide 5-12i by 13 to get \frac{5}{13}-\frac{12}{13}i.

$$\frac{5}{13}-\frac{12}{13}i$$

Real Part

$$\frac{5}{13} = 0.38461538461538464$$

Solution Steps

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

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

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

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

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

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

Multiply complex numbers 3-2i and 3-2i like you multiply binomials.

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

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

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

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

$$Re(\frac{9-6i-6i-4}{13})$$

Combine the real and imaginary parts in 9-6i-6i-4.

$$Re(\frac{9-4+\left(-6-6\right)i}{13})$$

Do the additions in 9-4+\left(-6-6\right)i.

$$Re(\frac{5-12i}{13})$$

Divide 5-12i by 13 to get \frac{5}{13}-\frac{12}{13}i.

$$Re(\frac{5}{13}-\frac{12}{13}i)$$

The real part of \frac{5}{13}-\frac{12}{13}i is \frac{5}{13}.

$$\frac{5}{13}$$