Solve 2-3i/2+3i=

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

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, 2-3i.

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

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

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

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

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

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

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

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

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

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

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

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

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

Do the additions in 4-9+\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{2-3i}{2+3i} by the complex conjugate of the denominator, 2-3i.

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

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

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

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

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

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

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

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

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

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

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

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

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

Do the additions in 4-9+\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}.

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