Solve 3-4i/3+4i

$$\frac { 3 - 4 i } { 3 + 4 i }$$

Evaluate

$$-\frac{7}{25}-\frac{24}{25}i=-0.28-0.96i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-4i.

$$\frac{\left(3-4i\right)\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)}$$

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

$$\frac{\left(3-4i\right)\left(3-4i\right)}{3^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(3-4i\right)\left(3-4i\right)}{25}$$

Multiply complex numbers 3-4i and 3-4i like you multiply binomials.

$$\frac{3\times 3+3\times \left(-4i\right)-4i\times 3-4\left(-4\right)i^{2}}{25}$$

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

$$\frac{3\times 3+3\times \left(-4i\right)-4i\times 3-4\left(-4\right)\left(-1\right)}{25}$$

Do the multiplications in 3\times 3+3\times \left(-4i\right)-4i\times 3-4\left(-4\right)\left(-1\right).

$$\frac{9-12i-12i-16}{25}$$

Combine the real and imaginary parts in 9-12i-12i-16.

$$\frac{9-16+\left(-12-12\right)i}{25}$$

Do the additions in 9-16+\left(-12-12\right)i.

$$\frac{-7-24i}{25}$$

Divide -7-24i by 25 to get -\frac{7}{25}-\frac{24}{25}i.

$$-\frac{7}{25}-\frac{24}{25}i$$

Real Part

$$-\frac{7}{25} = -0.28$$

Solution Steps

Multiply both numerator and denominator of \frac{3-4i}{3+4i} by the complex conjugate of the denominator, 3-4i.

$$Re(\frac{\left(3-4i\right)\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)})$$

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

$$Re(\frac{\left(3-4i\right)\left(3-4i\right)}{3^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(3-4i\right)\left(3-4i\right)}{25})$$

Multiply complex numbers 3-4i and 3-4i like you multiply binomials.

$$Re(\frac{3\times 3+3\times \left(-4i\right)-4i\times 3-4\left(-4\right)i^{2}}{25})$$

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

$$Re(\frac{3\times 3+3\times \left(-4i\right)-4i\times 3-4\left(-4\right)\left(-1\right)}{25})$$

Do the multiplications in 3\times 3+3\times \left(-4i\right)-4i\times 3-4\left(-4\right)\left(-1\right).

$$Re(\frac{9-12i-12i-16}{25})$$

Combine the real and imaginary parts in 9-12i-12i-16.

$$Re(\frac{9-16+\left(-12-12\right)i}{25})$$

Do the additions in 9-16+\left(-12-12\right)i.

$$Re(\frac{-7-24i}{25})$$

Divide -7-24i by 25 to get -\frac{7}{25}-\frac{24}{25}i.

$$Re(-\frac{7}{25}-\frac{24}{25}i)$$

The real part of -\frac{7}{25}-\frac{24}{25}i is -\frac{7}{25}.

$$-\frac{7}{25}$$