Solve -30i/80-30i

$$\frac { - 30 i } { 80 - 30 i }$$

Evaluate

$$\frac{9}{73}-\frac{24}{73}i\approx 0.123287671-0.328767123i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 80+30i.

$$\frac{-30i\left(80+30i\right)}{\left(80-30i\right)\left(80+30i\right)}$$

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

$$\frac{-30i\left(80+30i\right)}{80^{2}-30^{2}i^{2}}$$

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

$$\frac{-30i\left(80+30i\right)}{7300}$$

Multiply -30i times 80+30i.

$$\frac{-30i\times 80-30\times 30i^{2}}{7300}$$

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

$$\frac{-30i\times 80-30\times 30\left(-1\right)}{7300}$$

Do the multiplications in -30i\times 80-30\times 30\left(-1\right). Reorder the terms.

$$\frac{900-2400i}{7300}$$

Divide 900-2400i by 7300 to get \frac{9}{73}-\frac{24}{73}i.

$$\frac{9}{73}-\frac{24}{73}i$$

Real Part

$$\frac{9}{73} = 0.1232876712328767$$

Solution Steps

Multiply both numerator and denominator of \frac{-30i}{80-30i} by the complex conjugate of the denominator, 80+30i.

$$Re(\frac{-30i\left(80+30i\right)}{\left(80-30i\right)\left(80+30i\right)})$$

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

$$Re(\frac{-30i\left(80+30i\right)}{80^{2}-30^{2}i^{2}})$$

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

$$Re(\frac{-30i\left(80+30i\right)}{7300})$$

Multiply -30i times 80+30i.

$$Re(\frac{-30i\times 80-30\times 30i^{2}}{7300})$$

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

$$Re(\frac{-30i\times 80-30\times 30\left(-1\right)}{7300})$$

Do the multiplications in -30i\times 80-30\times 30\left(-1\right). Reorder the terms.

$$Re(\frac{900-2400i}{7300})$$

Divide 900-2400i by 7300 to get \frac{9}{73}-\frac{24}{73}i.

$$Re(\frac{9}{73}-\frac{24}{73}i)$$

The real part of \frac{9}{73}-\frac{24}{73}i is \frac{9}{73}.

$$\frac{9}{73}$$