Solve frac{20-30texttt{i}}{30-30texttt{i}}

$$\frac{ 20-30 \texttt{i} }{ 30-30 \texttt{i} }$$

Evaluate

$$\frac{5}{6}-\frac{1}{6}i\approx 0.833333333-0.166666667i$$

Solution Steps

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

$$\frac{\left(20-30i\right)\left(30+30i\right)}{\left(30-30i\right)\left(30+30i\right)}$$

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

$$\frac{\left(20-30i\right)\left(30+30i\right)}{30^{2}-30^{2}i^{2}}$$

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

$$\frac{\left(20-30i\right)\left(30+30i\right)}{1800}$$

Multiply complex numbers 20-30i and 30+30i like you multiply binomials.

$$\frac{20\times 30+20\times \left(30i\right)-30i\times 30-30\times 30i^{2}}{1800}$$

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

$$\frac{20\times 30+20\times \left(30i\right)-30i\times 30-30\times 30\left(-1\right)}{1800}$$

Do the multiplications in 20\times 30+20\times \left(30i\right)-30i\times 30-30\times 30\left(-1\right).

$$\frac{600+600i-900i+900}{1800}$$

Combine the real and imaginary parts in 600+600i-900i+900.

$$\frac{600+900+\left(600-900\right)i}{1800}$$

Do the additions in 600+900+\left(600-900\right)i.

$$\frac{1500-300i}{1800}$$

Divide 1500-300i by 1800 to get \frac{5}{6}-\frac{1}{6}i.

$$\frac{5}{6}-\frac{1}{6}i$$

Real Part

$$\frac{5}{6} = 0.8333333333333334$$

Solution Steps

Multiply both numerator and denominator of \frac{20-30i}{30-30i} by the complex conjugate of the denominator, 30+30i.

$$Re(\frac{\left(20-30i\right)\left(30+30i\right)}{\left(30-30i\right)\left(30+30i\right)})$$

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

$$Re(\frac{\left(20-30i\right)\left(30+30i\right)}{30^{2}-30^{2}i^{2}})$$

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

$$Re(\frac{\left(20-30i\right)\left(30+30i\right)}{1800})$$

Multiply complex numbers 20-30i and 30+30i like you multiply binomials.

$$Re(\frac{20\times 30+20\times \left(30i\right)-30i\times 30-30\times 30i^{2}}{1800})$$

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

$$Re(\frac{20\times 30+20\times \left(30i\right)-30i\times 30-30\times 30\left(-1\right)}{1800})$$

Do the multiplications in 20\times 30+20\times \left(30i\right)-30i\times 30-30\times 30\left(-1\right).

$$Re(\frac{600+600i-900i+900}{1800})$$

Combine the real and imaginary parts in 600+600i-900i+900.

$$Re(\frac{600+900+\left(600-900\right)i}{1800})$$

Do the additions in 600+900+\left(600-900\right)i.

$$Re(\frac{1500-300i}{1800})$$

Divide 1500-300i by 1800 to get \frac{5}{6}-\frac{1}{6}i.

$$Re(\frac{5}{6}-\frac{1}{6}i)$$

The real part of \frac{5}{6}-\frac{1}{6}i is \frac{5}{6}.

$$\frac{5}{6}$$