Solve 5-3i/10-4i

$$\frac { 5 - 3 i } { 10 - 4 i }$$

Evaluate

$$\frac{31}{58}-\frac{5}{58}i\approx 0.534482759-0.086206897i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 10+4i.

$$\frac{\left(5-3i\right)\left(10+4i\right)}{\left(10-4i\right)\left(10+4i\right)}$$

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

$$\frac{\left(5-3i\right)\left(10+4i\right)}{10^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(5-3i\right)\left(10+4i\right)}{116}$$

Multiply complex numbers 5-3i and 10+4i like you multiply binomials.

$$\frac{5\times 10+5\times \left(4i\right)-3i\times 10-3\times 4i^{2}}{116}$$

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

$$\frac{5\times 10+5\times \left(4i\right)-3i\times 10-3\times 4\left(-1\right)}{116}$$

Do the multiplications in 5\times 10+5\times \left(4i\right)-3i\times 10-3\times 4\left(-1\right).

$$\frac{50+20i-30i+12}{116}$$

Combine the real and imaginary parts in 50+20i-30i+12.

$$\frac{50+12+\left(20-30\right)i}{116}$$

Do the additions in 50+12+\left(20-30\right)i.

$$\frac{62-10i}{116}$$

Divide 62-10i by 116 to get \frac{31}{58}-\frac{5}{58}i.

$$\frac{31}{58}-\frac{5}{58}i$$

Real Part

$$\frac{31}{58} = 0.5344827586206896$$

Solution Steps

Multiply both numerator and denominator of \frac{5-3i}{10-4i} by the complex conjugate of the denominator, 10+4i.

$$Re(\frac{\left(5-3i\right)\left(10+4i\right)}{\left(10-4i\right)\left(10+4i\right)})$$

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

$$Re(\frac{\left(5-3i\right)\left(10+4i\right)}{10^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(5-3i\right)\left(10+4i\right)}{116})$$

Multiply complex numbers 5-3i and 10+4i like you multiply binomials.

$$Re(\frac{5\times 10+5\times \left(4i\right)-3i\times 10-3\times 4i^{2}}{116})$$

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

$$Re(\frac{5\times 10+5\times \left(4i\right)-3i\times 10-3\times 4\left(-1\right)}{116})$$

Do the multiplications in 5\times 10+5\times \left(4i\right)-3i\times 10-3\times 4\left(-1\right).

$$Re(\frac{50+20i-30i+12}{116})$$

Combine the real and imaginary parts in 50+20i-30i+12.

$$Re(\frac{50+12+\left(20-30\right)i}{116})$$

Do the additions in 50+12+\left(20-30\right)i.

$$Re(\frac{62-10i}{116})$$

Divide 62-10i by 116 to get \frac{31}{58}-\frac{5}{58}i.

$$Re(\frac{31}{58}-\frac{5}{58}i)$$

The real part of \frac{31}{58}-\frac{5}{58}i is \frac{31}{58}.

$$\frac{31}{58}$$