Solve 4+3i/-1+5i

$$\frac { 4 + 3 i } { - 1 + 5 i }$$

Evaluate

$$\frac{11}{26}-\frac{23}{26}i\approx 0.423076923-0.884615385i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -1-5i.

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

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

$$\frac{\left(4+3i\right)\left(-1-5i\right)}{\left(-1\right)^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(4+3i\right)\left(-1-5i\right)}{26}$$

Multiply complex numbers 4+3i and -1-5i like you multiply binomials.

$$\frac{4\left(-1\right)+4\times \left(-5i\right)+3i\left(-1\right)+3\left(-5\right)i^{2}}{26}$$

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

$$\frac{4\left(-1\right)+4\times \left(-5i\right)+3i\left(-1\right)+3\left(-5\right)\left(-1\right)}{26}$$

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

$$\frac{-4-20i-3i+15}{26}$$

Combine the real and imaginary parts in -4-20i-3i+15.

$$\frac{-4+15+\left(-20-3\right)i}{26}$$

Do the additions in -4+15+\left(-20-3\right)i.

$$\frac{11-23i}{26}$$

Divide 11-23i by 26 to get \frac{11}{26}-\frac{23}{26}i.

$$\frac{11}{26}-\frac{23}{26}i$$

Real Part

$$\frac{11}{26} = 0.4230769230769231$$

Solution Steps

Multiply both numerator and denominator of \frac{4+3i}{-1+5i} by the complex conjugate of the denominator, -1-5i.

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

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

$$Re(\frac{\left(4+3i\right)\left(-1-5i\right)}{\left(-1\right)^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(4+3i\right)\left(-1-5i\right)}{26})$$

Multiply complex numbers 4+3i and -1-5i like you multiply binomials.

$$Re(\frac{4\left(-1\right)+4\times \left(-5i\right)+3i\left(-1\right)+3\left(-5\right)i^{2}}{26})$$

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

$$Re(\frac{4\left(-1\right)+4\times \left(-5i\right)+3i\left(-1\right)+3\left(-5\right)\left(-1\right)}{26})$$

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

$$Re(\frac{-4-20i-3i+15}{26})$$

Combine the real and imaginary parts in -4-20i-3i+15.

$$Re(\frac{-4+15+\left(-20-3\right)i}{26})$$

Do the additions in -4+15+\left(-20-3\right)i.

$$Re(\frac{11-23i}{26})$$

Divide 11-23i by 26 to get \frac{11}{26}-\frac{23}{26}i.

$$Re(\frac{11}{26}-\frac{23}{26}i)$$

The real part of \frac{11}{26}-\frac{23}{26}i is \frac{11}{26}.

$$\frac{11}{26}$$