Solve 1+i/-1+5i

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

Evaluate

$$\frac{2}{13}-\frac{3}{13}i\approx 0.153846154-0.230769231i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{-1-5i-i+5}{26}$$

Combine the real and imaginary parts in -1-5i-i+5.

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

Do the additions in -1+5+\left(-5-1\right)i.

$$\frac{4-6i}{26}$$

Divide 4-6i by 26 to get \frac{2}{13}-\frac{3}{13}i.

$$\frac{2}{13}-\frac{3}{13}i$$

Real Part

$$\frac{2}{13} = 0.15384615384615385$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{-1-5i-i+5}{26})$$

Combine the real and imaginary parts in -1-5i-i+5.

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

Do the additions in -1+5+\left(-5-1\right)i.

$$Re(\frac{4-6i}{26})$$

Divide 4-6i by 26 to get \frac{2}{13}-\frac{3}{13}i.

$$Re(\frac{2}{13}-\frac{3}{13}i)$$

The real part of \frac{2}{13}-\frac{3}{13}i is \frac{2}{13}.

$$\frac{2}{13}$$