Solve 1+3i/7+7i

$$\frac { 1 + 3 i } { 7 + 7 i }$$

Evaluate

$$\frac{2}{7}+\frac{1}{7}i\approx 0.285714286+0.142857143i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-7i.

$$\frac{\left(1+3i\right)\left(7-7i\right)}{\left(7+7i\right)\left(7-7i\right)}$$

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

$$\frac{\left(1+3i\right)\left(7-7i\right)}{7^{2}-7^{2}i^{2}}$$

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

$$\frac{\left(1+3i\right)\left(7-7i\right)}{98}$$

Multiply complex numbers 1+3i and 7-7i like you multiply binomials.

$$\frac{1\times 7+1\times \left(-7i\right)+3i\times 7+3\left(-7\right)i^{2}}{98}$$

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

$$\frac{1\times 7+1\times \left(-7i\right)+3i\times 7+3\left(-7\right)\left(-1\right)}{98}$$

Do the multiplications in 1\times 7+1\times \left(-7i\right)+3i\times 7+3\left(-7\right)\left(-1\right).

$$\frac{7-7i+21i+21}{98}$$

Combine the real and imaginary parts in 7-7i+21i+21.

$$\frac{7+21+\left(-7+21\right)i}{98}$$

Do the additions in 7+21+\left(-7+21\right)i.

$$\frac{28+14i}{98}$$

Divide 28+14i by 98 to get \frac{2}{7}+\frac{1}{7}i.

$$\frac{2}{7}+\frac{1}{7}i$$

Real Part

$$\frac{2}{7} = 0.2857142857142857$$

Solution Steps

Multiply both numerator and denominator of \frac{1+3i}{7+7i} by the complex conjugate of the denominator, 7-7i.

$$Re(\frac{\left(1+3i\right)\left(7-7i\right)}{\left(7+7i\right)\left(7-7i\right)})$$

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

$$Re(\frac{\left(1+3i\right)\left(7-7i\right)}{7^{2}-7^{2}i^{2}})$$

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

$$Re(\frac{\left(1+3i\right)\left(7-7i\right)}{98})$$

Multiply complex numbers 1+3i and 7-7i like you multiply binomials.

$$Re(\frac{1\times 7+1\times \left(-7i\right)+3i\times 7+3\left(-7\right)i^{2}}{98})$$

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

$$Re(\frac{1\times 7+1\times \left(-7i\right)+3i\times 7+3\left(-7\right)\left(-1\right)}{98})$$

Do the multiplications in 1\times 7+1\times \left(-7i\right)+3i\times 7+3\left(-7\right)\left(-1\right).

$$Re(\frac{7-7i+21i+21}{98})$$

Combine the real and imaginary parts in 7-7i+21i+21.

$$Re(\frac{7+21+\left(-7+21\right)i}{98})$$

Do the additions in 7+21+\left(-7+21\right)i.

$$Re(\frac{28+14i}{98})$$

Divide 28+14i by 98 to get \frac{2}{7}+\frac{1}{7}i.

$$Re(\frac{2}{7}+\frac{1}{7}i)$$

The real part of \frac{2}{7}+\frac{1}{7}i is \frac{2}{7}.

$$\frac{2}{7}$$