Solve 1+2i/7+7i

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

Evaluate

$$\frac{3}{14}+\frac{1}{14}i\approx 0.214285714+0.071428571i$$

Solution Steps

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

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

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

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

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

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

Multiply complex numbers 1+2i and 7-7i like you multiply binomials.

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

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

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

Do the multiplications in 1\times 7+1\times \left(-7i\right)+2i\times 7+2\left(-7\right)\left(-1\right).

$$\frac{7-7i+14i+14}{98}$$

Combine the real and imaginary parts in 7-7i+14i+14.

$$\frac{7+14+\left(-7+14\right)i}{98}$$

Do the additions in 7+14+\left(-7+14\right)i.

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

Divide 21+7i by 98 to get \frac{3}{14}+\frac{1}{14}i.

$$\frac{3}{14}+\frac{1}{14}i$$

Real Part

$$\frac{3}{14} = 0.21428571428571427$$

Solution Steps

Multiply both numerator and denominator of \frac{1+2i}{7+7i} by the complex conjugate of the denominator, 7-7i.

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

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

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

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

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

Multiply complex numbers 1+2i and 7-7i like you multiply binomials.

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

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

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

Do the multiplications in 1\times 7+1\times \left(-7i\right)+2i\times 7+2\left(-7\right)\left(-1\right).

$$Re(\frac{7-7i+14i+14}{98})$$

Combine the real and imaginary parts in 7-7i+14i+14.

$$Re(\frac{7+14+\left(-7+14\right)i}{98})$$

Do the additions in 7+14+\left(-7+14\right)i.

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

Divide 21+7i by 98 to get \frac{3}{14}+\frac{1}{14}i.

$$Re(\frac{3}{14}+\frac{1}{14}i)$$

The real part of \frac{3}{14}+\frac{1}{14}i is \frac{3}{14}.

$$\frac{3}{14}$$