Solve 1+i/3+2i=

$$\frac { 1 + i } { 3 + 2 i } =$$

Evaluate

$$\frac{5}{13}+\frac{1}{13}i\approx 0.384615385+0.076923077i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-2i.

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

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

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

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

$$\frac{\left(1+i\right)\left(3-2i\right)}{13}$$

Multiply complex numbers 1+i and 3-2i like you multiply binomials.

$$\frac{1\times 3+1\times \left(-2i\right)+3i-2i^{2}}{13}$$

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

$$\frac{1\times 3+1\times \left(-2i\right)+3i-2\left(-1\right)}{13}$$

Do the multiplications in 1\times 3+1\times \left(-2i\right)+3i-2\left(-1\right).

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

Combine the real and imaginary parts in 3-2i+3i+2.

$$\frac{3+2+\left(-2+3\right)i}{13}$$

Do the additions in 3+2+\left(-2+3\right)i.

$$\frac{5+i}{13}$$

Divide 5+i by 13 to get \frac{5}{13}+\frac{1}{13}i.

$$\frac{5}{13}+\frac{1}{13}i$$

Real Part

$$\frac{5}{13} = 0.38461538461538464$$

Solution Steps

Multiply both numerator and denominator of \frac{1+i}{3+2i} by the complex conjugate of the denominator, 3-2i.

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

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

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

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

$$Re(\frac{\left(1+i\right)\left(3-2i\right)}{13})$$

Multiply complex numbers 1+i and 3-2i like you multiply binomials.

$$Re(\frac{1\times 3+1\times \left(-2i\right)+3i-2i^{2}}{13})$$

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

$$Re(\frac{1\times 3+1\times \left(-2i\right)+3i-2\left(-1\right)}{13})$$

Do the multiplications in 1\times 3+1\times \left(-2i\right)+3i-2\left(-1\right).

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

Combine the real and imaginary parts in 3-2i+3i+2.

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

Do the additions in 3+2+\left(-2+3\right)i.

$$Re(\frac{5+i}{13})$$

Divide 5+i by 13 to get \frac{5}{13}+\frac{1}{13}i.

$$Re(\frac{5}{13}+\frac{1}{13}i)$$

The real part of \frac{5}{13}+\frac{1}{13}i is \frac{5}{13}.

$$\frac{5}{13}$$