Solve 1+i/3-2i

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

Evaluate

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

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.

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

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

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

Real Part

$$\frac{1}{13} = 0.07692307692307693$$

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.

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

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

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

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

$$\frac{1}{13}$$