Solve 1+2i/3-i

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

Evaluate

$$\frac{1}{10}+\frac{7}{10}i=0.1+0.7i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(1+2i\right)\left(3+i\right)}{10}$$

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

$$\frac{1\times 3+i+2i\times 3+2i^{2}}{10}$$

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

$$\frac{1\times 3+i+2i\times 3+2\left(-1\right)}{10}$$

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

$$\frac{3+i+6i-2}{10}$$

Combine the real and imaginary parts in 3+i+6i-2.

$$\frac{3-2+\left(1+6\right)i}{10}$$

Do the additions in 3-2+\left(1+6\right)i.

$$\frac{1+7i}{10}$$

Divide 1+7i by 10 to get \frac{1}{10}+\frac{7}{10}i.

$$\frac{1}{10}+\frac{7}{10}i$$

Real Part

$$\frac{1}{10} = 0.1$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(1+2i\right)\left(3+i\right)}{10})$$

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

$$Re(\frac{1\times 3+i+2i\times 3+2i^{2}}{10})$$

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

$$Re(\frac{1\times 3+i+2i\times 3+2\left(-1\right)}{10})$$

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

$$Re(\frac{3+i+6i-2}{10})$$

Combine the real and imaginary parts in 3+i+6i-2.

$$Re(\frac{3-2+\left(1+6\right)i}{10})$$

Do the additions in 3-2+\left(1+6\right)i.

$$Re(\frac{1+7i}{10})$$

Divide 1+7i by 10 to get \frac{1}{10}+\frac{7}{10}i.

$$Re(\frac{1}{10}+\frac{7}{10}i)$$

The real part of \frac{1}{10}+\frac{7}{10}i is \frac{1}{10}.

$$\frac{1}{10}$$