Solve 3-i/1+2i

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

Evaluate

$$\frac{1}{5}-\frac{7}{5}i=0.2-1.4i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1-2i.

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

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

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

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

$$\frac{\left(3-i\right)\left(1-2i\right)}{5}$$

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

$$\frac{3\times 1+3\times \left(-2i\right)-i-\left(-2i^{2}\right)}{5}$$

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

$$\frac{3\times 1+3\times \left(-2i\right)-i-\left(-2\left(-1\right)\right)}{5}$$

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

$$\frac{3-6i-i-2}{5}$$

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

$$\frac{3-2+\left(-6-1\right)i}{5}$$

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

$$\frac{1-7i}{5}$$

Divide 1-7i by 5 to get \frac{1}{5}-\frac{7}{5}i.

$$\frac{1}{5}-\frac{7}{5}i$$

Real Part

$$\frac{1}{5} = 0.2$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(3-i\right)\left(1-2i\right)}{5})$$

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

$$Re(\frac{3\times 1+3\times \left(-2i\right)-i-\left(-2i^{2}\right)}{5})$$

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

$$Re(\frac{3\times 1+3\times \left(-2i\right)-i-\left(-2\left(-1\right)\right)}{5})$$

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

$$Re(\frac{3-6i-i-2}{5})$$

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

$$Re(\frac{3-2+\left(-6-1\right)i}{5})$$

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

$$Re(\frac{1-7i}{5})$$

Divide 1-7i by 5 to get \frac{1}{5}-\frac{7}{5}i.

$$Re(\frac{1}{5}-\frac{7}{5}i)$$

The real part of \frac{1}{5}-\frac{7}{5}i is \frac{1}{5}.

$$\frac{1}{5}$$