Solve 1-2i/1-3i

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

Evaluate

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

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+3i.

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

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

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

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

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

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

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

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

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

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

$$\frac{1+3i-2i+6}{10}$$

Combine the real and imaginary parts in 1+3i-2i+6.

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

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

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

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

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

Real Part

$$\frac{7}{10} = 0.7$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Combine the real and imaginary parts in 1+3i-2i+6.

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

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

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

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

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

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

$$\frac{7}{10}$$