Solve 21-3i/1+i

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

Evaluate

$$-2-4i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$2\times \frac{-2-4i}{2}$$

Divide -2-4i by 2 to get -1-2i.

$$2\left(-1-2i\right)$$

Multiply 2 times -1-2i.

$$2\left(-1\right)+2\times \left(-2i\right)$$

Do the multiplications.

$$-2-4i$$

Real Part

$$-2$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(2\times \frac{1-i-3i-3}{2})$$

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

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

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

$$Re(2\times \frac{-2-4i}{2})$$

Divide -2-4i by 2 to get -1-2i.

$$Re(2\left(-1-2i\right))$$

Multiply 2 times -1-2i.

$$Re(2\left(-1\right)+2\times \left(-2i\right))$$

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

$$Re(-2-4i)$$

The real part of -2-4i is -2.

$$-2$$