Solve 4+i/1-i+2-i/1+i

$$\frac { 4 + i } { 1 - i } + \frac { 2 - i } { 1 + i }$$

Evaluate

$$2+i$$

Solution Steps

Multiply both numerator and denominator of \frac{4+i}{1-i} by the complex conjugate of the denominator, 1+i.

$$\frac{\left(4+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}+\frac{2-i}{1+i}$$

Do the multiplications in \frac{\left(4+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$\frac{3+5i}{2}+\frac{2-i}{1+i}$$

Divide 3+5i by 2 to get \frac{3}{2}+\frac{5}{2}i.

$$\frac{3}{2}+\frac{5}{2}i+\frac{2-i}{1+i}$$

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

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

Do the multiplications in \frac{\left(2-i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}.

$$\frac{3}{2}+\frac{5}{2}i+\frac{1-3i}{2}$$

Divide 1-3i by 2 to get \frac{1}{2}-\frac{3}{2}i.

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

Add \frac{3}{2}+\frac{5}{2}i and \frac{1}{2}-\frac{3}{2}i to get 2+i.

$$2+i$$

Real Part

$$2$$

Solution Steps

Multiply both numerator and denominator of \frac{4+i}{1-i} by the complex conjugate of the denominator, 1+i.

$$Re(\frac{\left(4+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}+\frac{2-i}{1+i})$$

Do the multiplications in \frac{\left(4+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

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

Divide 3+5i by 2 to get \frac{3}{2}+\frac{5}{2}i.

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

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

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

Do the multiplications in \frac{\left(2-i\right)\left(1-i\right)}{\left(1+i\right)\left(1-i\right)}.

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

Divide 1-3i by 2 to get \frac{1}{2}-\frac{3}{2}i.

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

Add \frac{3}{2}+\frac{5}{2}i and \frac{1}{2}-\frac{3}{2}i to get 2+i.

$$Re(2+i)$$

The real part of 2+i is 2.

$$2$$