Solve 2-left(1+1right)i/1-i

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

Evaluate

$$2$$

Solution Steps

Add 1 and 1 to get 2.

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

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

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

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

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

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

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

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

$$\frac{2\times 1+2i-2i-2i^{2}}{2}$$

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

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

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

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

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

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

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

$$\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$2$$

Real Part

$$2$$

Solution Steps

Add 1 and 1 to get 2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{4}{2})$$

Divide 4 by 2 to get 2.

$$Re(2)$$

The real part of 2 is 2.

$$2$$