Solve 1-i/sqrt{2-i}

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

Evaluate

$$\sqrt{2}\left(\frac{1}{3}-\frac{1}{3}i\right)+\left(\frac{1}{3}+\frac{1}{3}i\right)\approx 0.804737854-0.138071187i$$

Solution Steps

Rationalize the denominator of \frac{1-i}{\sqrt{2}-i} by multiplying numerator and denominator by \sqrt{2}+i.

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

Consider \left(\sqrt{2}-i\right)\left(\sqrt{2}+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}.

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

Square \sqrt{2}. Square -i.

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

Subtract -1 from 2 to get 3.

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

Divide \left(1-i\right)\left(\sqrt{2}+i\right) by 3 to get \left(\frac{1}{3}-\frac{1}{3}i\right)\left(\sqrt{2}+i\right).

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

Use the distributive property to multiply \frac{1}{3}-\frac{1}{3}i by \sqrt{2}+i.

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

Real Part

$$\frac{\sqrt{2} + 1}{3} = 0.8047378541243649$$