Solve 1/1-i-1/12

$$\frac { 1 } { 1 - i } - \frac { 1 } { 12 }$$

Evaluate

$$\frac{5}{12}+\frac{1}{2}i\approx 0.416666667+0.5i$$

Solution Steps

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

$$\frac{1\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}-\frac{1}{12}$$

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

$$\frac{1\left(1+i\right)}{1^{2}-i^{2}}-\frac{1}{12}$$

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

$$\frac{1\left(1+i\right)}{2}-\frac{1}{12}$$

Multiply 1 and 1+i to get 1+i.

$$\frac{1+i}{2}-\frac{1}{12}$$

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

$$\frac{1}{2}+\frac{1}{2}i-\frac{1}{12}$$

Subtract \frac{1}{12} from \frac{1}{2}+\frac{1}{2}i by subtracting corresponding real and imaginary parts.

$$\frac{1}{2}-\frac{1}{12}+\frac{1}{2}i$$

Subtract \frac{1}{12} from \frac{1}{2} to get \frac{5}{12}.

$$\frac{5}{12}+\frac{1}{2}i$$

Real Part

$$\frac{5}{12} = 0.4166666666666667$$

Solution Steps

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

$$Re(\frac{1\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}-\frac{1}{12})$$

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

$$Re(\frac{1\left(1+i\right)}{1^{2}-i^{2}}-\frac{1}{12})$$

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

$$Re(\frac{1\left(1+i\right)}{2}-\frac{1}{12})$$

Multiply 1 and 1+i to get 1+i.

$$Re(\frac{1+i}{2}-\frac{1}{12})$$

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

$$Re(\frac{1}{2}+\frac{1}{2}i-\frac{1}{12})$$

Subtract \frac{1}{12} from \frac{1}{2}+\frac{1}{2}i by subtracting corresponding real and imaginary parts.

$$Re(\frac{1}{2}-\frac{1}{12}+\frac{1}{2}i)$$

Subtract \frac{1}{12} from \frac{1}{2} to get \frac{5}{12}.

$$Re(\frac{5}{12}+\frac{1}{2}i)$$

The real part of \frac{5}{12}+\frac{1}{2}i is \frac{5}{12}.

$$\frac{5}{12}$$