Solve 1+1/i-1/7-i

$$\frac { 1 + 1 } { i } - \frac { 1 } { 7 - i }$$

Evaluate

$$-\frac{7}{50}-\frac{101}{50}i=-0.14-2.02i$$

Solution Steps

Add 1 and 1 to get 2.

$$\frac{2}{i}-\frac{1}{7-i}$$

Multiply both numerator and denominator of \frac{2}{i} by imaginary unit i.

$$\frac{2i}{1i^{2}}-\frac{1}{7-i}$$

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

$$\frac{2i}{-1}-\frac{1}{7-i}$$

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

$$-2i-\frac{1}{7-i}$$

Multiply both numerator and denominator of \frac{1}{7-i} by the complex conjugate of the denominator, 7+i.

$$-2i-\frac{1\left(7+i\right)}{\left(7-i\right)\left(7+i\right)}$$

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

$$-2i-\frac{1\left(7+i\right)}{7^{2}-i^{2}}$$

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

$$-2i-\frac{1\left(7+i\right)}{50}$$

Multiply 1 and 7+i to get 7+i.

$$-2i-\frac{7+i}{50}$$

Divide 7+i by 50 to get \frac{7}{50}+\frac{1}{50}i.

$$-2i+\left(-\frac{7}{50}-\frac{1}{50}i\right)$$

Combine the real and imaginary parts in numbers -2i and -\frac{7}{50}-\frac{1}{50}i.

$$-\frac{7}{50}+\left(-2-\frac{1}{50}\right)i$$

Add -2 to -\frac{1}{50}.

$$-\frac{7}{50}-\frac{101}{50}i$$

Real Part

$$-\frac{7}{50} = -0.14$$

Solution Steps

Add 1 and 1 to get 2.

$$Re(\frac{2}{i}-\frac{1}{7-i})$$

Multiply both numerator and denominator of \frac{2}{i} by imaginary unit i.

$$Re(\frac{2i}{1i^{2}}-\frac{1}{7-i})$$

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

$$Re(\frac{2i}{-1}-\frac{1}{7-i})$$

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

$$Re(-2i-\frac{1}{7-i})$$

Multiply both numerator and denominator of \frac{1}{7-i} by the complex conjugate of the denominator, 7+i.

$$Re(-2i-\frac{1\left(7+i\right)}{\left(7-i\right)\left(7+i\right)})$$

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

$$Re(-2i-\frac{1\left(7+i\right)}{7^{2}-i^{2}})$$

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

$$Re(-2i-\frac{1\left(7+i\right)}{50})$$

Multiply 1 and 7+i to get 7+i.

$$Re(-2i-\frac{7+i}{50})$$

Divide 7+i by 50 to get \frac{7}{50}+\frac{1}{50}i.

$$Re(-2i+\left(-\frac{7}{50}-\frac{1}{50}i\right))$$

Combine the real and imaginary parts in numbers -2i and -\frac{7}{50}-\frac{1}{50}i.

$$Re(-\frac{7}{50}+\left(-2-\frac{1}{50}\right)i)$$

Add -2 to -\frac{1}{50}.

$$Re(-\frac{7}{50}-\frac{101}{50}i)$$

The real part of -\frac{7}{50}-\frac{101}{50}i is -\frac{7}{50}.

$$-\frac{7}{50}$$