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

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

Evaluate

$$\frac{1}{50}-\frac{57}{50}i=0.02-1.14i$$

Solution Steps

Multiply both numerator and denominator of \frac{1+i}{i} by imaginary unit i.

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

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

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

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

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

Do the multiplications in \frac{7\left(7+i\right)}{\left(7-i\right)\left(7+i\right)}.

$$1-i-\frac{49+7i}{50}$$

Divide 49+7i by 50 to get \frac{49}{50}+\frac{7}{50}i.

$$1-i+\left(-\frac{49}{50}-\frac{7}{50}i\right)$$

Add 1-i and -\frac{49}{50}-\frac{7}{50}i to get \frac{1}{50}-\frac{57}{50}i.

$$\frac{1}{50}-\frac{57}{50}i$$

Real Part

$$\frac{1}{50} = 0.02$$

Solution Steps

Multiply both numerator and denominator of \frac{1+i}{i} by imaginary unit i.

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

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

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

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

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

Do the multiplications in \frac{7\left(7+i\right)}{\left(7-i\right)\left(7+i\right)}.

$$Re(1-i-\frac{49+7i}{50})$$

Divide 49+7i by 50 to get \frac{49}{50}+\frac{7}{50}i.

$$Re(1-i+\left(-\frac{49}{50}-\frac{7}{50}i\right))$$

Add 1-i and -\frac{49}{50}-\frac{7}{50}i to get \frac{1}{50}-\frac{57}{50}i.

$$Re(\frac{1}{50}-\frac{57}{50}i)$$

The real part of \frac{1}{50}-\frac{57}{50}i is \frac{1}{50}.

$$\frac{1}{50}$$