Solve 1+i/i-3/4-i

$$\frac { 1 + i } { i } - \frac { 3 } { 4 - i }$$

Evaluate

$$\frac{5}{17}-\frac{20}{17}i\approx 0.294117647-1.176470588i$$

Solution Steps

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

$$\frac{-1+i}{-1}-\frac{3}{4-i}$$

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

$$1-i-\frac{3}{4-i}$$

Multiply both numerator and denominator of \frac{3}{4-i} by the complex conjugate of the denominator, 4+i.

$$1-i-\frac{3\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}$$

Do the multiplications in \frac{3\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}.

$$1-i-\frac{12+3i}{17}$$

Divide 12+3i by 17 to get \frac{12}{17}+\frac{3}{17}i.

$$1-i+\left(-\frac{12}{17}-\frac{3}{17}i\right)$$

Add 1-i and -\frac{12}{17}-\frac{3}{17}i to get \frac{5}{17}-\frac{20}{17}i.

$$\frac{5}{17}-\frac{20}{17}i$$

Real Part

$$\frac{5}{17} = 0.29411764705882354$$

Solution Steps

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

$$Re(\frac{-1+i}{-1}-\frac{3}{4-i})$$

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

$$Re(1-i-\frac{3}{4-i})$$

Multiply both numerator and denominator of \frac{3}{4-i} by the complex conjugate of the denominator, 4+i.

$$Re(1-i-\frac{3\left(4+i\right)}{\left(4-i\right)\left(4+i\right)})$$

Do the multiplications in \frac{3\left(4+i\right)}{\left(4-i\right)\left(4+i\right)}.

$$Re(1-i-\frac{12+3i}{17})$$

Divide 12+3i by 17 to get \frac{12}{17}+\frac{3}{17}i.

$$Re(1-i+\left(-\frac{12}{17}-\frac{3}{17}i\right))$$

Add 1-i and -\frac{12}{17}-\frac{3}{17}i to get \frac{5}{17}-\frac{20}{17}i.

$$Re(\frac{5}{17}-\frac{20}{17}i)$$

The real part of \frac{5}{17}-\frac{20}{17}i is \frac{5}{17}.

$$\frac{5}{17}$$