Solve 3-4i/1+i

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

Evaluate

$$-\frac{1}{2}-\frac{7}{2}i=-0.5-3.5i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1-i.

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

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

$$\frac{\left(3-4i\right)\left(1-i\right)}{1^{2}-i^{2}}$$

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

$$\frac{\left(3-4i\right)\left(1-i\right)}{2}$$

Multiply complex numbers 3-4i and 1-i like you multiply binomials.

$$\frac{3\times 1+3\left(-i\right)-4i-4\left(-1\right)i^{2}}{2}$$

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

$$\frac{3\times 1+3\left(-i\right)-4i-4\left(-1\right)\left(-1\right)}{2}$$

Do the multiplications in 3\times 1+3\left(-i\right)-4i-4\left(-1\right)\left(-1\right).

$$\frac{3-3i-4i-4}{2}$$

Combine the real and imaginary parts in 3-3i-4i-4.

$$\frac{3-4+\left(-3-4\right)i}{2}$$

Do the additions in 3-4+\left(-3-4\right)i.

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

Divide -1-7i by 2 to get -\frac{1}{2}-\frac{7}{2}i.

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

Real Part

$$-\frac{1}{2} = -0.5$$

Solution Steps

Multiply both numerator and denominator of \frac{3-4i}{1+i} by the complex conjugate of the denominator, 1-i.

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

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

$$Re(\frac{\left(3-4i\right)\left(1-i\right)}{1^{2}-i^{2}})$$

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

$$Re(\frac{\left(3-4i\right)\left(1-i\right)}{2})$$

Multiply complex numbers 3-4i and 1-i like you multiply binomials.

$$Re(\frac{3\times 1+3\left(-i\right)-4i-4\left(-1\right)i^{2}}{2})$$

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

$$Re(\frac{3\times 1+3\left(-i\right)-4i-4\left(-1\right)\left(-1\right)}{2})$$

Do the multiplications in 3\times 1+3\left(-i\right)-4i-4\left(-1\right)\left(-1\right).

$$Re(\frac{3-3i-4i-4}{2})$$

Combine the real and imaginary parts in 3-3i-4i-4.

$$Re(\frac{3-4+\left(-3-4\right)i}{2})$$

Do the additions in 3-4+\left(-3-4\right)i.

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

Divide -1-7i by 2 to get -\frac{1}{2}-\frac{7}{2}i.

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

The real part of -\frac{1}{2}-\frac{7}{2}i is -\frac{1}{2}.

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