Solve 4+3i/1-i

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

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.

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

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

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

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

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

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

$$\frac{4\times 1+4i+3i\times 1+3i^{2}}{2}$$

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

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

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

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

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

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

Do the additions in 4-3+\left(4+3\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{4+3i}{1-i} by the complex conjugate of the denominator, 1+i.

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

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

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

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

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

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

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

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

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

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

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

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

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

Do the additions in 4-3+\left(4+3\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}.

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