Solve 4+i/3-i

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

Evaluate

$$\frac{11}{10}+\frac{7}{10}i=1.1+0.7i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3+i.

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

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

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

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

$$\frac{\left(4+i\right)\left(3+i\right)}{10}$$

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

$$\frac{4\times 3+4i+3i+i^{2}}{10}$$

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

$$\frac{4\times 3+4i+3i-1}{10}$$

Do the multiplications in 4\times 3+4i+3i-1.

$$\frac{12+4i+3i-1}{10}$$

Combine the real and imaginary parts in 12+4i+3i-1.

$$\frac{12-1+\left(4+3\right)i}{10}$$

Do the additions in 12-1+\left(4+3\right)i.

$$\frac{11+7i}{10}$$

Divide 11+7i by 10 to get \frac{11}{10}+\frac{7}{10}i.

$$\frac{11}{10}+\frac{7}{10}i$$

Real Part

$$\frac{11}{10} = 1\frac{1}{10} = 1.1$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(4+i\right)\left(3+i\right)}{10})$$

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

$$Re(\frac{4\times 3+4i+3i+i^{2}}{10})$$

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

$$Re(\frac{4\times 3+4i+3i-1}{10})$$

Do the multiplications in 4\times 3+4i+3i-1.

$$Re(\frac{12+4i+3i-1}{10})$$

Combine the real and imaginary parts in 12+4i+3i-1.

$$Re(\frac{12-1+\left(4+3\right)i}{10})$$

Do the additions in 12-1+\left(4+3\right)i.

$$Re(\frac{11+7i}{10})$$

Divide 11+7i by 10 to get \frac{11}{10}+\frac{7}{10}i.

$$Re(\frac{11}{10}+\frac{7}{10}i)$$

The real part of \frac{11}{10}+\frac{7}{10}i is \frac{11}{10}.

$$\frac{11}{10}$$