Solve 4+i/4-i

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

Evaluate

$$\frac{15}{17}+\frac{8}{17}i\approx 0.882352941+0.470588235i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+i.

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

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

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

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

$$\frac{\left(4+i\right)\left(4+i\right)}{17}$$

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

$$\frac{4\times 4+4i+4i+i^{2}}{17}$$

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

$$\frac{4\times 4+4i+4i-1}{17}$$

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

$$\frac{16+4i+4i-1}{17}$$

Combine the real and imaginary parts in 16+4i+4i-1.

$$\frac{16-1+\left(4+4\right)i}{17}$$

Do the additions in 16-1+\left(4+4\right)i.

$$\frac{15+8i}{17}$$

Divide 15+8i by 17 to get \frac{15}{17}+\frac{8}{17}i.

$$\frac{15}{17}+\frac{8}{17}i$$

Real Part

$$\frac{15}{17} = 0.8823529411764706$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(4+i\right)\left(4+i\right)}{17})$$

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

$$Re(\frac{4\times 4+4i+4i+i^{2}}{17})$$

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

$$Re(\frac{4\times 4+4i+4i-1}{17})$$

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

$$Re(\frac{16+4i+4i-1}{17})$$

Combine the real and imaginary parts in 16+4i+4i-1.

$$Re(\frac{16-1+\left(4+4\right)i}{17})$$

Do the additions in 16-1+\left(4+4\right)i.

$$Re(\frac{15+8i}{17})$$

Divide 15+8i by 17 to get \frac{15}{17}+\frac{8}{17}i.

$$Re(\frac{15}{17}+\frac{8}{17}i)$$

The real part of \frac{15}{17}+\frac{8}{17}i is \frac{15}{17}.

$$\frac{15}{17}$$