Solve 6-i/6+i

$$\frac { 6 - i } { 6 + i }$$

Evaluate

$$\frac{35}{37}-\frac{12}{37}i\approx 0.945945946-0.324324324i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 6-i.

$$\frac{\left(6-i\right)\left(6-i\right)}{\left(6+i\right)\left(6-i\right)}$$

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

$$\frac{\left(6-i\right)\left(6-i\right)}{6^{2}-i^{2}}$$

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

$$\frac{\left(6-i\right)\left(6-i\right)}{37}$$

Multiply complex numbers 6-i and 6-i like you multiply binomials.

$$\frac{6\times 6+6\left(-i\right)-i\times 6-\left(-i^{2}\right)}{37}$$

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

$$\frac{6\times 6+6\left(-i\right)-i\times 6-\left(-\left(-1\right)\right)}{37}$$

Do the multiplications in 6\times 6+6\left(-i\right)-i\times 6-\left(-\left(-1\right)\right).

$$\frac{36-6i-6i-1}{37}$$

Combine the real and imaginary parts in 36-6i-6i-1.

$$\frac{36-1+\left(-6-6\right)i}{37}$$

Do the additions in 36-1+\left(-6-6\right)i.

$$\frac{35-12i}{37}$$

Divide 35-12i by 37 to get \frac{35}{37}-\frac{12}{37}i.

$$\frac{35}{37}-\frac{12}{37}i$$

Real Part

$$\frac{35}{37} = 0.9459459459459459$$

Solution Steps

Multiply both numerator and denominator of \frac{6-i}{6+i} by the complex conjugate of the denominator, 6-i.

$$Re(\frac{\left(6-i\right)\left(6-i\right)}{\left(6+i\right)\left(6-i\right)})$$

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

$$Re(\frac{\left(6-i\right)\left(6-i\right)}{6^{2}-i^{2}})$$

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

$$Re(\frac{\left(6-i\right)\left(6-i\right)}{37})$$

Multiply complex numbers 6-i and 6-i like you multiply binomials.

$$Re(\frac{6\times 6+6\left(-i\right)-i\times 6-\left(-i^{2}\right)}{37})$$

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

$$Re(\frac{6\times 6+6\left(-i\right)-i\times 6-\left(-\left(-1\right)\right)}{37})$$

Do the multiplications in 6\times 6+6\left(-i\right)-i\times 6-\left(-\left(-1\right)\right).

$$Re(\frac{36-6i-6i-1}{37})$$

Combine the real and imaginary parts in 36-6i-6i-1.

$$Re(\frac{36-1+\left(-6-6\right)i}{37})$$

Do the additions in 36-1+\left(-6-6\right)i.

$$Re(\frac{35-12i}{37})$$

Divide 35-12i by 37 to get \frac{35}{37}-\frac{12}{37}i.

$$Re(\frac{35}{37}-\frac{12}{37}i)$$

The real part of \frac{35}{37}-\frac{12}{37}i is \frac{35}{37}.

$$\frac{35}{37}$$