Solve (4-i)/(5-i)

$$\frac { ( 4 - i ) } { ( 5 - i ) }$$

Evaluate

$$\frac{21}{26}-\frac{1}{26}i\approx 0.807692308-0.038461538i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5+i.

$$\frac{\left(4-i\right)\left(5+i\right)}{\left(5-i\right)\left(5+i\right)}$$

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

$$\frac{\left(4-i\right)\left(5+i\right)}{5^{2}-i^{2}}$$

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

$$\frac{\left(4-i\right)\left(5+i\right)}{26}$$

Multiply complex numbers 4-i and 5+i like you multiply binomials.

$$\frac{4\times 5+4i-i\times 5-i^{2}}{26}$$

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

$$\frac{4\times 5+4i-i\times 5-\left(-1\right)}{26}$$

Do the multiplications in 4\times 5+4i-i\times 5-\left(-1\right).

$$\frac{20+4i-5i+1}{26}$$

Combine the real and imaginary parts in 20+4i-5i+1.

$$\frac{20+1+\left(4-5\right)i}{26}$$

Do the additions in 20+1+\left(4-5\right)i.

$$\frac{21-i}{26}$$

Divide 21-i by 26 to get \frac{21}{26}-\frac{1}{26}i.

$$\frac{21}{26}-\frac{1}{26}i$$

Real Part

$$\frac{21}{26} = 0.8076923076923077$$

Solution Steps

Multiply both numerator and denominator of \frac{4-i}{5-i} by the complex conjugate of the denominator, 5+i.

$$Re(\frac{\left(4-i\right)\left(5+i\right)}{\left(5-i\right)\left(5+i\right)})$$

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

$$Re(\frac{\left(4-i\right)\left(5+i\right)}{5^{2}-i^{2}})$$

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

$$Re(\frac{\left(4-i\right)\left(5+i\right)}{26})$$

Multiply complex numbers 4-i and 5+i like you multiply binomials.

$$Re(\frac{4\times 5+4i-i\times 5-i^{2}}{26})$$

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

$$Re(\frac{4\times 5+4i-i\times 5-\left(-1\right)}{26})$$

Do the multiplications in 4\times 5+4i-i\times 5-\left(-1\right).

$$Re(\frac{20+4i-5i+1}{26})$$

Combine the real and imaginary parts in 20+4i-5i+1.

$$Re(\frac{20+1+\left(4-5\right)i}{26})$$

Do the additions in 20+1+\left(4-5\right)i.

$$Re(\frac{21-i}{26})$$

Divide 21-i by 26 to get \frac{21}{26}-\frac{1}{26}i.

$$Re(\frac{21}{26}-\frac{1}{26}i)$$

The real part of \frac{21}{26}-\frac{1}{26}i is \frac{21}{26}.

$$\frac{21}{26}$$