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.

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

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

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

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

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

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

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

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

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