Solve 5+i/4+i

$$\frac { 5 + i } { 4 + i }$$

Evaluate

$$\frac{21}{17}-\frac{1}{17}i\approx 1.235294118-0.058823529i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(5+i\right)\left(4-i\right)}{17}$$

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

$$\frac{5\times 4+5\left(-i\right)+4i-i^{2}}{17}$$

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

$$\frac{5\times 4+5\left(-i\right)+4i-\left(-1\right)}{17}$$

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

$$\frac{20-5i+4i+1}{17}$$

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

$$\frac{20+1+\left(-5+4\right)i}{17}$$

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

$$\frac{21-i}{17}$$

Divide 21-i by 17 to get \frac{21}{17}-\frac{1}{17}i.

$$\frac{21}{17}-\frac{1}{17}i$$

Real Part

$$\frac{21}{17} = 1\frac{4}{17} = 1.2352941176470589$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(5+i\right)\left(4-i\right)}{17})$$

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

$$Re(\frac{5\times 4+5\left(-i\right)+4i-i^{2}}{17})$$

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

$$Re(\frac{5\times 4+5\left(-i\right)+4i-\left(-1\right)}{17})$$

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

$$Re(\frac{20-5i+4i+1}{17})$$

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

$$Re(\frac{20+1+\left(-5+4\right)i}{17})$$

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

$$Re(\frac{21-i}{17})$$

Divide 21-i by 17 to get \frac{21}{17}-\frac{1}{17}i.

$$Re(\frac{21}{17}-\frac{1}{17}i)$$

The real part of \frac{21}{17}-\frac{1}{17}i is \frac{21}{17}.

$$\frac{21}{17}$$