Solve 5+i4/2+i3

$$\frac { 5 + i 4 } { 2 + i 3 }$$

Evaluate

$$\frac{22}{13}-\frac{7}{13}i\approx 1.692307692-0.538461538i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 2-3i.

$$\frac{\left(5+4i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)}$$

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

$$\frac{\left(5+4i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(5+4i\right)\left(2-3i\right)}{13}$$

Multiply complex numbers 5+4i and 2-3i like you multiply binomials.

$$\frac{5\times 2+5\times \left(-3i\right)+4i\times 2+4\left(-3\right)i^{2}}{13}$$

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

$$\frac{5\times 2+5\times \left(-3i\right)+4i\times 2+4\left(-3\right)\left(-1\right)}{13}$$

Do the multiplications in 5\times 2+5\times \left(-3i\right)+4i\times 2+4\left(-3\right)\left(-1\right).

$$\frac{10-15i+8i+12}{13}$$

Combine the real and imaginary parts in 10-15i+8i+12.

$$\frac{10+12+\left(-15+8\right)i}{13}$$

Do the additions in 10+12+\left(-15+8\right)i.

$$\frac{22-7i}{13}$$

Divide 22-7i by 13 to get \frac{22}{13}-\frac{7}{13}i.

$$\frac{22}{13}-\frac{7}{13}i$$

Real Part

$$\frac{22}{13} = 1\frac{9}{13} = 1.6923076923076923$$

Solution Steps

Multiply both numerator and denominator of \frac{5+4i}{2+3i} by the complex conjugate of the denominator, 2-3i.

$$Re(\frac{\left(5+4i\right)\left(2-3i\right)}{\left(2+3i\right)\left(2-3i\right)})$$

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

$$Re(\frac{\left(5+4i\right)\left(2-3i\right)}{2^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(5+4i\right)\left(2-3i\right)}{13})$$

Multiply complex numbers 5+4i and 2-3i like you multiply binomials.

$$Re(\frac{5\times 2+5\times \left(-3i\right)+4i\times 2+4\left(-3\right)i^{2}}{13})$$

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

$$Re(\frac{5\times 2+5\times \left(-3i\right)+4i\times 2+4\left(-3\right)\left(-1\right)}{13})$$

Do the multiplications in 5\times 2+5\times \left(-3i\right)+4i\times 2+4\left(-3\right)\left(-1\right).

$$Re(\frac{10-15i+8i+12}{13})$$

Combine the real and imaginary parts in 10-15i+8i+12.

$$Re(\frac{10+12+\left(-15+8\right)i}{13})$$

Do the additions in 10+12+\left(-15+8\right)i.

$$Re(\frac{22-7i}{13})$$

Divide 22-7i by 13 to get \frac{22}{13}-\frac{7}{13}i.

$$Re(\frac{22}{13}-\frac{7}{13}i)$$

The real part of \frac{22}{13}-\frac{7}{13}i is \frac{22}{13}.

$$\frac{22}{13}$$