Solve 2+3i/5+4i

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

Evaluate

$$\frac{22}{41}+\frac{7}{41}i\approx 0.536585366+0.170731707i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5-4i.

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

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

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

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

$$\frac{\left(2+3i\right)\left(5-4i\right)}{41}$$

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

$$\frac{2\times 5+2\times \left(-4i\right)+3i\times 5+3\left(-4\right)i^{2}}{41}$$

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

$$\frac{2\times 5+2\times \left(-4i\right)+3i\times 5+3\left(-4\right)\left(-1\right)}{41}$$

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

$$\frac{10-8i+15i+12}{41}$$

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

$$\frac{10+12+\left(-8+15\right)i}{41}$$

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

$$\frac{22+7i}{41}$$

Divide 22+7i by 41 to get \frac{22}{41}+\frac{7}{41}i.

$$\frac{22}{41}+\frac{7}{41}i$$

Real Part

$$\frac{22}{41} = 0.5365853658536586$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(2+3i\right)\left(5-4i\right)}{41})$$

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

$$Re(\frac{2\times 5+2\times \left(-4i\right)+3i\times 5+3\left(-4\right)i^{2}}{41})$$

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

$$Re(\frac{2\times 5+2\times \left(-4i\right)+3i\times 5+3\left(-4\right)\left(-1\right)}{41})$$

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

$$Re(\frac{10-8i+15i+12}{41})$$

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

$$Re(\frac{10+12+\left(-8+15\right)i}{41})$$

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

$$Re(\frac{22+7i}{41})$$

Divide 22+7i by 41 to get \frac{22}{41}+\frac{7}{41}i.

$$Re(\frac{22}{41}+\frac{7}{41}i)$$

The real part of \frac{22}{41}+\frac{7}{41}i is \frac{22}{41}.

$$\frac{22}{41}$$