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.

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

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

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

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

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

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

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

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

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