Solve 2-3i/4-7i

$$\frac{ 2-3i }{ 4-7i }$$

Evaluate

$$\frac{29}{65}+\frac{2}{65}i\approx 0.446153846+0.030769231i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4+7i.

$$\frac{\left(2-3i\right)\left(4+7i\right)}{\left(4-7i\right)\left(4+7i\right)}$$

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

$$\frac{\left(2-3i\right)\left(4+7i\right)}{4^{2}-7^{2}i^{2}}$$

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

$$\frac{\left(2-3i\right)\left(4+7i\right)}{65}$$

Multiply complex numbers 2-3i and 4+7i like you multiply binomials.

$$\frac{2\times 4+2\times \left(7i\right)-3i\times 4-3\times 7i^{2}}{65}$$

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

$$\frac{2\times 4+2\times \left(7i\right)-3i\times 4-3\times 7\left(-1\right)}{65}$$

Do the multiplications in 2\times 4+2\times \left(7i\right)-3i\times 4-3\times 7\left(-1\right).

$$\frac{8+14i-12i+21}{65}$$

Combine the real and imaginary parts in 8+14i-12i+21.

$$\frac{8+21+\left(14-12\right)i}{65}$$

Do the additions in 8+21+\left(14-12\right)i.

$$\frac{29+2i}{65}$$

Divide 29+2i by 65 to get \frac{29}{65}+\frac{2}{65}i.

$$\frac{29}{65}+\frac{2}{65}i$$

Real Part

$$\frac{29}{65} = 0.4461538461538462$$

Solution Steps

Multiply both numerator and denominator of \frac{2-3i}{4-7i} by the complex conjugate of the denominator, 4+7i.

$$Re(\frac{\left(2-3i\right)\left(4+7i\right)}{\left(4-7i\right)\left(4+7i\right)})$$

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

$$Re(\frac{\left(2-3i\right)\left(4+7i\right)}{4^{2}-7^{2}i^{2}})$$

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

$$Re(\frac{\left(2-3i\right)\left(4+7i\right)}{65})$$

Multiply complex numbers 2-3i and 4+7i like you multiply binomials.

$$Re(\frac{2\times 4+2\times \left(7i\right)-3i\times 4-3\times 7i^{2}}{65})$$

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

$$Re(\frac{2\times 4+2\times \left(7i\right)-3i\times 4-3\times 7\left(-1\right)}{65})$$

Do the multiplications in 2\times 4+2\times \left(7i\right)-3i\times 4-3\times 7\left(-1\right).

$$Re(\frac{8+14i-12i+21}{65})$$

Combine the real and imaginary parts in 8+14i-12i+21.

$$Re(\frac{8+21+\left(14-12\right)i}{65})$$

Do the additions in 8+21+\left(14-12\right)i.

$$Re(\frac{29+2i}{65})$$

Divide 29+2i by 65 to get \frac{29}{65}+\frac{2}{65}i.

$$Re(\frac{29}{65}+\frac{2}{65}i)$$

The real part of \frac{29}{65}+\frac{2}{65}i is \frac{29}{65}.

$$\frac{29}{65}$$