Solve 12-i/7+4i

$$\frac { 12 - i } { 7 + 4 i }$$

Evaluate

$$\frac{16}{13}-\frac{11}{13}i\approx 1.230769231-0.846153846i$$

Solution Steps

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

$$\frac{\left(12-i\right)\left(7-4i\right)}{\left(7+4i\right)\left(7-4i\right)}$$

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

$$\frac{\left(12-i\right)\left(7-4i\right)}{7^{2}-4^{2}i^{2}}$$

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

$$\frac{\left(12-i\right)\left(7-4i\right)}{65}$$

Multiply complex numbers 12-i and 7-4i like you multiply binomials.

$$\frac{12\times 7+12\times \left(-4i\right)-i\times 7-\left(-4i^{2}\right)}{65}$$

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

$$\frac{12\times 7+12\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right)}{65}$$

Do the multiplications in 12\times 7+12\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right).

$$\frac{84-48i-7i-4}{65}$$

Combine the real and imaginary parts in 84-48i-7i-4.

$$\frac{84-4+\left(-48-7\right)i}{65}$$

Do the additions in 84-4+\left(-48-7\right)i.

$$\frac{80-55i}{65}$$

Divide 80-55i by 65 to get \frac{16}{13}-\frac{11}{13}i.

$$\frac{16}{13}-\frac{11}{13}i$$

Real Part

$$\frac{16}{13} = 1\frac{3}{13} = 1.2307692307692308$$

Solution Steps

Multiply both numerator and denominator of \frac{12-i}{7+4i} by the complex conjugate of the denominator, 7-4i.

$$Re(\frac{\left(12-i\right)\left(7-4i\right)}{\left(7+4i\right)\left(7-4i\right)})$$

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

$$Re(\frac{\left(12-i\right)\left(7-4i\right)}{7^{2}-4^{2}i^{2}})$$

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

$$Re(\frac{\left(12-i\right)\left(7-4i\right)}{65})$$

Multiply complex numbers 12-i and 7-4i like you multiply binomials.

$$Re(\frac{12\times 7+12\times \left(-4i\right)-i\times 7-\left(-4i^{2}\right)}{65})$$

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

$$Re(\frac{12\times 7+12\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right)}{65})$$

Do the multiplications in 12\times 7+12\times \left(-4i\right)-i\times 7-\left(-4\left(-1\right)\right).

$$Re(\frac{84-48i-7i-4}{65})$$

Combine the real and imaginary parts in 84-48i-7i-4.

$$Re(\frac{84-4+\left(-48-7\right)i}{65})$$

Do the additions in 84-4+\left(-48-7\right)i.

$$Re(\frac{80-55i}{65})$$

Divide 80-55i by 65 to get \frac{16}{13}-\frac{11}{13}i.

$$Re(\frac{16}{13}-\frac{11}{13}i)$$

The real part of \frac{16}{13}-\frac{11}{13}i is \frac{16}{13}.

$$\frac{16}{13}$$