Solve 4-i/3+7i

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

Evaluate

$$\frac{5}{58}-\frac{31}{58}i\approx 0.086206897-0.534482759i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-7i.

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

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

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

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

$$\frac{\left(4-i\right)\left(3-7i\right)}{58}$$

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

$$\frac{4\times 3+4\times \left(-7i\right)-i\times 3-\left(-7i^{2}\right)}{58}$$

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

$$\frac{4\times 3+4\times \left(-7i\right)-i\times 3-\left(-7\left(-1\right)\right)}{58}$$

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

$$\frac{12-28i-3i-7}{58}$$

Combine the real and imaginary parts in 12-28i-3i-7.

$$\frac{12-7+\left(-28-3\right)i}{58}$$

Do the additions in 12-7+\left(-28-3\right)i.

$$\frac{5-31i}{58}$$

Divide 5-31i by 58 to get \frac{5}{58}-\frac{31}{58}i.

$$\frac{5}{58}-\frac{31}{58}i$$

Real Part

$$\frac{5}{58} = 0.08620689655172414$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(4-i\right)\left(3-7i\right)}{58})$$

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

$$Re(\frac{4\times 3+4\times \left(-7i\right)-i\times 3-\left(-7i^{2}\right)}{58})$$

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

$$Re(\frac{4\times 3+4\times \left(-7i\right)-i\times 3-\left(-7\left(-1\right)\right)}{58})$$

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

$$Re(\frac{12-28i-3i-7}{58})$$

Combine the real and imaginary parts in 12-28i-3i-7.

$$Re(\frac{12-7+\left(-28-3\right)i}{58})$$

Do the additions in 12-7+\left(-28-3\right)i.

$$Re(\frac{5-31i}{58})$$

Divide 5-31i by 58 to get \frac{5}{58}-\frac{31}{58}i.

$$Re(\frac{5}{58}-\frac{31}{58}i)$$

The real part of \frac{5}{58}-\frac{31}{58}i is \frac{5}{58}.

$$\frac{5}{58}$$