Solve -1-4i/-5-9i

$$\frac { - 1 - 4 i } { - 5 - 9 i }$$

Evaluate

$$\frac{41}{106}+\frac{11}{106}i\approx 0.386792453+0.103773585i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -5+9i.

$$\frac{\left(-1-4i\right)\left(-5+9i\right)}{\left(-5-9i\right)\left(-5+9i\right)}$$

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

$$\frac{\left(-1-4i\right)\left(-5+9i\right)}{\left(-5\right)^{2}-9^{2}i^{2}}$$

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

$$\frac{\left(-1-4i\right)\left(-5+9i\right)}{106}$$

Multiply complex numbers -1-4i and -5+9i like you multiply binomials.

$$\frac{-\left(-5\right)-9i-4i\left(-5\right)-4\times 9i^{2}}{106}$$

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

$$\frac{-\left(-5\right)-9i-4i\left(-5\right)-4\times 9\left(-1\right)}{106}$$

Do the multiplications in -\left(-5\right)-9i-4i\left(-5\right)-4\times 9\left(-1\right).

$$\frac{5-9i+20i+36}{106}$$

Combine the real and imaginary parts in 5-9i+20i+36.

$$\frac{5+36+\left(-9+20\right)i}{106}$$

Do the additions in 5+36+\left(-9+20\right)i.

$$\frac{41+11i}{106}$$

Divide 41+11i by 106 to get \frac{41}{106}+\frac{11}{106}i.

$$\frac{41}{106}+\frac{11}{106}i$$

Real Part

$$\frac{41}{106} = 0.3867924528301887$$

Solution Steps

Multiply both numerator and denominator of \frac{-1-4i}{-5-9i} by the complex conjugate of the denominator, -5+9i.

$$Re(\frac{\left(-1-4i\right)\left(-5+9i\right)}{\left(-5-9i\right)\left(-5+9i\right)})$$

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

$$Re(\frac{\left(-1-4i\right)\left(-5+9i\right)}{\left(-5\right)^{2}-9^{2}i^{2}})$$

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

$$Re(\frac{\left(-1-4i\right)\left(-5+9i\right)}{106})$$

Multiply complex numbers -1-4i and -5+9i like you multiply binomials.

$$Re(\frac{-\left(-5\right)-9i-4i\left(-5\right)-4\times 9i^{2}}{106})$$

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

$$Re(\frac{-\left(-5\right)-9i-4i\left(-5\right)-4\times 9\left(-1\right)}{106})$$

Do the multiplications in -\left(-5\right)-9i-4i\left(-5\right)-4\times 9\left(-1\right).

$$Re(\frac{5-9i+20i+36}{106})$$

Combine the real and imaginary parts in 5-9i+20i+36.

$$Re(\frac{5+36+\left(-9+20\right)i}{106})$$

Do the additions in 5+36+\left(-9+20\right)i.

$$Re(\frac{41+11i}{106})$$

Divide 41+11i by 106 to get \frac{41}{106}+\frac{11}{106}i.

$$Re(\frac{41}{106}+\frac{11}{106}i)$$

The real part of \frac{41}{106}+\frac{11}{106}i is \frac{41}{106}.

$$\frac{41}{106}$$