Solve 2+3i/3+5i

$$\frac { 2 + 3 i } { 3 + 5 i }$$

Evaluate

$$\frac{21}{34}-\frac{1}{34}i\approx 0.617647059-0.029411765i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-5i.

$$\frac{\left(2+3i\right)\left(3-5i\right)}{\left(3+5i\right)\left(3-5i\right)}$$

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

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

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

$$\frac{\left(2+3i\right)\left(3-5i\right)}{34}$$

Multiply complex numbers 2+3i and 3-5i like you multiply binomials.

$$\frac{2\times 3+2\times \left(-5i\right)+3i\times 3+3\left(-5\right)i^{2}}{34}$$

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

$$\frac{2\times 3+2\times \left(-5i\right)+3i\times 3+3\left(-5\right)\left(-1\right)}{34}$$

Do the multiplications in 2\times 3+2\times \left(-5i\right)+3i\times 3+3\left(-5\right)\left(-1\right).

$$\frac{6-10i+9i+15}{34}$$

Combine the real and imaginary parts in 6-10i+9i+15.

$$\frac{6+15+\left(-10+9\right)i}{34}$$

Do the additions in 6+15+\left(-10+9\right)i.

$$\frac{21-i}{34}$$

Divide 21-i by 34 to get \frac{21}{34}-\frac{1}{34}i.

$$\frac{21}{34}-\frac{1}{34}i$$

Real Part

$$\frac{21}{34} = 0.6176470588235294$$

Solution Steps

Multiply both numerator and denominator of \frac{2+3i}{3+5i} by the complex conjugate of the denominator, 3-5i.

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

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

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

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

$$Re(\frac{\left(2+3i\right)\left(3-5i\right)}{34})$$

Multiply complex numbers 2+3i and 3-5i like you multiply binomials.

$$Re(\frac{2\times 3+2\times \left(-5i\right)+3i\times 3+3\left(-5\right)i^{2}}{34})$$

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

$$Re(\frac{2\times 3+2\times \left(-5i\right)+3i\times 3+3\left(-5\right)\left(-1\right)}{34})$$

Do the multiplications in 2\times 3+2\times \left(-5i\right)+3i\times 3+3\left(-5\right)\left(-1\right).

$$Re(\frac{6-10i+9i+15}{34})$$

Combine the real and imaginary parts in 6-10i+9i+15.

$$Re(\frac{6+15+\left(-10+9\right)i}{34})$$

Do the additions in 6+15+\left(-10+9\right)i.

$$Re(\frac{21-i}{34})$$

Divide 21-i by 34 to get \frac{21}{34}-\frac{1}{34}i.

$$Re(\frac{21}{34}-\frac{1}{34}i)$$

The real part of \frac{21}{34}-\frac{1}{34}i is \frac{21}{34}.

$$\frac{21}{34}$$