Solve 1+4i/4+4i

$$\frac { 1 + 4 i } { 4 + 4 i }$$

Evaluate

$$\frac{5}{8}+\frac{3}{8}i=0.625+0.375i$$

Solution Steps

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

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

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

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

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

$$\frac{\left(1+4i\right)\left(4-4i\right)}{32}$$

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

$$\frac{1\times 4+1\times \left(-4i\right)+4i\times 4+4\left(-4\right)i^{2}}{32}$$

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

$$\frac{1\times 4+1\times \left(-4i\right)+4i\times 4+4\left(-4\right)\left(-1\right)}{32}$$

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

$$\frac{4-4i+16i+16}{32}$$

Combine the real and imaginary parts in 4-4i+16i+16.

$$\frac{4+16+\left(-4+16\right)i}{32}$$

Do the additions in 4+16+\left(-4+16\right)i.

$$\frac{20+12i}{32}$$

Divide 20+12i by 32 to get \frac{5}{8}+\frac{3}{8}i.

$$\frac{5}{8}+\frac{3}{8}i$$

Real Part

$$\frac{5}{8} = 0.625$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(1+4i\right)\left(4-4i\right)}{32})$$

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

$$Re(\frac{1\times 4+1\times \left(-4i\right)+4i\times 4+4\left(-4\right)i^{2}}{32})$$

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

$$Re(\frac{1\times 4+1\times \left(-4i\right)+4i\times 4+4\left(-4\right)\left(-1\right)}{32})$$

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

$$Re(\frac{4-4i+16i+16}{32})$$

Combine the real and imaginary parts in 4-4i+16i+16.

$$Re(\frac{4+16+\left(-4+16\right)i}{32})$$

Do the additions in 4+16+\left(-4+16\right)i.

$$Re(\frac{20+12i}{32})$$

Divide 20+12i by 32 to get \frac{5}{8}+\frac{3}{8}i.

$$Re(\frac{5}{8}+\frac{3}{8}i)$$

The real part of \frac{5}{8}+\frac{3}{8}i is \frac{5}{8}.

$$\frac{5}{8}$$