1 Divided By Complex Number

The division of complex numbers is solved past multiplying both the numerator and denominator by the conjugate of the circuitous number in the denominator. This will permit us to obtain a real number in the denominator and nosotros will obtain the effect to the division.

Here, we will acquire how to divide complex numbers using their conjugate. Also, we will wait at several examples with answers to look at the application of this procedure.

ALGEBRA

Relevant for…

Learning to dissever circuitous numbers with examples.

See examples

ALGEBRA

Relevant for…

Learning to dissever complex numbers with examples.

See examples

How to divide complex numbers?

To divide complex numbers, we have to start by writing the trouble in fractional form. Then, we have to multiply both the numerator and denominator by the conjugate of the denominator.

Remember that to find the conjugate of the denominator, we simply accept to modify the sign to the imaginary component. For example, the conjugate of $latex a+bi$ is $latex a-bi$.

Therefore, if we want to split the number $latex a+bi$ by the number $latex c+bi$, we form the following expression:

$latex \frac{a+bi}{c+di}=\frac{a+bi}{c+di}\times \frac{c-di}{c-di}$

Nosotros solve this expression by distributing the multiplication in both the numerator and denominator. Then, nosotros simplify the powers ofi, specifically, we call back that $latex {{i}^ii}$ is equal to $latex -one$.

And so, we combine like terms to simplify the obtained expression, and finally, we write the answer in the form $latex a+bi$.

Partition of circuitous numbers – Examples with answers

The following exercises use the process detailed above to solve the divisions of complex numbers. Each case has its respective solution, merely information technology is recommended that you lot try to solve the exercises yourself earlier looking at the answer.

Example 1

Divide the complex numbers: $latex (2+4i) \div (i+2i)$.

Solution

We have to start by writing the original problem in fractional course:

⇒ $latex \frac{two+4i}{one+2i}$

At present, we multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of the denominator is $latex i-2i$. Therefore, we accept:

$latex \frac{2+4i}{1+2i}=\frac{2+4i}{1+2i}\times \frac{1-2i}{1-2i}$

We aggrandize the multiplication of the numerator and denominator and simplify by combining similar terms:

$latex = \frac{2-4i+4i-8{{i}^2}}{one-2i+2i-2{{i}^2}}$

$latex = \frac{2-8{{i}^2}}{1-2{{i}^two}}$

We call up that $latex {{i}^2}$ is equal to $latex -1$:

$latex = \frac{2-8(-1)}{1-2(-1)}$

$latex = \frac{10}{3}$

EXAMPLE 2

Split up the circuitous numbers: $latex (5+10i) \div (iv+3i)$.

Solution

We rewrite complex numbers in fractional course:

⇒ $latex \frac{5+10i}{four+3i}$

Hither, the cohabit of the denominator is $latex 4-3i$. Therefore, we multiply the numerator and denominator by this conjugate:

$latex \frac{v+10i}{4+3i}=\frac{five+10i}{4+3i}\times \frac{4-3i}{iv-3i}$

Nosotros solve the multiplications in the numerator and denominator and simplify:

$latex = \frac{20-15i+40i-30{{i}^2}}{xvi-12i+12i-9{{i}^ii}}$

$latex = \frac{20+25i-30{{i}^2}}{16-9{{i}^2}}$

Nosotros recall that $latex {{i}^two}$ is equal to $latex -1$:

$latex = \frac{20+25i-30(-1)}{xvi-9(-1)}$

$latex = \frac{l+25i}{25}$

Nosotros already got the answer, simply we have to write it in the form $latex a+bi$. Thus, we accept:

$latex =\frac{fifty}{25}+\frac{25}{25}i$

$latex =2+i$

Instance iii

What is the consequence of the division $latex (iv-6i)\div (-2-4i)$?

Solution

We have to write the division in fractional form:

⇒ $latex \frac{four-6i}{-2-4i}$

The cohabit of the denominator is $latex -ii+4i$. Therefore, we have:

$latex \frac{4-6i}{-ii-4i}=\frac{4-6i}{-2-4i}\times \frac{-ii+4i}{-2+4i}$

Now, we solve the multiplications in the numerator and denominator and simplify:

$latex = \frac{-eight+16i+12i-24{{i}^2}}{4-8i+8i-16{{i}^2}}$

$latex = \frac{-eight+28i-24{{i}^ii}}{iv-16{{i}^ii}}$

We retrieve that $latex {{i}^2}$ is equal to $latex -1$:

$latex = \frac{-8+28i-24(-one)}{4-16(-i)}$

$latex = \frac{16+28i}{20}$

Writing in the form $latex a+bi$, we have:

$latex =\frac{16}{20}+\frac{28i}{20}$

$latex =\frac{4}{5}+\frac{7i}{5}$

Case 4

What is the upshot of the sectionalisation $latex (-4-4i) \div (-four+4i)$?

Solution

The division in fractional form is:

⇒ $latex \frac{-4-4i}{-4+4i}$

If nosotros multiply both the numerator and the denominator by the conjugate of the denominator, which is $latex -iv-4i$, we take:

$latex \frac{-4-4i}{-4+4i}=\frac{-4-4i}{-iv+4i}\times \frac{-4-4i}{-4-4i}$

We aggrandize the multiplication of the numerator and denominator and simplify past combining similar terms:

$latex = \frac{xvi+16i+16i+16{{i}^2}}{16+16i-16i-xvi{{i}^2}}$

$latex = \frac{xvi+32i+16{{i}^ii}}{xvi-16{{i}^2}}$

Nosotros call back that $latex {{i}^two}$ is equal to $latex -1$:

$latex = \frac{16+32i+sixteen(-1)}{16-16(-1)}$

$latex = \frac{32i}{32}$

$latex =i$

EXAMPLE 5

Solve the segmentation $latex \frac{10-2i}{-iv+5i}$.

Solution

In this case, we already accept the division written in partial form, so nosotros outset past multiplying both the numerator and the denominator by the conjugate of the denominator.

In this case, the conjugate of the denominator is $latex -4-5i$. Therefore, we have:

$latex \frac{ten-2i}{-4+5i}=\frac{10-2i}{-4+5i}\times \frac{-4-5i}{-4-5i}$

We solve the multiplications in the numerator and denominator:

$latex = \frac{-xl-50i+8i+10{{i}^2}}{xvi+20i-20i-25{{i}^2}}$

$latex = \frac{-40-32i+10{{i}^2}}{16-25{{i}^ii}}$

We use the fact that $latex {{i}^2}$ is equal to $latex -1$:

$latex = \frac{-40-32i+ten(-1)}{16-25(-1)}$

$latex = \frac{-50-32i}{41}$

$latex =-\frac{l}{41}-\frac{32}{41}i$

EXAMPLE 6

Solve the partition $latex \frac{-4}{one-i}$.

Solution

Here we already accept a division in fractional course, and then we offset past multiplying both the numerator and the denominator by the conjugate of the denominator.

In this case, the cohabit of the denominator is $latex ane+i$. Therefore, we have:

$latex \frac{-four}{1-i}=\frac{-4}{1-i}\times \frac{1+i}{one+i}$

Now, nosotros distribute to the multiplications and simplify:

$latex = \frac{-4-4i}{i+i-i-{{i}^2}}$

$latex = \frac{-4-4i}{1-{{i}^2}}$

Recall that $latex {{i}^ii}$ is equal to $latex -one$:

$latex = \frac{-4-4i}{1+1}$

$latex = \frac{-4-4i}{2}$

$latex =-\frac{four}{2}-\frac{four}{ii}i$

$latex =-two-2i$

Segmentation of complex numbers – Do problems

Practice what you have learned about the division of complex numbers with the post-obit problems. If y'all need help with these problems, yous tin can wait at the solved examples above.

What is the result of $latex \frac{3+2i}{4-3i}$?

Cull an answer

$latex \frac{3}{25}+\frac{7}{25}i$

$latex \frac{6}{25}+\frac{17}{25}i$

$latex \frac{10}{9}+\frac{7}{ix}i$

$latex \frac{5}{9}+\frac{17}{nine}i$

What is the result of $latex \frac{one-3i}{one+2i}$?

Choose an answer

$latex 1+i$

$latex 1-i$

$latex -ane-i$

$latex -1+i$

Solve the sectionalization $latex \frac{iv+5i}{2+6i}$.

Choose an answer

$latex \frac{19}{20}-\frac{7}{20}i$

$latex \frac{nine}{10}-\frac{7}{twenty}i$

$latex \frac{seven}{10}-\frac{3}{x}i$

$latex \frac{nine}{vii}-\frac{7}{11}i$

Solve the division $latex \frac{-six-3i}{4+6i}$.

Cull an answer

$latex -\frac{11}{26}-\frac{3}{13}i$

$latex \frac{ane}{6}-\frac{iv}{13}i$

$latex -\frac{11}{16}-\frac{5}{3}i$

$latex -\frac{21}{26}-\frac{vi}{13}i$