# 2.4 Complex numbers  (Page 3/8)

 Page 3 / 8

## Multiplying a complex number by a complex number

Multiply: $\text{\hspace{0.17em}}\left(4+3i\right)\left(2-5i\right).$

$\begin{array}{ccc}\hfill \left(4+3i\right)\left(2-5i\right)& =& 4\left(2\right)-4\left(5i\right)+3i\left(2\right)-\left(3i\right)\left(5i\right)\hfill \\ & =& 8-20i+6i-15\left({i}^{2}\right)\hfill \\ & =& \left(8+15\right)+\left(-20+6\right)i\hfill \\ & =& 23-14i\hfill \end{array}$

Multiply: $\text{\hspace{0.17em}}\left(3-4i\right)\left(2+3i\right).$

$18+i$

## Dividing complex numbers

Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form $\text{\hspace{0.17em}}a+bi.\text{\hspace{0.17em}}$ We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}a-bi.\text{\hspace{0.17em}}$ For example, the product of $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}a-bi\text{\hspace{0.17em}}$ is

$\begin{array}{ccc}\hfill \left(a+bi\right)\left(a-bi\right)& =& {a}^{2}-abi+abi-{b}^{2}{i}^{2}\\ & =& {a}^{2}+{b}^{2}\hfill \end{array}$

The result is a real number.

Note that complex conjugates have an opposite relationship: The complex conjugate of $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}a-bi,$ and the complex conjugate of $\text{\hspace{0.17em}}a-bi\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}a+bi.\text{\hspace{0.17em}}$ Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

Suppose we want to divide $\text{\hspace{0.17em}}c+di\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}a+bi,$ where neither $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ nor $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

Multiply the numerator and denominator by the complex conjugate of the denominator.

$\frac{\left(c+di\right)}{\left(a+bi\right)}\cdot \frac{\left(a-bi\right)}{\left(a-bi\right)}=\frac{\left(c+di\right)\left(a-bi\right)}{\left(a+bi\right)\left(a-bi\right)}$

Apply the distributive property.

$=\frac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}$

Simplify, remembering that $\text{\hspace{0.17em}}{i}^{2}=-1.$

$\begin{array}{l}=\frac{ca-cbi+adi-bd\left(-1\right)}{{a}^{2}-abi+abi-{b}^{2}\left(-1\right)}\hfill \\ =\frac{\left(ca+bd\right)+\left(ad-cb\right)i}{{a}^{2}+{b}^{2}}\hfill \end{array}$

## The complex conjugate

The complex conjugate    of a complex number $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}a-bi.\text{\hspace{0.17em}}$ It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

• When a complex number is multiplied by its complex conjugate, the result is a real number.
• When a complex number is added to its complex conjugate, the result is a real number.

## Finding complex conjugates

Find the complex conjugate of each number.

1. $2+i\sqrt{5}$
2. $-\frac{1}{2}i$
1. The number is already in the form $\text{\hspace{0.17em}}a+bi.\text{\hspace{0.17em}}$ The complex conjugate is $\text{\hspace{0.17em}}a-bi,$ or $\text{\hspace{0.17em}}2-i\sqrt{5}.$
2. We can rewrite this number in the form $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}0-\frac{1}{2}i.\text{\hspace{0.17em}}$ The complex conjugate is $\text{\hspace{0.17em}}a-bi,$ or $\text{\hspace{0.17em}}0+\frac{1}{2}i.$ This can be written simply as $\text{\hspace{0.17em}}\frac{1}{2}i.$

Find the complex conjugate of $\text{\hspace{0.17em}}-3+4i.$

$-3-4i$

Given two complex numbers, divide one by the other.

1. Write the division problem as a fraction.
2. Determine the complex conjugate of the denominator.
3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
4. Simplify.

## Dividing complex numbers

Divide: $\text{\hspace{0.17em}}\left(2+5i\right)\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}\left(4-i\right).$

We begin by writing the problem as a fraction.

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

Then we multiply the numerator and denominator by the complex conjugate of the denominator.

$\frac{\left(2+5i\right)}{\left(4-i\right)}\text{\hspace{0.17em}}\cdot \frac{\left(4+i\right)}{\left(4+i\right)}$

To multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).

Note that this expresses the quotient in standard form.

root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
the indicated sum of a sequence is known as
how do I attempted a trig number as a starter
cos 18 ____ sin 72 evaluate