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Dividing complex numbers

Divide ( 2 + 5 i ) by ( 4 i ) .

We begin by writing the problem as a fraction.

( 2 + 5 i ) ( 4 i )

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

( 2 + 5 i ) ( 4 i ) ( 4 + i ) ( 4 + i )

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

( 2 + 5 i ) ( 4 i ) ( 4 + i ) ( 4 + i ) = 8 + 2 i + 20 i + 5 i 2 16 + 4 i 4 i i 2                             = 8 + 2 i + 20 i + 5 ( 1 ) 16 + 4 i 4 i ( 1 ) Because    i 2 = 1                             = 3 + 22 i 17                             = 3 17 + 22 17 i Separate real and imaginary parts .

Note that this expresses the quotient in standard form.

Substituting a complex number into a polynomial function

Let f ( x ) = x 2 5 x + 2. Evaluate f ( 3 + i ) .

Substitute x = 3 + i into the function f ( x ) = x 2 5 x + 2 and simplify.

Let f ( x ) = 2 x 2 3 x . Evaluate f ( 8 i ) .

102 29 i

Substituting an imaginary number in a rational function

Let f ( x ) = 2 + x x + 3 . Evaluate f ( 10 i ) .

Substitute x = 10 i and simplify.

2 + 10 i 10 i + 3 Substitute  10 i  for  x . 2 + 10 i 3 + 10 i Rewrite the denominator in standard form . 2 + 10 i 3 + 10 i 3 10 i 3 10 i Prepare to multiply the numerator and denominator by the complex conjugate of the denominator . 6 20 i + 30 i 100 i 2 9 30 i + 30 i 100 i 2 Multiply using the distributive property or the FOIL method . 6 20 i + 30 i 100 ( 1 ) 9 30 i + 30 i 100 ( 1 ) Substitute –1 for   i 2 . 106 + 10 i 109 Simplify . 106 109 + 10 109 i Separate the real and imaginary parts .

Let f ( x ) = x + 1 x 4 . Evaluate f ( i ) .

3 17 + 5 i 17

Simplifying powers of i

The powers of i are cyclic. Let’s look at what happens when we raise i to increasing powers.

i 1 = i i 2 = 1 i 3 = i 2 i = 1 i = i i 4 = i 3 i = i i = i 2 = ( 1 ) = 1 i 5 = i 4 i = 1 i = i

We can see that when we get to the fifth power of i , it is equal to the first power. As we continue to multiply i by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of i .

i 6 = i 5 i = i i = i 2 = 1 i 7 = i 6 i = i 2 i = i 3 = i i 8 = i 7 i = i 3 i = i 4 = 1 i 9 = i 8 i = i 4 i = i 5 = i

Simplifying powers of i

Evaluate i 35 .

Since i 4 = 1 , we can simplify the problem by factoring out as many factors of i 4 as possible. To do so, first determine how many times 4 goes into 35: 35 = 4 8 + 3.

i 35 = i 4 8 + 3 = i 4 8 i 3 = ( i 4 ) 8 i 3 = 1 8 i 3 = i 3 = i

Can we write i 35 in other helpful ways?

As we saw in [link] , we reduced i 35 to i 3 by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of i 35 may be more useful. [link] shows some other possible factorizations.

Factorization of i 35 i 34 i i 33 i 2 i 31 i 4 i 19 i 16
Reduced form ( i 2 ) 17 i i 33 ( 1 ) i 31 1 i 19 ( i 4 ) 4
Simplified form ( 1 ) 17 i i 33 i 31 i 19

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

Access these online resources for additional instruction and practice with complex numbers.

Key concepts

  • The square root of any negative number can be written as a multiple of i . See [link] .
  • To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See [link] .
  • Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See [link] .
  • Complex numbers can be multiplied and divided.
  • To multiply complex numbers, distribute just as with polynomials. See [link] , [link] , and [link] .
  • To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator. See [link] , [link] , and [link] .
  • The powers of i are cyclic, repeating every fourth one. See [link] .

Verbal

Explain how to add complex numbers.

Add the real parts together and the imaginary parts together.

What is the basic principle in multiplication of complex numbers?

Give an example to show the product of two imaginary numbers is not always imaginary.

i times i equals –1, which is not imaginary. (answers vary)

What is a characteristic of the plot of a real number in the complex plane?

Algebraic

For the following exercises, evaluate the algebraic expressions.

If  f ( x ) = x 2 + x 4 , evaluate f ( 2 i ) .

8 + 2 i

If  f ( x ) = x 3 2 , evaluate f ( i ) .

If  f ( x ) = x 2 + 3 x + 5 , evaluate f ( 2 + i ) .

14 + 7 i

If  f ( x ) = 2 x 2 + x 3 , evaluate f ( 2 3 i ) .

If  f ( x ) = x + 1 2 x , evaluate f ( 5 i ) .

23 29 + 15 29 i

If  f ( x ) = 1 + 2 x x + 3 , evaluate f ( 4 i ) .

Graphical

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

Graph of a parabola intersecting the real axis.

2 real and 0 nonreal

Graph of a parabola not intersecting the real axis.

For the following exercises, plot the complex numbers on the complex plane.

1 2 i

Graph of the plotted point, 1-2i.

2 + 3 i

i

Graph of the plotted point, i.

3 4 i

Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

( 3 + 2 i ) + ( 5 3 i )

8 i

( 2 4 i ) + ( 1 + 6 i )

( 5 + 3 i ) ( 6 i )

11 + 4 i

( 2 3 i ) ( 3 + 2 i )

( 4 + 4 i ) ( 6 + 9 i )

2 5 i

( 2 + 3 i ) ( 4 i )

( 5 2 i ) ( 3 i )

6 + 15 i

( 6 2 i ) ( 5 )

( 2 + 4 i ) ( 8 )

16 + 32 i

( 2 + 3 i ) ( 4 i )

( 1 + 2 i ) ( 2 + 3 i )

4 7 i

( 4 2 i ) ( 4 + 2 i )

( 3 + 4 i ) ( 3 4 i )

25

3 + 4 i 2

6 2 i 3

2 2 3 i

5 + 3 i 2 i

6 + 4 i i

4 6 i

2 3 i 4 + 3 i

3 + 4 i 2 i

2 5 + 11 5 i

2 + 3 i 2 3 i

9 + 3 16

15 i

4 4 25

2 + 12 2

1 + i 3

4 + 20 2

i 8

1

i 15

i 22

1

Technology

For the following exercises, use a calculator to help answer the questions.

Evaluate ( 1 + i ) k for k = 4, 8, and 12 . Predict the value if k = 16.

Evaluate ( 1 i ) k for k = 2, 6, and 10 . Predict the value if k = 14.

128i

Evaluate ( 1 + i ) k ( 1 i ) k for k = 4, 8, and 12 . Predict the value for k = 16.

Show that a solution of x 6 + 1 = 0 is 3 2 + 1 2 i .

( 3 2 + 1 2 i ) 6 = 1

Show that a solution of x 8 1 = 0 is 2 2 + 2 2 i .

Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

1 i + 4 i 3

3 i

1 i 11 1 i 21

i 7 ( 1 + i 2 )

0

i −3 + 5 i 7

( 2 + i ) ( 4 2 i ) ( 1 + i )

5 – 5i

( 1 + 3 i ) ( 2 4 i ) ( 1 + 2 i )

( 3 + i ) 2 ( 1 + 2 i ) 2

2 i

3 + 2 i 2 + i + ( 4 + 3 i )

4 + i i + 3 4 i 1 i

9 2 9 2 i

3 + 2 i 1 + 2 i 2 3 i 3 + i

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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