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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.

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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.

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

102 29 i

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

3 17 + 5 i 17

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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
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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.

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What is the basic principle in multiplication of complex numbers?

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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)

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

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Algebraic

For the following exercises, evaluate the algebraic expressions.

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

8 + 2 i

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

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

14 + 7 i

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

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

23 29 + 15 29 i

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

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Graphical

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

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

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

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

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( 5 + 3 i ) ( 6 i )

11 + 4 i

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( 2 3 i ) ( 3 + 2 i )

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

2 5 i

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( 5 2 i ) ( 3 i )

6 + 15 i

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( 2 + 4 i ) ( 8 )

16 + 32 i

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

4 7 i

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( 4 2 i ) ( 4 + 2 i )

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( 3 + 4 i ) ( 3 4 i )

25

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3 + 4 i 2 i

2 5 + 11 5 i

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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.

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

128i

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

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

( 3 2 + 1 2 i ) 6 = 1

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

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Extensions

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

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

5 – 5i

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

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

2 i

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

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4 + i i + 3 4 i 1 i

9 2 9 2 i

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

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Questions & Answers

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
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Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
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meena
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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