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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

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
how to solve the sums
meena
hello guys
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
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
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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