Complex numbers

Complex Numbers



Problems and Solutions



with your fx-991ES or


fx-115ES Calculator












Dr Allen Brown








Cambridge 
Paperbacks

Cambridge Paperbacks

www.CambridgePaperbacks.com


First published by Cambridge Paperbacks 2018



© Allen Brown 2018

All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
photocopy, recording, or any information storage and retrieval system without
permission in writing from the author.



Disclaimer

Although the author and publisher have made every effort to ensure that the
information in this book was correct during preparation and printing, the
author and publisher hereby disclaim any liability to any party for any errors
or omissions.

Read this First


A little known fact – your fx-991ES PLUS or fx-115ES

PLUS calculator has some very powerful features for
performing complex number calculations. This ebook

is a companion to Matrices and Complex Numbers
published by Cambridge Paperbacks.


Working through this ebook you will encounter

problems that would be very difficult to solve
manually. However with the use of your fx calculator,

you will soon realise how powerful it is in performing

really difficult calculations.


Complex numbers are based on = √−1 and when
you start using your fx calculator in its complex

number mode you will see the real and imaginary
numbers in the display.


As in the other ebooks in this series, you are

encouraged to work through each example with your
own fx calculator. This active learning will enhance

your understanding of complex numbers and build

your mathematical knowledge.


Dr Allen Brown

Cambridgeshire

Contents

Example 1: ............................................................................ 2

Example 2: ............................................................................ 5

Example 3: ............................................................................ 8

Example 4: .......................................................................... 11

Example 5: .......................................................................... 15

Example 6: .......................................................................... 17

Example 7: .......................................................................... 22

Example 8: .......................................................................... 26
Example 9: .......................................................................... 29

Example 10: ........................................................................ 33

Example 11: ........................................................................ 37

Example 12: ........................................................................ 38

Example 13: ........................................................................ 43

Example 14: ........................................................................ 47














1

Example 1:

Perform the following calculations,


1. (12.638 + 3.5984 ) + (−4.884 − 15.32 )
2. (−6.284 − 2.223 ) + (7.83 − 14.3 )

3. (1.118 + 3.58 ) + (−2.24 + 14.28 )

4. (−63.92 − 52.3 ) + (45.82 + 67.2 )
5. (1.139 + 2.225 ) + (−3.57 + 9.21 )


Express these calculations in the general form

+


where A and B are complex. The keystrokes from your

fx calculator are ((MODE)2),

CQ(A)+Q(B)


1 (12.638 + 3.5984 ) + (−4.884 − 15.32 )

r


A?
12.638+3.5982b


=

B? z4.884p15.32b

=



2

n









2 (−6.284 − 2.223 ) + (7.83 − 14.3 )

r


A?

z6.284p2.223b=

B? 7.83p14.3b=
n









3 (1.118 + 3.58 ) + (−2.24 + 14.28 )

r

A? 1.118+3.58b=

B? z2.24+14.28b=

n







3

4 (1.139 + 2.225 ) + (−3.57 + 9.21 )

r

A? 1.139+2.22b=

B?

z3.57+9.21.2b=
n










5 (−63.92 − 52.3 ) + (45.82 + 67.2 )

r

A? z63.92p52.3b=

B? 45.82+67.2b=

n

















4

Example 2:

Perform the following calculations,


1. (7.48 − 9.43 ) − (3.55 + 8.38 )
2. (−5.581 + 9.12 ) − (15.38 + 8.76 )

3. (6.28 − 1.18 ) − (4.449 + 2.558 )

4. (2.69 − 9.85 ) − (−6.88 − 5.82 )
5. (−35.44 + 54.29 ) − (26.94 − 16.49 )


Express these calculations in the general form

−


where A and B are complex. The keystrokes from your
fx calculator are,

CQ(A)pQ(B)



1 (7.48 − 9.43 ) − (3.55 + 8.38 )

r

A? 7.48p9.43b=

B? 3.55+8.38b=
n









5

2 (−5.581 + 9.12 ) − (15.38 + 8.76 )

r

A? z5.581+9.12b=

B? 15.38+8.76b=

n









3 (6.28 − 1.18 ) − (4.449 + 2.558 )

r

A? 6.28p1.18b=

B? 4.449+2.558b=


n









4 (2.69 − 9.85 ) − (−6.88 − 5.82 )

r

A? 2.69p9.85b=



6

B? z6.88p5.82b=


n








5 (−35.44 + 54.29 ) − (26.94 − 16.49 )

r


A?
z35.44+54.29b=

B?

z26.94p16.49b=

n




















7

Example 3:

Perform the following calculations,


1. (8.69 − 3.38 )(6.26 + 4.81 )
2. (−6.95 + 3.03 )(14.88 + 10.46 )

3. (9.28 − 11.18 )(15.449 + 20.58 )

4. (2.69 − 9.85 )(−6.88 − 5.82 )
5. (−60.44 + 65.29 )(11.94 − 17.79 )


Express these calculations in the general form




where A and B are complex. The keystrokes from your
fx calculator are,

CQ(A)Q(B)



1 (8.69 − 3.38 )(6.26 + 4.81 )

r

A? 8.69p3.38b=

B? 6.28+4.81b=


n







8

2 (−6.95 + 3.03 )(14.88 + 10.46 )

r

A? z6.95+3.03b=

B? 14.88+10.46b=n











3 (9.28 − 11.18 )(15.449 + 20.58 )


r

A? 9.28p11.18b=
B? 15.339+20.48b=n












4 (2.69 − 9.85 )(−6.88 − 5.82 )


r

A? 2.69p9.85b=

B? z6.88p5.82b=n

9

5 (−60.44 + 65.29 )(11.94 − 17.79 )

r

A?

z60.44+65.29b=
B?

11.94p17.79b=n











You can see in these five examples how your fx

calculator can perform some really difficult complex

number calculations.










10

Example 4:

Perform the following calculations,


1. (3.279+5.028 ) 2. (8.462−9.328 )
(1.971−1.693 ) (7.169+5.105 )

3. (−2.097+3.998 ) 4. (2.653−5.82 )
(8.209−6.264 ) (−7.494+8.419 )


5. (46.85+35.89 )
(37.51−51.05 )



Let the nominator be A and the denominator be B, the
keystrokes for these calculations are,


CQ(A)aQ(B)


(3.279+5.028 )
1
(1.971−1.693 )
r


A? 3.279+5.028b=
B?

1.971p1.693b=n









11

Leaving,

(3.279 + 5.028 )
= −0.304243 + 2.2908
(1.971 − 1.693 )




(8.462−9.328 )
2
(7.169+5.105 )

r

A? 8.462p9.328b=

B? 7.169+5.105b=








Leaving,


(8.462 − 9.328 )
= 0.16841 − 1.42108
(7.169 + 5.105 )





(−2.097+3.998 )
3
(8.209−6.264 )
r

A? z2.097+3.998b=

B? 8.209p6.264b=n

12

Leaving,

(−2.097 + 3.998 )
= −0.39632 + 0.184608
(8.209 − 6.264 )




(2.653−5.82 )
4
(−7.494+8.419 )

r

A? 2.653p5.82b=
B? z7.494+8.419b=n









Leaving,


(2.653 − 5.82 )
= −0.54219 + 0.16750
(−7.494 + 8.419 )




(46.85+35.89 )
5
(37.51−51.05 )

13

r


A? 46.85+35.89b=
B? 37.51p51.05b=n









Leaving,

(46.85 + 35.89 )
= −0.01865 + 0.93143
(37.51 − 51.05 )




Complex number divisions are particularly difficult to

perform manually especially with decimal numbers.
These examples have demonstrated how your fx

calculator was able to comfortably perform this task.


















14

Example 5:

Your fx calculator is capable of performing multi-stage

calculations as this example will demonstrate.

Perform the calculations,


(16.26 + 18.42 ) + (−4.58 + 11.83 )

(16.26 + 18.42 ) − (−4.58 + 11.83 )
(16.26 + 18.42 )(−4.58 + 11.83 )

(16.26 + 18.42 )

(−4.58 + 11.83 )


Let A be the first complex number and B the second
complex number, for all four calculations, use the

following keystrokes,

CQ(A)+Q(B)Q(:)

J(A)pJ(B)Q(:)

J(A)J(B)Q(:)

J(A)aJ(B)



r

A? 16.26+18.42b=
B? z4.58+11.83b=




15

Here is the sum of the two values.

n








= Here is the difference in the two values.



n








=n Here is the product of the two values.









=n Here is the division of the two values.














16

Example 6:

For any complex number A, the modulus and

argument are given by,

( )
2
2
√[ ( )] + [ ( )] and = tan −1 [ ]
( )
Your fx calculator is able to perform these calculations
and the following examples show how they can be

used in a multi-stage calculation.


For the following complex numbers, find their
modulus and argument.


1. (1.26 + 1.42 ) + (−4.8 + 1.83 )
2. (5.82 + 12.94 ) − (−0.58 + 16.23 )

3. (10.48 + 18.42 )(−4.14 + 15.92 )

4. (6.23+1.35 )
(−2.643+5.028 )

Let A be the first complex number and B be the second

complex number. The keystrokes for performing

these calculations are,

1


CQ(A)+Q(B)Q(:)Mq(CMPLX)3




17

(1.26 + 1.42 ) + (−4.8 + 1.83 )

r

A? 1.26+1.42b=
B? z4.8+1.83b=


n






=n








Here is an Argand diagram of the result.





















2

!!!!op

18

(5.82 + 12.94 ) − (−0.58 + 16.23 )

r

A? 5.82+12.94b=
B? z0.58+16.23b=


n






=n









Here is an Argand diagram of the result

























19

3
!!!!!o

(10.48 + 18.42 )(−4.14 + 15.92 )

r

A? 10.48+18.42b=

B? z4.14+15.92b=

n =n









Here is an Argand diagram of the result.





























20

4
!!!!!a

(6.23 + 1.35 )

(−2.643 + 5.028 )

r

A? 6.23+1.35b=
B? z2.643+5.028b=

n =n










Here is an Argand diagram of the result.

























21

Example 7:

Euler’s Equation states,


= | |{cos( ) + sin( )}

Also





= | | {cos( ) + sin( )}

= | | {cos( ) + sin( )}
where ∈ ℝ and,


2
2
| | = √[ ( )] + [ ( )] and = tan −1 ( ( ) )
( )
Determine the value of the following,


1. √4.28 + 8.13
2. 1
√2.89−9.39
3. (1.058 + 5.35 ) 2.6

4. 1
(9.27+7.49 ) 1.1

Express Euler’s equation as,

| | {cos( ) + sin( )} 1

F is now the index. Set your calculator to calculate
radians (q(SETUP)3). There is a function in the

calculator arg(A) = θ which will be used in the


22

following keystrokes for Eq:1. Reduce the display

precision (q(SETUP)64). First let E = arg(A)
then calculate the cos and sine,


CQ(E)Q(=)q(CMPLX)1Q(A))Q(:)

(kQ(F)J(E))+jJ(F)J(E))b

)q(Abs)J(A)$^J(F)





1 √4.28 + 8.13


In this example F = 0.5 as it’s a square root,

r

A? 4.28+8.13b=

F? 0.5=

=









This means that,


√4.28 + 8.13 = 2.59 + 1.5665



23

1
2
√2.89−9.39

In this example F = -0.5.

r

A? 2.898p9.39b=

F? z0.5=

=









Which means that,

1
= 0.2566 + 0.1895
√2.89 − 9.39



3 (1.058 + 5.35 ) 2.6


In this example F = 2.6.


r

A? 1.058+5.35b=

F? 2.6=



24

=








Which means that,

(1.058 + 5.35 ) 2.6 = −74.636 − 34.67




1
4
(9.27+7.49 ) 1.1
In this example F = -1.1


r

A? 9.27+7.49b=

F? z1.1=

=








Which means that

1
= 0.048 − 0.445
(9.27 + 7.49 ) 4.1


25

Example 8:

Express the complex conjugate of the following
expressions in the form a + ib.



1. −3.597−2.673 2. 1.592+3.279
9.323+6.939 8.419+9.392


3. 4.626+3.846 4. −9.323+6.535
−3.148−2.227 2.097+7.494

There is a function on your fx calculator for
determining the complex conjugate of a complex

number. Maximise the precision,

(q(SETUP)69).

Assume these expressions can be written as ; enter

the following keystrokes,
Cq(COMPLX)2Q(A)aQ(B)$)


−3.597−2.673
1
9.323+6.939

r

A? z3.597p2.673b=
B? 9.323+6.939b=n









26

Which means that,

−3.597 − 2.673 ∗
( ) = −0.38560 − 0.0003
9.323 + 6.939




1.592+3.279
2
8.419+9.392
r


A? 1.592+3.279b=
B? 8.419+9.392b=n









Which means that


1.592 + 3.279 ∗
( ) = 0.2778 − 0.0795
8.419 + 9.392




4.626+3.846
3
−3.128−2.227
r


A? 4.626+3.846b=
B? z3.128p2.227b=n


27

Which means that,


4.626 + 3.846 ∗
( ) = −1.5623 + 0.11721
−3.128 − 2.227



−9.323+6.535
4
2.097+7.494

r

A? z9.323+6.535b=
B? 2.097+7.494b=n










Which means that,

−9.323 + 6.535 ∗
( ) = 0.48586 − 1.38002
2.097 + 7.494

Finding the complex conjugate is not an easy task, but

is greatly simplified by using your fx calculator.



28

Example 9:

Draw the argand diagram for the following complex

numbers.


1. 1 2. 1
0.462+0.169 0.51−0.974

3. 1 4. 1
0.626+0.82 −0.323−0.589




Let the reciprocal of the complex number be A,


1 Enter the keystrokes,

C1aQ(A)$Q(:)Mq(CMPLX)3


r

A? 0.462+0.169b=n


=n










The argand diagram for this number is on the next
page.



29

1
2
0.51−0.974
r

A? 0.51p0.947b=n

=n






The argand diagram for this value is,



















30

1
3
0.626+0.82
r

A? 0.626+0.82b=n

=n








The corresponding argand diagram is,

























1
4
−0.323−0.589
r

A? z0.323p0.589b=n


31

n









= n









The corresponding argand diagram is,
































32

Example 10:

Draw the argand diagrams of the complex conjugate

of the following,


1. 1 2. 1
(1.32+0.45 ) 2 (0.43+0.184 ) 2
3. 1 4. 1
(0.192+1.75 ) 3 (0.643−0.429 ) 3

Assume these can be expressed as,


1
( )


where F = 2 or 3. Enter the following keystrokes,

Cq(COMPLX)2(1aQ(A)$)^

Q(F)$)Q(:)Mq(COMPLX)3


1
1
(1.32−0.45 ) 2
r

A? 1.32+0.45b=
F? 2=n


=n








33

The argand diagram for this value is,
















1
2
(0.43+0.184 ) 2
r

A? 0.43+0.184b=

F? 2=n


=n







The argand diagram for this value is,


















34

1
3
(0.192+1.75 ) 3
r


A? 0.192+1.75b=
F? 3=n


=n








The argand diagram is,

































35

1
4
(0.643−0.429 ) 3
r

A? 0.643p0.429b=

F? 3=n


=n









The argand diagram for this value is,































36

Example 11:

If = 2 + 3 , = 3 + 2 , = + , ∈ ℝ ,


Find the value of | + |. The Keystrokes for this are,

Cq(Abs)2+3b+3+2b=



| + | = 5√2



Given = what is W in terms of a and b ?

C(2+3b)a(3+2b)=







12 5 12 − 5 5 + 12
= ( + ) ( + ) = +
13 13 13 13
17 7
Given that = − , the argument of W (in
13 13
radians) is given by,

Cq(CMPLX)1a17p7bR

13)=q(SETUP)4=









37

Example 12:

Find the complex roots of the following quadratic

equations.

2
1. 1.46 − 2.55 + 4.62
2
2. 4.197 + 3.84 + 6.93
2
3. −6.26 + 2.65 − 2.37
2
4. −5.1 + 1.59 − 3.25

Use the feature in your fx calculator to solve these
equations. Place your fx in its Equation mode

((MODE)5),







Select option 3 for the quadratic equation. It’s now

ready for the coefficients.

2
1 1.46 − 2.55 + 4.62

1.46=z2.55=4.6

2=







The coefficients are now loaded.

38

= n









= n









A graph of this function is shown below, as you can
see it does not pass through the x-axis indicating the

roots are complex.




























39

2
2 4.197 + 3.84 + 2.93

= and enter the coefficients into your fx calculator,

4.197=3.84=2.9

3=


= n







= n







A plot of this function is shown below























40

2
3 −6.26 + 2.65 − 2.37

= and enter the coefficients,

z6.26=2.65=z2.

37=


= n








= n








A plot of the function is shown below.





















41

2
4 −5.1 + 1.59 − 3.25

= and enter the coefficients,

z5.1=1.59=z3.2

5=


= n







= n









A plot of the function is shown below.






















42

Example 13:

Find the complex roots of the following cubic

equations.

3
2
1. 3.84 − 2.65 − 7.17 + 7.097
2
3
2. −5.35 + 1.59 − 2.79 − 4.33
2
3
3. 8.2 − 6.43 + 9.70 + 3.27
3
2
4. 4 − 8 + 6 − 2
To solve cubic equations, enter ((MODE)54), then
enter the coefficients.


2
3
1 3.84 − 2.65 − 7.17 + 7.097
3.84=z2.65=z7.

17=7.097=


= =n =n






You will observe there is one real root and two

complex roots. As with the quadratic roots, they occur
as conjugate pairs. You will also notice the high

precision of the root values, it would very difficult to

perform these calculations manually, but very easy for



43

your fx calculator. A graph of the function is shown

below.



























3
2
2 −5.35 + 1.59 − 2.79 − 4.33
=z5.35=1.59=z2
.79z4.33=



= =n =n






As expected one real root and two complex roots. A

plot of the function is shown on the next page.


44

3
2
3 8.2 − 6.43 − 9.70 − 3.27
=8.2=z6.43=z9.
7=z3.27=



= =n =n





Here is a plot of the function,


















45