HP 15c Manual

							 Section 11: Calculating With Complex Numbers 131 
 
One-Number Functions 
The following functions operate on both the real and imaginary parts of the 
number  in  the  X-register,  and  place  the  real  and  imaginary  parts  of  the 
answer back into those registers. 
¤ x N o ∕ @  a : ; 
All trigonometric and hyperbolic functions and their inverses also belong to 
this group.* 
The a function  gives  the  magnitude  of  the  number  in  the  X-registers 
(the  square  root  of  the  sum  of  the  squares  of  the  real  and  imaginary  parts); 
the imaginary part of the magnitude is zero. 
: converts  to  polar  form  and ; converts  to  rectangular  form, 
as described later in this section (page 133). 
For the trigonometric functions, the calculator considers numbers in the real 
and  imaginary  X-registers  to  be  expressed  in radians—regardless  of  the 
current trigonometric  mode. To calculate trigonometric  functions for values 
given  in  degrees,  use r to  convert  those  values  to  radians  before 
executing the trigonometric function. 
Two-Number Functions 
The following functions operate on both the real and imaginary parts of the 
numbers in the X- and Y-registers, and place the real and imaginary parts of 
the  answer into the  X-registers. Both  stacks drop, just as the  ordinary  stack 
drops after a two-number function not in Complex mode. 
+ - * ÷ y 
Stack Manipulation Functions 
When  the  calculator  is  in  Complex  mode,  the  following  functions 
simultaneously  manipulate  both  the  real  and  imaginary  stacks  in  the  same 
way  as  they  manipulate  the  ordinary  stack  when  the  calculator  is  not in 
Complex  mode.  The ® function.  for  instance,  will  exchange both the 
real and imaginary parts of the numbers in the X- and Y-registers. 
® ) ( v K 
* Refer  to  the HP-15C  Advanced Functions Handbook for  definitions of complex trigonometric functions and further information about doing calculations in Complex mode.  
						

							132 Section 11: Calculating With Complex Numbers 
 
Conditional Tests 
For programming, the four conditional tests below will work in the complex 
sense: ~ and T 0  compare  the complex number  in  the  (real  and 
imaginary)  X-registers  to  0  +  0i,  while T 5  and T 6  compare  the 
complex numbers  in  the  (real  and  imaginary)  X- and  Y-registers.  All  other 
conditional tests besides those listed below ignore the imaginary stack. 
~  T 0 (x ≠ 0)  T 5 (x = y) T 6 (x ≠ y) 
Example:  Complex  Arithmetic. The  characteristic  impedance  of  a  ladder 
network is given by an equation of the form 
, 
where A and B are  complex  numbers.  Find Z0 for  the  hypothetical  values 
A = 1.2 + 4.7i and B = 2.7 + 3.2i. 
Keystrokes Display  
1.2 v 4.7 ´ V 1.2000 Enters A into real and 
imaginary X-registers. 
2.7 v 3.2 ´ V 2.7000 Enters B into real and 
imaginary X-registers, 
moving A into real and 
imaginary Y-registers. 
÷ 1.0428 Calculates A/B. 
¤ 1.0491 Calculates Z0 and 
displays real part. 
´ %  (hold) 0.2406 Displays imaginary part 
of Z0 while % is held 
down. 
(release) 1.0491 Again displays real part 
of Z0. B
AZ0  
						

							 Section 11: Calculating With Complex Numbers 133 
 
Complex Results from Real Numbers 
In  the  preceding  examples,  the  entry  of  complex  numbers  had  ensured  the 
(automatic)  activation  of  Complex  mode.  There  will  be  times,  however, 
when  you  will  need  Complex  mode  to  perform  certain  operations  on real 
numbers, such as . (Without Complex mode, such as operation would 
result  in  an Error 0 – improper  math  function.)  To  activate  Complex  mode 
at  any  time and  without  disturbing  the  stack  contents, set  flag  8  before 
executing the function in question.* 
Example: The arc sine (sin-1) of 2.404 normally would result in an Error 0. 
Assuming  2.404  in  the  X-register,  the  complex  value  arc  sin  2.404  can  be 
calculated as follows: 
Keystrokes Display  
| F 8  Activates Complex Mode. 
| ,  1.5708 Real part of 
arc sin 2.404. 
´ %  (hold) -1.5239 Imaginary part of 
arc sin 2.404. 
(release)  1.5708 Display shows real part 
again when % is 
released. 
Polar and Rectangular Coordinate Conversions 
In  many  applications,  complex  numbers  are  represented  in  polar  form, 
sometimes  using  phasor  notation.  However,  the  HP-15C  assumes  that  any 
complex numbers are in rectangular form. Therefore, any numbers in polar 
or  phasor  form  must  be  converted  to  rectangular  form  before  performing  a 
function in Complex mode. 
                                                           * Pressing ´ } twice will accomplish the same thing. The sequence ´ V is not used because it would combine any numbers, in the real X-. and Y-registers into a single complex number. 5  
						

							134 Section 11: Calculating With Complex Numbers 
 
 
a + ib = 
r (cos θ + i sin θ) = reiθ (polar) 
rθ (phasor) 
 
; and : can be used to interconvert the rectangular and polar forms of 
a complex number. They operate in Complex mode as follows: 
´ 
; 
converts  the  polar  (or  phasor)  form  of  a  complex  number  to  its 
rectangular  form  by  replacing  the  magnitude r in  the  real  X-
register  with a, and  replacing  the  angle θ in  the  imaginary  X-
register with b. 
| 
: 
converts  the  rectangular  coordinates  of  a  complex  number  to  the 
polar  (or  phasor)  form  by  replacing  the real  part a in  the  real  X-
register  with r, and  replacing  the  imaginary  part b in  the 
imaginary X-register with θ. 
 
These  are  the  only  functions  in  Complex  mode  that  are  affected  by  the 
current  trigonometric  mode  setting. That  is,  the  angular  units  for θ must 
correspond  to  the  trigonometric  mode  indicated  by  the  annunciator 
(or absence thereof).     
						

							 Section 11: Calculating With Complex Numbers 135 
 
Example: Find the  sum 2(cos 65° + i sin 65°)  +  3(cos 40°  + i sin 40°)  and 
express  the  result  in  polar  form,  (In  phasor  form,  evaluate  265° + 
340°.) 
Keystrokes Display  
| D  Sets Degrees mode for any polar-
rectangular conversions. 
2 v 2.0000  
65 ´ V 2.0000 C annunciator displayed; 
Complex mode activated.   
´ ; 0.8452 Converts polar to rectangular 
form; real part (a) displayed. 
3 v 3.0000  
40 ´ V 3.0000  
´ ; 2.2981 Converts polar to rectangular 
form; real part (a) displayed. 
+ 3.1434  
| : 4.8863 Converts rectangular to polar 
form; r displayed. 
´%   (hold) 49.9612 θ (in degrees). 
(release) 4.8863  
Problems 
By working  through  the  following  problems,  you  will  see  that  calculating 
with  complex  numbers  on  the  HP-15C  is  as  easy  as  calculating  with  real 
numbers.  In  fact,  once  your  numbers  are  entered,  most  mathematical 
operations will use exactly the same keystrokes. Try it and see! 
1. Evaluate:    )54(2 ) 52(4
3)68( 2
ii
ii

  
						

							136 Section 11: Calculating With Complex Numbers 
 
 
Keystrokes Display  
2 ´ }  0.0000 2i. Display shows real part. 
8 “ v -8.0000  
6 ´ V -8.0000 -8 + 6i. 
3 Y  352.0000 (-8 + 6i)3. 
* -1.872.0000 2 i (-8 + 6i)3. 
4 v  4.0000  
5 ¤  2.2361  
2 “ * -4.4721 . 
´ V  4.0000 . 
÷ -295.4551 . 
2 v 5 ¤  2.2361  
4 “ * -8.9443  
´ V  2.0000 . 
÷  9.3982 Real part of result. 
´ % -35.1344 Answer: 9.3982 -35.1344i.   9.3982  
2. Write  a  program  to  evaluate  the  function  for  different 
values of z. (represents a linear fractional transformation, a class of 
conformal mappings.) Evaluate for z = l+2i. 
 (Answer: 0.3902 + 0.0122i. One possible keystroke  sequence is: ´ 
b A v v 2 * 1 + ® 5 * 3 + ÷ 
¦ ´ } | n.) 
3. Try  your  hand  at  a  complex  polynomial  and  rework  the  example  on 
page  80. You can use  the  same  program to evaluate  P(z) = 5z4 +  2z3, 
where z is some complex number. 
 Load  the  stack  with z =  7  +  0i and  see  if  you  get  the  same  answer  as 
before. (Answer: 12,691.0000 + 0.0000i.) 
  
 Now run the program for z = 1 + i. (Answer -24.0000 + 4.0000i.) 52 i524 i
ii
5 2 - 4
3)6  (-82 i542 35
12

z
zω ω ω  
						

							 Section 11: Calculating With Complex Numbers 137 
 
For Further Information 
The HP-15C  Advanced  Functions  Handbook presents  more  detailed  and 
technical  aspects  of  using  complex  numbers  in  various  functions  with  the 
HP-15C. Applications are included. The topics include: 
 Accuracy considerations. 
 Principal branches of multi-valued functions. 
 Complex contour integrals. 
 Complex potentials. 
 Storing and recalling complex numbers using a matrix. 
 Calculating the nth roots of a complex number. 
 Solving an equation for its complex roots. 
 Using _ and f in Complex mode.  
						

							 
138 
Section 12 
Calculating With Matrices 
The  HP-15C  enables  you  to  perform  matrix  calculations,  giving  you  the 
capability to  handle  advanced problems  with ease. The  calculator can  work 
with  up  to  five  matrices,  which  are  named A through E since  they  are 
accessed  using the  corresponding A through E keys. The  HP-15C lets 
you  specify  the  size  of  each  matrix,  store  and  recall  the  values  of  matrix 
elements,  and  perform  matrix  operations – for  matrices  with  real  or 
complex  elements.  (A  summary  of  matrix  functions  is  listed  at  the  end  of 
this section.) 
A  common  application  of  matrix  calculations  is  solving  a  system  of  linear 
equations. For example, consider the equations 
3.8x1 + 7.2x2 =  16.5 
1.3x1  - 0.9x2 = -22.1 
for which you must determine the values of x1 and x2. 
These equations can be expressed in matrix form as AX = B, where 
, and   . 
The  following  keystrokes  show  how  easily  you  can  solve  this  matrix 
problem  using  your  HP-15C.  (The  matrix  operations  used  in  this  example 
are explained in detail later in this section.) 
First, dimension  the  two  known  matrices, A and B,  and  enter  the  values  of 
their  elements,  from  left  to  right  along  each  row  from  the  first  row  to  the 
last.  Also,  designate  matrix C as  the  matrix  that  you  will  use  to  store  the 
result of your matrix calculation (C = X). 







2
1     , 0.91.3
7.2   3.8
x
xXA 



22.1
16.5   B  
						

							 Section 12: Calculating with Matrices 139 
 
 
Keystrokes Display  
|  8  Deactivates Complex 
mode. 
2 v ´ m A  2.0000 Dimensions matrix A 
to be 2×2. 
´ > 1  2.0000 Prepares for automatic 
entry of matrix 
elements in User mode. 
´ U  2.0000 (Turns on the USER 
annunciator.) 
3.8 O A  A     1,1 Denotes matrix A, row 
1, column 1. (A display 
like this appears 
momentarily as you 
enter each element and 
remains as long as you 
hold the letter key.) 
  3.8000 Stores a11. 
7.2 O A  7.2000 Stores a12. 
1.3 O A  1.3000 Stores a21. 
.9 “ O A -0.9000 Stores a22. 
2 v 1 ´ m 
B 
 1.0000 Dimensions matrix B to 
be 2×l. 
16.5 O B  16.5000 Stores b11. 
22.1 “ O B -22.1000 Stores b21. 
´ < C -22.1000 Designates matrix C 
for storing the result. 
Using matrix notation, the solution of the matrix equation AX = B is 
X = A-1B 
where A–1 is  the  inverse  of  matrix A.  You  can  perform  this  operation  by 
entering  the  ―descriptors‖  for  matrices B and A into  the  Y- and  X-registers 
and  then  pressing ÷.  (A  descriptor  shows  the  name  and  dimensions  of  a 
matrix.) Note that if A and B were numbers, you could calculate the answer 
in a similar manner.   
						

							140 Section 12: Calculating with Matrices 
 
Keystrokes Display  
l > B b 2 1 Enters descriptor for B, the 2×1 
constant matrix. 
l > A A 2 2 Enters descriptor for A, the 2×2 
coefficient matrix, into the X-
register, moving the descriptor 
for B into the Y-register. 
÷ running Temporary display while A-1B is 
being calculated and stored in 
matrix C. 
 C 2 1 Descriptor for the result matrix, 
C, a 2×1 matrix. 
Now  recall  the  elements  of  matrix C – the  solution  to  the  matrix  equation. 
(Also remove the calculator from User mode and clear all matrices.) 
Keystrokes Display  
l C C     1,1 Denotes matrix C, row 1, column 
1. 
 -11.2887 Value of c11 (x1). 
l C 8.2496 Value of c21 (x2). 
´ U 8.2496 Deactivates User mode. 
´>0 8.2496 Clears all matrices. 
The solution to the system of equations is x1 = -11.2887 and x2 = 8.2496. 
Note: The  description  of  matrix  calculations  in  this  section 
presumes  that  you  are  already  familiar  with  matrix  theory  and 
matrix algebra. 
Matrix Dimensions 
Up  to  64  matrix  elements can  be  stored  in  memory.  You  can  use  all 
64 elements in one matrix or distribute them among up to five matrices.