HP 15c Manual

							 Section 11: Calculating With Complex Numbers 121 
 
Complex mode is activated 
1) automatically, when executing ´ V or ´ }; or 
2) by setting flag 8, the Complex mode flag (|F 8). 
When  the  calculator  is  in  Complex  mode,  the C annunciator  in  the  display 
is lit. This tells you that  flag 8 is set and the complex stack exists. In or out 
of Complex mode, the number appearing in the display is the number in the 
real X-register. 
Note: In Complex mode (signified by the C annunciator), the HP-
15C  performs all trigonometric  functions  using radians. The 
trigonometric  mode  annunciator  in  the  display (RAD, GRAD, or 
blank  for  Degrees) applies  to  two  functions  only: ; and : 
(as explained later in this section). 
Deactivating Complex Mode 
Since Complex mode requires the allocation of five registers from memory, 
you will have more memory available for programming and other advanced 
functions  if  you  deactivate  Complex  mode  when  you  are  working  solely 
with real numbers. 
To  deactivate  Complex  mode,  clear  flag  8  (keystroke  sequence: |  
8). The C annunciator will disappear. 
Complex  mode  is  also deactivated  when  Continuous  Memory  is  reset  (as 
described  on  page  63).  In  any  case,  deactivating  Complex  mode  dissolves 
the imaginary stack, and all imaginary numbers there are lost. 
Complex Numbers and the Stack 
Entering Complex Numbers 
To enter a number with real and imaginary parts; 
1. Key the real part of the number into the display. 
2. Press v 
3. Key the imaginary part of the number into the display. 
4. Press ´ V.  (If  not  already  in  Complex  mode,  this  creates  the 
imaginary stack and displays the C annunciator.)  
						

							122 Section 11: Calculating With Complex Numbers 
 
Example: Add 2 + 3i and 4 + 5i. (The operations are illustrated in the stack 
diagrams following the keystroke listing.) 
Keystrokes Display  
´ • 4   
2 v 2.0000 Keys real part of first number 
into (real) Y-register. 
3 3 Keys imaginary part of first 
number into (real)  
X-register. 
´ V 2.0000 Creates imaginary stack; moves 
the 3 into the imaginary X-
register, and drops the 2 into the 
real X-register. 
4 v 4.0000 Keys real part of second number 
into (real) Y-register. 
5 5 Keys imaginary part of second 
number into (real) X-register. 
´ V 4.0000 Copies 5 from real  
X-register into imaginary  
X-register, copies 4 from real Y-
register into real X-register, and 
drops stack. 
+ 6.0000 Real part of sum. 
´ % (hold) 8.0000 Displays imaginary part 
            (release) 6.0000 of sum while the % key is held. 
(This also terminates digit entry.) 
The  operation  of  the  real  and  imaginary  stacks  during  this  process  is 
illustrated below. (Assume that the stack registers have been loaded already 
with  the  numbers  shown  as  the  result  of  previous  calculations).  Note  that 
the  imaginary  stack,  which  is  shown  below  at  the  right  of  the  real  stack,  is 
not  created  until ´ V is  pressed.  (Recall  also  that  the  shading  of  the 
stack  indicates  that  those  contents  will  be  written  over  when  the  next 
number is keyed in or recalled.)  
						

							 Section 11: Calculating With Complex Numbers 123 
 
 
 Re Im  Re Im  Re Im  Re Im  Re Im 
T 9   8   7   7   7 0 
Z 8   7   6   6   7 0 
Y 7   6   2   2   6 0 
X 6   2   2   3   2 3 
Keys: 2 v 3 ´ V 
The execution of ´ V causes the entire stack to drop, the T contents to 
duplicate, and the real X contents to move to the imaginary X-register. 
When  the  second  complex  number  is  entered,  the  stacks  operate  as  shown 
below. Note that v lifts both stacks. 
 
 Re Im  Re Im  Re Im  Re Im 
T 7 0  7 0  6 0  6 0 
Z 7 0  6 0  2 3  2 3 
Y 6 0  2 3  4 0  4 0 
X 2 3  4 0  4 0  5 0 
Keys: 4 v 5 
 
 Re Im  Re Im  Re Im 
T 6 0  6 0  6 0 
Z 2 3  6 0  6 0 
Y 4 0  2 3  6 0 
X 5 0  4 5  6 8 
 Keys: ´ V +  
A  second  method  of  entering  complex  numbers  is  to  enter  the  imaginary 
part first,  then  use } and −.  This  method  is  illustrated  under 
Entering Complex Numbers With −, page 127.  
						

							124 Section 11: Calculating With Complex Numbers 
 
Stack Lift in Complex Mode 
Stack  lift  operates  on  the  imaginary  stack  as  it  does  on  the  real  stack  (the 
real  stack  behaves  identically  in  and  out  of  Complex  mode). The  same 
functions that enable, disable, or are  neutral to lifting of the  real stack will 
enable,  disable,  or  be  neutral  to  lifting  of  the  imaginary  stack. (These 
processes are explained in detail in section 3 and appendix B.) 
In  addition, every nonneutral  function,  except − and ` causes  the 
clearing of the  imaginary  X-register when the  next number is entered. That 
is,  these  functions  cause  a  zero  to  be  placed  in  the  imaginary  X-register 
when  the  next  number  is  keyed  in  or  recalled. Refer  to  the  stack  diagrams 
above  for  illustrations.  This  feature  allows  you  to  execute  calculator 
operations  using  the  same  key  sequences  you  use  outside  of  Complex 
mode.* 
Manipulating the Real and Imaginary Stacks 
} (real  exchange  imaginary). Pressing ´ } will  exchange 
the  contents  of  the  real  and  imaginary  X-registers,  thereby  converting  the 
imaginary  part  of  the  number  into  the  real  part  and  vice-versa.  The  Y-,  Z-, 
and T-registers are not affected. Press ´ } twice restore  a  number 
to its original form. 
} also activates Complex mode if it is not already activated. 
Temporary  Display  of  the  Imaginary  X-Register. Press ´ % to 
momentarily display  the  imaginary  part  of  the  number  in  the  X-register 
without actually switching the real and imaginary parts. Hold the key down 
to maintain the display. 
Changing Signs 
In Complex mode, the “ function affects only the number in the real X-
register – the  imaginary  X-register  does  not  change.  This  enables  you  to 
change the sign of the real or imaginary part without affecting the other. To 
key in a  negative  real or imaginary part,  change  the  sign of that  part as you 
enter it. 
If you want to find the additive inverse of a complex number already in the 
X-register, however,  you  cannot  simply  press “ as  you  would  outside 
                                                           * Except for the : and ; functions, as explained in this section (page 133).  
						

							 Section 11: Calculating With Complex Numbers 125 
 
of Complex mode. Instead, you can do either of the following: 
 Multiply by -1. 
 If  you  dont  want  to  disturb  the  rest  of  the  stack,  press “ ´ 
} “ ´ }. 
To find the negative of only one part of a complex number in the X-register: 
 Press “ to negate the real part only. 
 Press ´ } “ ´ } to  negate  the imaginary 
part only, forming the complex conjugate. 
Clearing a Complex Number 
Inevitably you will need to clear a complex number. You can clear only one 
part at a time, but you can then write over both parts (since − and ` 
disable the stack). 
Clearing  the  Real  X-Register. Pressing − (or | `)  with  the 
calculator in Complex mode clears only the number in the real X-register; it 
does not clear the number in the imaginary X-register. 
Example: Change  6  +  8i to  7  +  8i and  subtract  it  from  the  previous  entry. 
(Use ´ } or ´ % to view the imaginary part in X.) Assume a, 
b, c and d represent parts of complex numbers. 
 Re Im  Re Im  Re Im  Re Im 
T a b  a b  a b  a b 
Z c d  c d  c d  a b 
Y 6 0  6 0  6 0  c d 
X 6 8  0 8  7 8  -1 -8 
Keys: −  7 -  (or other 
operation) 
Since clearing disables the stack (as explained above), the next number you 
enter will replace the cleared value. If you want to replace the real part with 
zero,  after  clearing  use v or  any  other  function  to  terminate  digit 
entry  (otherwise  the  next  number  you  enter  will  write  over  the  zero);  the 
imaginary  part  will  remain  unchanged.  You  can  then  continue  with  any 
calculator function.  
						

							126 Section 11: Calculating With Complex Numbers 
 
Clearing the Imaginary X-Register. To clear the number in the imaginary 
X-register, press ´ }, then press −. Press ´ } again to 
return the zero, or any new number keyed in, to the imaginary X-register. 
Example: Replace -1 -8i by -1 + 5i. 
 Re Im  Re Im  Re Im  Re Im  Re Im 
T a b  a b  a b  a b  a b 
Z c d  c d  c d  c d  c d 
Y e f  e f  e f  e f  e f 
X -1 -8  -8 -1  0 -1  5 -1  -1 5 
Keys: } − 5 } 
(continue with 
any operation) 
Clearing  the  Real  and  Imaginary  X-Registers. If  you  want  to  clear  or 
replace both the  real  and  imaginary  parts  of  the  number  in  the  X-register, 
simply press −, which will disable the stack, and enter your new number. 
(Enter  zeros  if  you  want  the  X-register  to  contain  zeros.)  Alternatively,  if 
the new number will be purely real (including 0 + 0i), you can quickly clear 
or  replace  the  old,  complex  number  by  pressing ) followed  by  zero  or 
the new, real number. 
Example: Replace -1 + 5i with 4 + 7i. 
 Re Im  Re Im  Re Im  Re Im  Re Im 
T a b  a b  c d  c d  c d 
Z c d  c d  e f  e f  c d 
Y e f  e f  4 5  4 5  e f 
X -1 5  0 5  4 5  7 0  4 7 
Keys: − 4 v 7 ´ V 
(continue with 
any operation)   
						

							 Section 11: Calculating With Complex Numbers 127 
 
Entering  Complex  Numbers  with −.  The  clearing  functions − and 
` can  also  be  used  with } as  an  alternative  method  of  entering 
(and  clearing)  complex  numbers.  Using  this  method,  you  can  enter  a 
complex  number using only  the  X-register, without affecting the  rest of the 
stack.  (This  is  possible  because − and ` disable  stack  lift.) 
Executing } will also create  an imaginary stack if one  is not already 
present. 
Example: Enter 9 + 8i without moving the stack and then find its square. 
Keystrokes Display  
(−) (0.0000) Prevents stack lift when the 
next digit (8) is keyed in. 
Omit this step if youd rather 
save whats in X and lose 
whats in T. 
  
8 8 Enter imaginary part first. 
´ }  7.0000 Displays real part; Complex 
mode activated. 
− 0.0000 Disables stack. (Otherwise, it 
would lift following }.) 
9 9 Enters real part (digit entry not 
terminated). 
| x 17.0000 Real part. 
´ % (hold) 
(release) 
144.0000 Imaginary part. 
17.0000  
 Re Im  Re Im  Re Im  Re Im 
T a b  a b  a b  a b 
Z c d  c d  c d  c d 
Y e f  e f  e f  e f 
X 4 7  0 7  8 7  7 8 
Keys: − 8 ´ }  
						

							128 Section 11: Calculating With Complex Numbers 
 
 
 Re Im  Re Im  Re Im  Re Im 
T a b  a b  a b  a b 
Z c d  c d  c d  c d 
Y e f  e f  e f  e f 
X 7 8  0 8  9 8  17 144 
Keys: − 9 | x 
Entering a Real Number 
You have already  seen two  ways of entering a  complex  number. There  is a 
shorter  way  to  enter  a  real  number:  simply  key  it  (or  recall  it)  into  the 
display  just  as  you  would  if  the  calculator  were  not  in  Complex  mode.  As 
you do  so, a  zero  will be  placed in  the imaginary X-register (as long as the 
previous operation was not − or `, as explained on page 124). 
The  operation  of  the  real  and  imaginary  stacks  during  this  process  is 
illustrated  below.  (Assume  the  last  key  pressed  was  not − or ` and 
the contents remain from the previous example.) 
 Re Im  Re Im  Re Im 
T a b  c d  e f 
Z c d  e f  17 144 
Y e f  17 144  4 0 
X 17 144  4 0  4 0 
Keys: 4 v (Followed by 
another number.)  
						

							 Section 11: Calculating With Complex Numbers 129 
 
Entering a Pure Imaginary Number 
There is a shortcut for entering a pure imaginary number into the X-register 
when you are already in Complex mode: key in the (imaginary) number and 
press ´ } 
Example: Enter  0  +  10i (assuming  the  last  function  executed  was  not − 
or `. 
Keystrokes Display  
10 10 Keys 10 into the displayed 
real X-register and zero into 
the imaginary X-register. 
´ } 0.0000 Exchanges numbers in real 
and imaginary X-registers. 
Display again shows that the 
number in the real X-
register is zero —=as it 
should be for a pure=
imaginary numberK=
The  operation  of= the  real= and  imaginary  stacks= during= this= process  is=
illustrated below. (Assume the stack registers contain the= numbers resulting 
from the preceding examples.F=
 
Re Im  Re Im  Re Im 
T e f  e f  e f 
Z 17 144  17 144  17 144 
Y 4 0  4 0  4 0 
X 4 0  10 0  0 10 
Keys: 10 ´} (Continue with 
any operation.) 
Note  that  pressing ´ } simply  exchanges  the  numbers  in  the  real 
and imaginary X-registers and not those in the remaining stack registers.  
						

							130 Section 11: Calculating With Complex Numbers 
 
Storing and Recalling Complex Numbers 
The O and l functions  act  on  the real  X-register  only; therefore, 
the  imaginary  part  of  a  complex  number  must  be  stored  or  recalled 
separately.  The  keystrokes  to  do  this  can  be  entered  as  part  of  a  program 
and executed automatically.* 
To  store a  +  ib from  the  complex  X-register  to  R1 and  R2,  you  can  use  the 
sequence 
O 1  ´}  O 2 
You  can  follow  this  by ´ } to  return  the  stack  to  its  original 
condition  if  desired.  To  recall a  +  ib from  R1 and  R2 you  can  use  the 
sequence 
l 1    l 2    ´ V 
If  you  wish  to  avoid  disturbing  the  rest  of  the  stack,  you  can  recall  the 
number using the sequence 
l 2    ´ }   −   l 1 
(In Program mode, use | ` instead of −.) 
Operations With Complex Numbers 
Almost all functions performed on real numbers will yield the same answer 
whether  executed  in  or  out  of  Complex  mode,† assuming  the  result  is  also 
real. In  other  words,  Complex  mode  does  not  restrict  your  ability  to 
calculate with real numbers. 
Any  functions  not  mentioned  below  or  in  the  rest  of  this  section 
(Calculating With Complex Numbers) ignore the imaginary stack. 
 
* You  can  use  the  HP-15C  matrix  function,  described  in  section  12,  to  make  storing  and recalling  complex  numbers  more  convenient.  By  dimensioning  a  matrix  to  be n×2, n complex  numbers  can  be stored  as  rows  of  the  matrix.  (This  technique  is  demonstrated  in the HP-15C Advanced Functions Handbook, section 3, under Applications.) 
† The  exceptions  are : and ;,  which  operate  differently  in  Complex  mode  in  order  to facilitate converting complex numbers to polar form (page 133).