Complex number capability in non-HP machines


In a May 17 thread "HP 35s successor?" Rodger Rosenbaum asked:

Has there ever been a non-HP calculator with excellent complex number capability?

The ones I've seen have been abysmal.

Since no one else has responded to Roger's question with respect to non-HP calculators I decided to try. My answer is limited to machines to which I have access. I wrote separately to Rodger noting that several machines in the TI product line had what I thought were substantial complex number capabilities. He responded
I have a TI86, and as you say, it has comprehensive complex variable
capability. But, my experience with calculators other than HP or TI
has been that if they have any complex capability at all, it is rudimentary.

My Casio fx-115D offers the four arithmetic functions and 1/x.

My Casio fx-115ES (currently available) offers the four arithmetic functions, 1/x, x^2, x^3, argument and conjugate.

My Sharp EL-515S has a complex mode which offers only the four arithmetic functions.

My Sharp PC-1261 Pocket Computer does not offer a built-in complex capability. The PC-1260/1261 Instruction Manual includes fifteen sample programs none of which provide complex number capabilities. Sharp's Application Software for PC-1250/CE-125 includes twenty BASIC programs. Again, none of them provide complex number capabilities. The book Mathematics Library - Application Software for the Sharp EL-5500 and PC-1403 Scientific Computers by Maurice E. T. Swinnen and David Thomas (Systems Publications, 1987) offers five BASIC programs for complex number calculations:

Program 1. Sine, cosine and tangent and the inverses.

Program 2. The four arithmetic functions plus polar to rectangular and rectangular to polar conversions.

Program 3. Square, square root, reciprocal, natural log, e^x, plus polar to rectangular and rectangular conversions.

Program 4. y^x, y^(1/x), log to the base y of x, plus polar to rectangular and rectangular to polar conversions.

Program 5. Complex Roots

These programs should be compatible with the PC-1261. I have successfully run parts of the first three on my PC-1261. I have successfully run the Simultaneous Equation program from the book on my PC-1261 and converted it for use with my Radio Shack Model 100 and with my Texas Instruments CC-40.

The book provides example calculations which can be used to verify that the programs have been entered correctly. For the first program the sine of 3 - 5i is calculated as 3.883848618 -74.10151768i which does NOT agree with the result obtained with my TI-85 and HP-28S. The first line of the program sets degree mode. If I set radian mode instead I get results which agree with my TI-85 and HP-28S. Degree mode is also set at the beginning of programs 3 and 4. As a result the example calculation for any function which uses real trigonometric functions as a part of its solution will not agree with results from machines which use radian mode. Setting radian mode will result in agreement. I leave any discussion over the merits of the use of degree or radian mode to others more skilled in the use of complex functions than I am.

To demonstrate the arcsine fnction the book asks the user to enter 3.8838 - 74.1i (a truncated version of the sine of 3 - 5i when using degree mode) and see 3.000023834 - 4.99997943i as the answer. There is a better way to demonstrate the arcsine after having calculated the sine. The input routine stores the real part in R and the imaginary part in M. Each complex trigonometric function stops with the real part in O and the imaginary part in P. Since the PC-1261 accepts variables as the response to INPUT statements the user can enter the complete answer from a previous trigonometric function by responding with O and P for the real and imaginary inputs. A user who does that to calculate the arcsine after having calculated the sine of 3 - 5i in degree mode will see the result 3.000000101 - 4.999999881 .

The arithmetic functions in the second program stop with the result in the locations where the first complex number is entered for an arithmetic function. The INPUT statement of the PC-1261 does not display those values; however, if ENTER is pressed the INPUT statement essentailly acts as a non-operation and the existing values in the input locations are preserved. The INPUT statement on my Radio Shack Model 100 reacts in the same way. To preserve the previously stored values on the CC-40 the user must use the response Shift ENTER.

I have not worked with the fifth program.

The TI Machines:

Rodger noted that the TI-86 has a comprehensive capability. Several other TI machines also offer comprehensive capability.
The TI-58/58C/59 offer complex calculations if either the Master Library module or the Electrical Engineering module is in place. Three sub-programs offer the four arithmetic functions, x^2, square root, 1/x, e^x, ln x and sin, cos, tan and the
inverses. The functions are selected from sub-programs with the user defined keys (A through E and A' through E'). Prompting to the display is not available since the TI-59 does not offer alphanumerics in the display. The user can use the label card in the slot above the top row of keys. That requires shuffling the appropriate cards in and out of the slot. Frank Fujimoto recognized the various limitations of that method and wrote a Complex Keyboard program which was published in the Volume 4 Number 4 (July/August 1980) issue of PPX Exchange. His program offers access to all of the complex functions in the memory module. Seven complex memory registers are provided which can be accessed with Store, Recall, Exchange, Sum, Inverse Sum, Product and Inverse Product functions. I recommend that the user change step 026 of the program from B to INV and step 117 of the program from INV to B. Then the user can perform complex mathematics with the same keyboard sequences that would be used for real functions except that SBR must be pressed before each complex function.

The CC-40, the TI-74 and the TI-95 all offer an extended capability as part of their Mathematics Library modules including x^2, SQRT, 1/x, LN, EXP, SIN, COS, TAN, ASIN, ACOS, ATAN, y^x and x root of y. The CC-40 and the TI-95 also offer LOGyX. The TI-74 does not. A curious addendum to the TI-74 manual states:

The Complex Functions Program (COMPF) contains two complex logarithm options, LN(X) and LOGyX . LOGyX calculations are needed less often than LN(X) calculations. To prevent you from accidentally selecting LOGyX when you intend to calculate LN(X), th LOGyX option (Option 20) has been disabled. Selecting this option results in an 'E1 Syntax' error message.

The addendum then goes on to offer a program to calculate LOGyX. Somehow, I believe that a user of the complex capability will be able to keep track of which function to use.

The TI-95 also offers the hyperbolics and their inverses. The CC-40 and the TI-74 offer a Complex Systems program (COMPS) which solve a system of nxn simultaneous equations with complex coefficients. The baseline 6K versions of the machines will handle up to 12x12 complex systems. The program is easy to use with excellent prompting.

The first page of the section on complex numbers in the TI-68 Guidebook states: "A major innovation of the TI-68 is its extensive complex number capability. You have the freedom to enter complex numbers wherever they are applicable without 'mode' limitations. Few other calculators evn approach the versatility of complex numbers provided by the TI-68." How much salesmanship and how much actual capability? I haven't done any substantial amount of work with the TI-68. I do know that the method of entering and displaying is similar to the HP-48. I can't compare with the HP-15 or the HP-42 since I don't have either of those.

Many of the TI graphic machines offer complex number capabilities. The TI-83 offers a relatively wide range of functions but it does not offer the trigonometric functions. I have not tested my TI-84. The TI-85 offers all the capabilites of the TI-83 and adds the trigonometric and hyperbolic functions and their inverses, polynomial solutions, simultaneous equation solutions, sorting based on modulus, and the sum or product of the elements of a list.

A Question on HP Capabilities

I haven't been able to use memory register arithmetic with complex functions in Algebraic mode on the HP-35S. I haven't found a reference in the manual which indicates that should be so. I may be doing something incorrectly. If anyone knows how to do that please let me know.


Hi, Palmer --

You've done a commenable amount of investigation, and confirmed Rodger's suspicions: Complex-number capability in the Sharp and Casio models you described is either rudimentary or not built-in.

In this post of mine from 2004, I questioned whether any TI model had a definitive complex-number capability, because none of the HP models had all of what I believed to be useful. I did subsequently experiment with a TI-89, but was not fully pleased with the way the complex-number functionality worked.

-- KS

Edited: 6 July 2008, 10:53 p.m. after one or more responses were posted


Hi everyone,

This is my first post on the forum, although I'd been keeping an eye on it for quite some time.

I've been a Sharp user from the early 80s. Sharp has been "poor man's HP" in those days in Europe, much more popular than in the US, it seems. As a student of electrical engineering in late 80s, I was quite happy to find that the mighty Sharp PC-E500 had some built-in support for complex numbers. True, it didn't do transcendental functions on complex numbers, and the complex number support was a separate application, isolated from the calculator mode just like the matrix mode. But COMPLX application was at least available out of the box.

Somehow, Sharp pocket computers consistently failed to deliver in calculator department, where HP 42s from the same period was awesome. On the other hand, it often took just a few lines of Sharp Basic code to do pretty much the same as what hundreds of keystrokes of 42s RPN would do (or at least it looked like that to me at the time), and E500 had peripherals, connected to the PC, had a really big screen... (Now I understand I've been comparing a proper calculator to a proper pocket computer and the two concepts were not really directly comparable, although I've seen comparison debates on this forum.)

Back to HP, I was really pleased to see the "i" button on HP 35s last year, only to find out that 35s didn't do exactly what I was hoping for when I typed in 1i2 and tried to calculate a square root of it. Why o why did they have to do the complex support on 35s so wrong?

Will there ever be a simple calculator that does complex numbers really "transparently"?


Will there ever be a simple calculator that does complex numbers really "transparently"?

Isn't this a contradiction in itself? Well, I guess what you want is a calc where the input of

1 i 2 SQRT COS results in 0.39 - 0.83i

So far, I don't know any HP calc doing exactly this - only virtual ones like this or this ;) Won't help in the short run, though.


That'd definitely be both simple and transparent. No contradiction then :)


On HP-42S you would type

1 ENTER 2 COMPLEX (to enter 1 i 2)

and it gives 0.3901 -i0.8311. To me it looks exactly what was asked for (transparent).

48/49/50 series and 71b with Math rom do the same - they all support complex data type seamlessly integrated into the basic operating system, so all operations normally work with complex arguments (or return complex results when required, for example 2 ACOS on 42s gives 0.0000 i1.3170). This is unlike 35s which unfortunately only returns complex results for some functions (such as -2 ENTER 0.5 y^x) while others return an error (such as -2 SQRT).


On the hp49/50, it would be


which is about as simple as you can get.



... and so we have

- on 48/49/50:       L_shift (   1  SPC  2  ENTER  SQRT  COS

- on 42s: 1 ENTER 2 Shift COMPLEX SQRT COS

- while we wanted: 1 i 2 SQRT COS

I admit the first 2 lines are

about as simple as you can get.

(emphasis added) d;-)


- on 48/49/50:       L_shift (   1  SPC  2  ENTER  SQRT  COS

Can be:
- on 48/49/50:       L_shift (   1  SPC  2  SQRT  COS


Just to make one thing clear here, about this functionality on RPN (not RPL) machines.
Unlike the 35s, the 42s initially uses two stack registers to construct a complex number. So yes, in terms of number of keystrokes it is about as simple as it gets, but comes at a higher price.

(I am aware that this has been a subject of discussions on this forum a few times...)


You've done a commenable amount of investigation, and confirmed Rodger's suspicions: Complex-number capability in the Sharp and Casio models you described is either rudimentary or not built-in.

I did all of that because Rodger's original statement which appeared in the Forum had not included TI machines with HP machines as having substantial complex number capability. And, of course, eveyone knows that deep, down inside I am a TI guy.

In introspective communities such as this forum such statements as Rodger's original one tend to generate a life of their own. I will offer a non complex number example. If you go to the beginning of the section on Casio graphing calculators at Viktor Toth's site you will find the following statement:

Casio invented the graphing calculator. No, it wasn't Hewlett-Packard; much to my surprise, I found out that Casio's first graphing model, the fx-7000G, preceded HP's first graphing machines by at least a year or more.

I have surprised a number HP aficianodos with that information, and surprised them even more when I told them that Casio had offered several graphing calculator models before HP released the 28C.

You also wrote:

I did subsequently experiment with a TI-89, but was not fully pleased with the way the complex-number functionality worked.

I have a TI-89 but hardly ever use it. The menus seem to go on forever and ever, and the reduced size letters are really difficult for these old eyes to read. I much prefer to use the TI-85 or TI-86 if those machines can do what I want to do.

The thing I really like about the TI-85 and TI-86, or the HP-28S for that matter, is that, for example, a user calculates the function of a quantity in the same way whether the quantity is real or complex. I don't know when that started. Was it the HP-15? It didn't appear in the Math Pac for the HP-41 or in the Master Library for the TI-59, although Fujimoto's progam was a big step in the right direction for the TI community.


The thing I really like about the TI-85 and TI-86, or the HP-28S for that matter, is that, for example, a user calculates the function of a quantity in the same way whether the quantity is real or complex. I don't know when that started. Was it the HP-15?

I think you're right: the 15c was the first HP with this feature.

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