Hyperbolic Curiousity Revisited


The thread "Hyperbolic Curiousity" initiated by Bill Triplett on 14 October 2008 referred to an article on slide rules which included the problem sinh(0.243 + 53.5i) as a test case. Several correspondents reported inability to solve the problem with a TI-89. I was surprised by that. I was unable to respond at the time because I did not have my TI-89 with me. Once I had it available I experienced little difficulty if only I recognized that the TI-89 mechanization requires that the machine be in Radian mode for hyperbolic calculations. If the machine is in Degree or Gradian mode the machine returns the message "Domain Error". Thus, with my TI-89 in Float 12 mode, and in the Real or Rectangular options for Complex calculations

In Degree mode I get a "Domain error" message.

In Radian mode: sinh(.243 + 53.5 i) = -.244339804809 - .095544333408 i

In Radian mode: sinh(.243 + 53.5*pi/180 i) = .145968652453 + .827707349094 i

What's so hard about that? Actually, what's so hard is that the printed manual for the TI-89 is of no real help. But that's a problem with many machines. On my TI-85

In Degree or Radian mode: sinh(.243,53.5) = (-.244339804809,-0.095544333408)

In Degree or Radian mode: sinh(.243,53.5*pi/180) (.145968652453,.827707349094)

so I conclude that the TI-85 calculates complex numbers as if the second element is in radians. But, I find no mention of that in the manual.

Karl Schneider's message #41 in the thread stated in part

I find it curious to enter a complex-valued argument in rectangular coordinates, but with the imaginary part as an angle. However, that's the input form the author clearly indicates. Maybe that's how these slide rules were designed.

Karl was correct. Consider the Pickett slide rule manual how to use... dual base log log SLIDE RULES written by Maurice L. Hartung. For the calculation of sinh(x + jy) page 71 of the manual states
... First, it should be noted that the real number y may be expressed in radians or angular degrees. Since the graduations on the S and T scales are in terms of degrees, this measure is more convenient. A value of y given in radian measure should therefore first be converted to degree measure. ...

The 93 page manual contains three pages on "Hyperbolic Functions of Real Values", thirteen pages on "Hyperbolic Functions of Complex Arguments", four pages on "Inverse Hyperbolic Functions of Complex Arguments" and three pages on "Circular Functions of Complex Arguments". It includes specific examples for the Pickett Model N4 Vector type slide rule. I do not have an N4 slide rule, but there is an illustration of that device on page 3 of the manual. There are three hyperbolic scales:

An upper sinh(x) scale for values of x between approximately 0.1 and 0.885

A lower sinh(x) scale for values of x between approximately 0.885 and 3

A tanh(x) scale for values of x between 1 and 3

There is no cosh(x) scale. To obtain the cosh(x) the user must divide the sinh(x) by the cosh(x). The inverse hyperbolic functions arcsinh(x) and arctanh(x) are readily obtained. I have not yet figured out how to get the arccosh(x) and I haven't found an example in the manual. Other methods must be used to obtain the hyperbolics for values of x outside the ranges described above. For example, page 69 of the manual states

For values of x greater than 3, both sinh x and cosh x are approximately equal to ex. Hence, if x is set on DF/M, then ex may be read on LL4+ and divided by 2 mentally. As noted above, tanh x in this case is approximately 1.

The bold type is mine. The manual continues on with approximations for small values of x.

I mention all of this because the thread "What is the smallest full trig calculator in production?" initiated by Bill Triplett on 28 Oct 2008 stated in part

In a few discussion threads, people have noted here that some math functions are not built into some of the high end calculators. I was just wondering which calculator would be the smallest machine in the world that would be capable of directly evaluating all of the trig functions and their inverses, both circular and hyperbolic, plus logarithms, and powers, and roots, with complex numbers?

I think it is likely that you can do all of the listed complex number functions with an antique HP-15C without needing to add any custom functionality with user programs. Of course, you can carry along a large machine like the HP-50g, and you can perform all of the listed functions, but I would prefer to see all of the functions available in a machine that I would not mind carrying along in a vest pocket.

Strangely, you can perform all of the listed math functions with a "vector" slide rule from yesteryear. Pickett even made a tiny 1/10 inch thick six inch long pocket version of their N4 that would perform all of the mentioned functions with complex numbers.

Again, the bold type is mine. The material that I extracted above from the Pickett manual suggests that the process of obtaining the hyperbolic functions with vector slide rules may not be all that easy relative to modern low cost machines. For example, on my five dollar LeWORLD 2209 to find the cosh(x) I simply enter the value x, press the hyp key and then the cos key and Voilla! the answer appears in the display. No dividing of the sinh(x) by the tanh(x). No messing around with worrying about the allowable range of input values to decide on which method to use for the calculation.

I admit that the 2209 won't return the hyperbolics for complex arguments that easily. But a quick look at the slide rule manual mentioned above suggests that it isn't all that easy with the Pickett N4 slide rule. Since I don't have an N4 or another "standard" (whatever that means) vector slide rule with me it is difficult for me to follow the instructions in the manual.

I do have a Pickett N16 with me but I don't have the manual and I can't directly use the manual mentioned above because it doesn't seem to be a "standard" vector slide rule. For example, the Sh and Th scales are on the slide of the N4 but are on the frame of the N16. The N16 has unfamiliar renditions (at least to me) for other scales as well. That may be because it is a "Chan Street" mechanization. Trying to use the N16 without a manual it is akin to trying to use an RPL calculator without a manual when one is only familiar with RPN. I tried to do that several years ago when I adquired an HP-28S with only the reference manual. I couldn't have gone very far with it without the help of Gene Wright and Viktor Toth.



ha! a test for the uWatch:

53.5 +/- . . 243 menu menu sin .. +/- *

gives -0.244339804809 – i0.09554433341

no problem! smaller than the Pickett, the Ti-89 and the 15c !



This is a thoughtful and complete post. I admire the spirit of adventure that can prompt a person to actually take out a slide rule, and figure out how to do the operations. I keep tinkering with a few of them as time permits.

Compared to a good modern calculator, it certainly is not quick to perform the many necessary steps when using a slide rule to do hyperbolic trig funtions. The big thing about the advent of the vector slide rules was that they made it possible for people to get the work done much more quickly than by using tables and interpolations. William Robinson was kind enough to clock himself both ways, and report the results in his article at this link:


He made the speed comparisons using several models of vector slide rules. He noticed that some slide rules were configured better for preventing the user from needing to write down intermediate steps. Attention to this required a fair bit of brilliance on the part of the slide rule designers.

My favorite is the Pickett N4. If you don't have one to play with, check out this simulation:


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