Calculator test

[Commie]

(12-08-2024 06:31 PM)Thomas Klemm Wrote:  From the HP 35s User's Guide:

Quote:Operations with Complex Numbers

It has operations for complex arithmetic (+, -, ×, ÷), complex trigonometry (sin, cos, tan), and the mathematics functions \(-z\), \(1/z\), \(z_1^{z_2}\), \(\ln z\), and \(e^z\). (where \(z_1\) and \(z_2\) are complex numbers).

In addition to that \(\text{ABS}(z)\) and \(\text{ARG}(z)\) are supported.

No luck with \(\sqrt{z}\), \(10^z\), \(\log(z)\), \(\sin^{-1}(z)\), \(\cos^{-1}(z)\) or \(\tan^{-1}(z)\).

Thanks for your post Thomas, at least it's better than the ti84+, I originally payed about £45 for my hp35s, 15 years ago, it's not up there with the 15c but at the same time not too bad.
I am deeply dissapointed with my ti84 collection though, oh well.

This thread, with the help of various members, has opened up a whole new branch of mathematics for me to explore by which I am very grateful.

Cheers
Darren

[Thomas Klemm]

(12-08-2024 07:35 PM)Commie Wrote:  I am deeply disappointed with my ti84 collection though, oh well.

First I thought that this could be something for you:
RPN83P: RPN calculator for TI-83+ TI-84+ inspired by HP-42S

But then I noticed:

• extended regular functions: +, -, *, /, 1/X, X^2, SQRT, Y^X, X^3, 3ROOTX, XROOTY, LOG, LN, 10^X, E^X, 2^X, LOG2, LOGB
• complex specific functions: REAL, IMAG, CONJ, CABS, CANG
• unsupported: trigonometric and hyperbolic functions (not supported by TI-OS)
  

Bummer. It turns out that they are not that easy to calculate:
[HP-48] calculating complex arccos and arcsin functions

[Johnh]

While the achievements of HP, and the way these advanced functions were included more than 40 years ago must be celebrated, let's also acknowledge TI.

By May 1977, five years before the Hp15c, you could buy a TI-58 or TI59, with its Master ROM module included. TI58 had 490 steps (960 on a TI59) and 100 registers, and with the Module, it could deal with complex arithmetic, and complex maths functions including trig and natural log. The Master library also included solve, integrate and matrix operations.

I've long-ago parted with my old TI58, but I have the RCL-58 emulator on my phone, which also emulates the ROM module. Using module program ML-05 , I ran ln(-1), and got 0 + i.Pi to 10 significant figures!

Can anyone test this on a real TI58 or 59?

(in ML-05, key in -1 + i.0 by pressing 1 +/- A 0 A and then the natural log is engaged with 'A. Press x<>t to reveal the imaginary part.)

[dm319]

(12-08-2024 06:37 PM)naddy Wrote:  The 15C is a mean machine.

It really is so impressive. Even from the perspective of 42 years in the future.

(12-08-2024 06:37 PM)naddy Wrote:  I was too young then, but I think video terminals were still scarce and if undergrad engineering students wanted/had to use a computer, that meant submitting a stack of punch cards and waiting for a printout.

Was there software around at the time to do complex calculations on a PC or Unix? What do people use these days (and please don't say wolfram alpha)?

[Thomas Klemm]

(12-08-2024 08:13 PM)dm319 Wrote:  What do people use these days (and please don't say wolfram alpha)?

Python:

Code:

from cmath import log
log(-1)

3.141592653589793j

[dm319]

(12-08-2024 08:18 PM)Thomas Klemm Wrote:  3.141592653589793j

Of course the answer is python. The answer is always python.

[carey]

(12-08-2024 05:34 PM)Commie Wrote:  

(12-08-2024 03:15 PM)carey Wrote:  Hmm…from the TI-84 Plus manual for the sine function: “Returns the sine of a real number, expression, or list.” Same description for the other trig functions and their inverses.

It seems you are correct, my ti84+ throws an error when trying to solve complex trig functions. I assumed it would work because ln(z) functions correctly.Thanks for highlighting this carey.

There is a workaround for the TI-84 Plus inability to accept complex arguments for its trig functions and their inverses. Since the TI-84 Plus exponential function (e^) does accept complex arguments, sine and cosine can be defined in terms of e^ using Euler ‘s formulas:

\[\cos(z) = \frac{e^{iz} + e^{-iz}}{2}\]
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]


In Mode, set angle to radians and select complex polar. After calculation, select complex rectangular for conversion to rectangular form. The LHS (which can’t be input directly) can be obtained by working on the RHS.
\[ \sin(1+2i) = \frac{e^{i *(1 + 2i)} - e^{-i *(1 + 2i)}}{2i}\]
\[ \sin(1+2i) = 3.165778 + 1.959601i \]
It’s cumbersome, but works!

A similar workaround applies to the Casio 9860gii and CG50 calculators as they also have complex argument limitations for trig functions but not for e^.

[Commie]

(12-08-2024 08:38 PM)carey Wrote:  There is a workaround for the TI-84 Plus inability to accept complex arguments for its trig functions and their inverses. Since the TI-84 Plus exponential function (e^) does accept complex arguments, sine and cosine can be defined in terms of e^ using Euler ‘s formulas:

\[\cos(z) = \frac{e^{iz} + e^{-iz}}{2}\]
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]


In Mode, set angle to radians and select complex polar. After calculation, select complex rectangular for conversion to rectangular form. The LHS (which can’t be input directly) can be obtained by working on the RHS.
\[ \sin(1+2i) = \frac{e^{i *(1 + 2i)} - e^{-i *(1 + 2i)}}{2i}\]
\[ \sin(1+2i) = 3.165778 + 1.959601i \]
It’s cumbersome, but works!

A similar workaround applies to the Casio 9860gii and CG50 calculators as they also have complex argument limitations for trig functions but not for e^.

Thanks Carey, I know about these equations and I know how to derive them from Eulers identity, however until today they mean a great deal more now, especially with your input to confirm.
Could I just ask, what overlay are you using, in your browser, which makes math easy to print/read?

Cheers
Darren

[Gil]

I have no TI 84, but the real part of sin (1+2*i) should be
3.165779, or 3.16577851322, and not 3.165778.

[Thomas Klemm]

(12-08-2024 08:52 PM)Commie Wrote:  Could I just ask, what overlay are you using, in your browser, which makes math easy to print/read?

Not sure if I understand the question correctly but this forum allows the use of LaTeX in our posts.

In case of Carey's formulas it looks like this:

Code:

\[\cos(z) = \frac{e^{iz} + e^{-iz}}{2}\]
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]

You could have used LaTeX in your initial post:

Darren Wrote:Euler, in the 18th century wrote down a very important equation, this being in one form can be written:

\(e^{\pi \cdot i}=\cos(\pi)+i \cdot \sin(\pi)\) which equates to \(e^{\pi \cdot i}=-1\)

Which can be written as \(e^{\pi \cdot i}+1=0\)

Now we can also deduce that \(\pi \cdot i=\ln(-1)\)

You can use the button to look at the source.

[AnnoyedOne]

(12-08-2024 08:13 PM)dm319 Wrote:  Was there software around at the time to do complex calculations on a PC or Unix?

I has access to two mainframes in 1980. A DEC-10 (punchcards) and PDP-11/70 (terminals).

We were taught FORTRAN (mostly for solving matrix operations) and BASIC. In my final year assembler on Zilog Z80 based CP/M 2.2 systems with two 8" single-sided floppy drives. I was running multitasking/multiuser OS-9 on my Motorola 6809 system with one 5.25" floppy drive at home and was already writing software in 6809 assembler. There was also the BASIC09 compiler for OS/9 but I never had that. I never used a UNIX machine but perhaps those had math software.

The home computers of the time were too slow/crude for serious work. Used mostly for games. Hence the HP-15C for math. I learned about PC/Apple computers at the time of their release but they were uncommon, had little to no software available, and were slow. The first PC I used (a PC/XT clone with a 4.77MHz 8086) was in 1986. I never used it for math (but did for PCB layout).

Later engineering software for PC's became available (a circuit simulator in 1989?).

As it turned out the HP-15C and desktop computers came out at around the same time. After a few years the latter were in common use. Calculators weren't really needed after that (except for HS level math). By the time the Intel 386-DX (on-chip FPU) came out it was "all over". HP got into the PC business in earnest around that time. Calculators (including HP ones) were marketed to HS students (think graphing and CAS).

These days serious math is done on a computer and calculators are used when that isn't possible/convenient.

A1

PS: As I recall there was a room full of "dumb" Televideo 910 terminals at the campus "computing center".

https://terminals-wiki.org/wiki/images/f...9462-1.jpg

Also large viewing windows into the (sealed) computer room. And other terminals scattered around the campus.

https://c8.alamy.com/comp/BRA4HK/1970-19...BRA4HK.jpg

[Maximilian Hohmann]

Hello,

(12-08-2024 08:13 PM)dm319 Wrote:  Was there software around at the time to do complex calculations on a PC or Unix? What do people use these days (and please don't say wolfram alpha)?

I am talking only about the early 1980ies now: I did my first programming course at university in FORTRAN (which natively features the COMPLEX data type) using punch cards. There were several mainframe computers and there were many terminals with "time share" access, but not for ordinary students. Then there was the first generation of personal computers like Commodore PET, Apple II, Atari 800 or the HP-85. They all came with BASIC interpreters. To my knowledge, none of these BASIC versions came with complex data natively (not even the HP-85 - I just looked it up), but it is fairly straightforward to program your own complex routines if you need them.
Later in the 1980ies I worked on unix workstations, mostly programming in C. There is no native support for complex numbers in C either. When I needed complex numbers, I used the routines from the book "Numerical Recipes in C".
In one of my student jobs around the mid 1980ies there was an early IBM PC with a FORTRAN compiler (that fitted on a single low-density floppy disk!) that also had the complex data type.

Regards
Max

[gentzel]

Macsyma was around, but may have only run on Lisp Machines in the early 80s.

EDIT: Looks like it ran anywhere Maclisp was supported. In the early 80s that would have included most DEC hardware, including PDP-10s and VAXen. I do remember seeing on the TOPS-20 systems at CMU at the time (in addition the Lisp Machines).

[dm319]

(12-08-2024 07:35 PM)Commie Wrote:  Thanks for your post Thomas, at least it's better than the ti84+, I originally payed about £45 for my hp35s, 15 years ago, it's not up there with the 15c but at the same time not too bad.
I am deeply dissapointed with my ti84 collection though, oh well.

So the HP-35s beats the TI-84+? And that was not developed by HP?

Also curious what you mean when you say you are disappointed - as in with the calculators or the size of the collection?

(12-08-2024 07:35 PM)Commie Wrote:  This thread, with the help of various members, has opened up a whole new branch of mathematics for me to explore by which I am very grateful.

Yes, me too. I'm probably more a complex-number-denier but I think I'm starting to come round.

[naddy]

(12-09-2024 11:44 PM)dm319 Wrote:  I'm probably more a complex-number-denier but I think I'm starting to come round.

I think for most people even in technical fields, complex numbers are an algebraic curiosity that was briefly touched on in some math class and then more or less forgotten. They are extensively used in electrical engineering, though, principally because electrical engineers deal with sinusoidal functions all the time, and moving those into the complex plane via Euler's formula turns out to be more convenient than lugging around sines and cosines.

There can be some mild cases of culture shock, like when computer science students are sent to the EE department's introductory signal processing class, and the lecturer, while scribbling the first formula on the blackboard, casually mentions that everything will be done in complex of course.

Buying an HP-15C CE and revisiting the Owner's Handbook made me realize that I'm still comfortable dealing with complex numbers, but frankly, I haven't had any practical use for them in decades.

Getting back to the test at the start of the thread: You generally know if the problem you're working involves complex numbers, and if it doesn't and you're suddenly looking at a complex result, then best case you skip it because it isn't physically interpretable, but more likely it signals that there's a mistake somewhere, so in practical terms that's like getting an ERROR.

[Idnarn]

(12-08-2024 08:26 PM)dm319 Wrote:  

(12-08-2024 08:18 PM)Thomas Klemm Wrote:  3.141592653589793j

Of course the answer is python. The answer is always python.

[vaklaff]

(12-08-2024 06:37 PM)naddy Wrote:  The 15C is a mean machine.

Sorry for OT, just trying to grasp the language used here, English isn't my mother tongue. Mean here means something like tough or badass, do I get it right? Is there a better synonym?

[RPNerd]

(12-10-2024 09:42 AM)vaklaff Wrote:  Mean here means something like tough or badass, do I get it right? Is there a better synonym?

This is 100% correct!

[carey]

(12-09-2024 11:44 PM)dm319 Wrote:  I'm probably more a complex-number-denier but I think I'm starting to come round.

What did it for me in the debate whether complex numbers are a necessity or a mathematical convenience occurred in high school when my chemistry teacher introduced the Schrödinger equation (thank you Mr. Cardulla!) and there it was, i, in the equation itself, suggesting a fundamental role of complex numbers in nature. That didn't stop diehard deniers of the necessity of complex numbers from pointing out that "complex numbers are just ordered pairs of real numbers." However the argument that complex numbers can be expressed using real numbers behaving like complex numbers doesn't appear to me to undercut the necessary role of the complex number structure.

[Johnh]

TBH, I've not seen complex numbers in 45 years of structural engineering. But I'm almost as equally interested in electronics, and for anything analogue, they are such an elegant and intuitive way of representing signals with phase that they are compelling. Even as basic a building block as a simple RC low pass or high pass filter only becomes fully clear with complex numbers
We are taught how the -3dB frequency (half power or about 1/sqrt2 voltage) happens when the capacitive reactance = the resistance. But plot it out on a complex plane and you can see it!