Calculator test

[naddy]

(12-10-2024 10:53 AM)carey Wrote:  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.

The designations "real" and "imaginary" are not to be taken literally. Like natural, integer, rational, or real numbers, complex numbers are a mathematical construct that happens to have proven useful for modeling some aspects of physical reality. Whether that implies that any numbers "exist" in a meaningful way is one for the philosophers and oddly irrelevant for mathematicians and physicists/engineers.

It's also worth pointing out that unless you are into pure mathematics, all the math you will ever encounter is from a subset selected because we have found applications for it, with the parts underlying cryptography the most esoteric because they are purely structural and don't model physical systems. That doesn't mean that this math is more "real" than the one we haven't found applications for.

[AnnoyedOne]

(12-10-2024 12:34 AM)naddy Wrote:  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.

Exactly. Simply a mathemetical method to avoid extra work

(12-10-2024 12:34 AM)naddy Wrote:  ...I'm still comfortable dealing with complex numbers, but frankly, I haven't had any practical use for them in decades.

The same here. And for that I'm grateful. Was that 7 sheep or 7i (or 7j) sheep? ZZZzzz.

(12-10-2024 12:34 AM)naddy Wrote:  ...so in practical terms that's like getting an ERROR.

Precisely! Unless you're a mathematician a complex number result probably means that you made a mistake somewhere.

To me kind of like "irrational numbers". You probably can't actually measure them to more than a few decimal places so why bother calculating them to a million? Unless you're in to that sort of thing. Each to his/her own.

A1

[AnnoyedOne]

(12-10-2024 12:28 PM)Johnh Wrote:  But plot it out on a complex plane and you can see it!

And "for real" on an oscilloscope.

"Hey, look at that phase shift between voltage (v) and current (i)."

A1

[Commie]

(12-09-2024 11:44 PM)dm319 Wrote:  So the HP-35s beats the TI-84+?
Also curious what you mean when you say you are disappointed - as in with the calculators or the size of the collection?

The way I see the ti84+ is its equivalent to a 1980's home computer and given you can buy them sh of Ebay for about £40 or less represents to me a good deal. However, its limitations with complex functions has been exposed. With hindsight, this shouldn't come as a surprise because the ti84+ was designed for college students doing higher education, so it had to be designed with constraints to be passed by the various examination boards.That said, the ti84+ is a good calculator no doubt.

I really like the hp 35s, especially the equation mode and its programming instructions are not just keycodes but actual function strings making programming/debugging a bit simpler.So, in terms of complex functions available on the hp 35s, it is better than the ti84+.Definitely a keeper.

I've been over at Amazon, just recently perusing the complex analysis books there are hundred's to choose from, knowing which one to buy is a nightmare. Anyone?

Cheers
Darren

[AnnoyedOne]

(12-10-2024 02:49 PM)Commie Wrote:  ...just recently perusing the complex analysis books there are hundred's to choose from, knowing which one to buy is a nightmare. Anyone?

I had a pure math textbook while studying but I don't recall ever referring to it for complex numbers. Mostly simple algebra.

What is your purpose? A pure math book will likely give you the theory and ignore any practical uses. I think that the polar representation is more common in mathematics.

Electronic and mechanical engineers typically use the rectangular form (unless the polar form is easier) and will usually refer to some practical use.

Mathematicians refer to a pi/2...-pi/2 rotation around the y-axis. Engineers more often use degrees (e.g. a phase shift of +/-90deg).

A1

[John Keith]

(12-10-2024 12:44 PM)naddy Wrote:  The designations "real" and "imaginary" are not to be taken literally.

I do think that the names are one of the major problems that students or newcomers have in dealing with complex numbers. When people see the words "complex" and "imaginary" they think "complicated" and "nonsense". Real-world examples (e.g. R-L-C circuits) and visual examples (in both Cartesian and polar coordinates) can be really helpful.

[carey]

(12-10-2024 12:44 PM)naddy Wrote:  

(12-10-2024 10:53 AM)carey Wrote:  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.

The designations "real" and "imaginary" are not to be taken literally. Like natural, integer, rational, or real numbers, complex numbers are a mathematical construct that happens to have proven useful for modeling some aspects of physical reality. Whether that implies that any numbers "exist" in a meaningful way is one for the philosophers and oddly irrelevant for mathematicians and physicists/engineers.

I appreciate your comments but I think they refer to a different debate (i.e., the philosophical debate whether "real" or "imaginary" are to be taken literally or "exist") vs the pragmatic debate whether complex numbers are necessary to describe nature. For example, many types of numbers can be defined (e.g., Abraham Robinson's hyperreals) and can be useful, but strictly speaking, may not be necessary. However, the square root of -1 occurs in the Schrödinger equation, the fundamental equation describing atoms and other systems, hence complex numbers (irrespective of the meaning of the name "complex" :) are essential in physics.

[Commie]

Hi Guys,

Although not new to me, when I first saw Carey's post, the penny dropped, he stated that there is a workaround for complex trig. for the ti 84+ this being....



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

These two equations are the true equations which describe sin(x) and cos(x), i.e., if you want it to output only real, then simply leave the real part of z and set coefficient of i to zero, and hey presto,
it gives the same answer as sin(x) real. And, since the hp 15c(e) can do this for any function(more or less), means everything is complex, real is just a subset of complex.

I've checked my Derive 6.2 and it works for most functions.

Cheers Guys

[carey]

(12-10-2024 02:49 PM)Commie Wrote:  I've been over at Amazon, just recently perusing the complex analysis books there are hundred's to choose from, knowing which one to buy is a nightmare. Anyone?

You've identified an area of math where, despite many books, it's difficult to find one that isn't too abstract.

Below are a couple of personal favorites:

1) In keeping with the numerical theme of this forum, A. David Wunsch's book: "A MatLab® Companion to Complex Variables" is all about calculating with complex numbers. While written as a supplement to other textbooks (including Wunch's own "Complex Variables with Applications" - find a less expensive used 1st or 2nd edition), it stands fairly well on its own. Open-source GNU Octave can be used to do the calculations or it might be fun translating commands to 15C or 42S RPN or to RPL.

2) Some of the most practical examples of complex analysis can be found in the complex variables chapter of (old) books on Laplace transforms, e.g., "Introduction to Laplace transforms for radio and electronic engineers" by W. D. Day - a vacuum tube era book using British notation (p instead of s for the transform variable) but with a nice graphical example of complex integration.

[Idnarn]

(12-10-2024 05:44 PM)Commie Wrote:  \[\cos(z) = \frac{e^{iz} + e^{-iz}}{2}\]
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]

\[\sin(a+b\times i) = \sin(a)\times\cosh(b)+\cos(a)\times \sinh(b)\times i\]
\[\cos(a+b\times i) = \cos(a)\times\cosh(b)-\sin(a)\times \sinh(b)\times i\]

If you have a calculator capable of sinh and cosh on real numbers, you can compute it.

[Commie]

(12-10-2024 06:49 PM)Idnarn Wrote:  \[\sin(a+b\times i) = \sin(a)\times\cosh(b)+\cos(a)\times \sinh(b)\times i\]
\[\cos(a+b\times i) = \cos(a)\times\cosh(b)-\sin(a)\times \sinh(b)\times i\]

If you have a calculator capable of sinh and cosh on real numbers, you can compute it.

Hi Idnarn,

Thanks for your input, I must admit, this is the first time I've seen this equation. I'm also interested in scientific calculator development using 8 bit mcu's and I can see it's usefulness in that area. Carey's original Post was calculatable using the ti84+ because e^(i.z) and ln(iz) are indeed supported on the ti84+.I've also just found out square roots, log10 and a few other functions are also supported on the ti84+, but no complex trig.
Again, thanks for your input, much obliged.

[Commie]

(12-10-2024 06:12 PM)carey Wrote:  ...despite many books, it's difficult to find one that isn't too abstract.

Hi Carey,

Yes, precisely why I asked, I've bought math books from Amazon before and found some of them too abstract to follow, well I have some advanced books in my bookshelf so I'll see if I can find anything of use. Anyhow, thanks for your reply.

Cheers
Darren

[dm319]

From Wikipedia:

In Hellenistic Egypt, the Greek mathematician Diophantus in the 3rd century AD referred to an equation that was equivalent to \[4 x + 20 = 4\] (which has a negative solution) in Arithmetica, saying that the equation was absurd.[24] For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others.

Also from Wikipedia, casus irreducibilis:

It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843.

As my maths isn't good enough, I can at least extrapolate conceptionally from negative numbers to i.

[dm319]

Back to calculator testing. I only realised from this thread that the TI-83+ doesn't do trig on complex numbers.

So my question is:

What calculators can do trig on complex numbers (other than 15c and successors)?

[Commie]

(12-10-2024 10:58 PM)dm319 Wrote:  Back to calculator testing. I only realised from this thread that the TI-83+ doesn't do trig on complex numbers.

So my question is:

What calculators can do trig on complex numbers (other than 15c and successors)?

The hp35s will do sin(x),cos(x) and tan(x) but the inverses it can't do.

Math package like Derive 6 can do everything that I've thrown at it.

Cheers
Darren

[naddy]

(12-10-2024 10:58 PM)dm319 Wrote:  What calculators can do trig on complex numbers (other than 15c and successors)?

The whole line of 32S, 32SII, 33S, and 35S supports sin, cos, and tan for complex numbers, but neither the inverse nor the hyperbolic functions.

[Thomas Klemm]

The Math Pac for the HP-71 supports SIN, COS, TAN, SINH, COSH, TANH for complex numbers but not their inverse functions.

[Johnh]

TI58 and TI59 (1977) did complex trig through the included Master Library ROM module, program 06 , with sin, cos and tan and also their inverses.

[AnnoyedOne]

(12-10-2024 10:26 PM)dm319 Wrote:  ...I can at least extrapolate conceptionally from negative numbers to i.

Negative numbers are an interesting concept. There's really no such thing thing since you can't have -x of anything. You might not have something which equates to zero (0). That said something can be negative with respect to zero (if you use it as your reference).

As an example in EE you can have negative current (flowing in "reverse"), negative voltage etc. It refers to something real--a relationship. That is something is backwards with respect to a reference (usually 0).

By contrast complex numbers have no referent since they're a mathematical method. Useful but they don't actually exist. That is probably why the i/j part of complex numbers is called "imaginary".

In EE messy differential equations are avoided using Laplace transforms using the symbol 's'.

https://en.wikipedia.org/wiki/Laplace_transform

Again it is a method that makes calculations easier. The underlying reality hasn't changed at all.

A1

[Commie]

Hi Guys,

Here is one of the equations for tan(x+iy):
\[\tan(x+iy) = \frac{sin{2x} + {i}sinh{2y}}{cosh{2y}+cos{2x}}\]

Tested on my ti30x pro, and confirmed the answer on my hp35s.

Cheers Guys