09-09-2013, 11:56 AM

When raising complex numbers to some power, the x root y won't work nor will x^2, yet y^x works. 3 root x fails for complex numbers, but x^ 1/3 works. Why?

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09-09-2013, 11:56 AM

When raising complex numbers to some power, the x root y won't work nor will x^2, yet y^x works. 3 root x fails for complex numbers, but x^ 1/3 works. Why?

09-09-2013, 02:46 PM

Because this is the way the 35s was designed. Some functions work in the complex domain, others don't. For instance, the root keys are for real arguments only, while the y^x key can handle complex input as well.

Take a look at the manual. It has a list of all functions that will work with complex numbers.

Dieter

09-09-2013, 03:13 PM

Quote:

It has operations for complex arithmetic (+, -, *, /), complex trigonometry (sin, cos, tan), and the mathematics functions -z, 1/z, z1^{z2}, ln z, and e^{z}. (where z1 and z2 are complex numbers).

However this doesn't give the answer to why they only implemented this subset. Karl Schneider gave a plausible explanation.

Cheers

Thomas

*Edited: 9 Sept 2013, 3:20 p.m. *

09-09-2013, 03:23 PM

Quote:This should be answered by the people responsible for the specs sheet - HP.

Why?

I might be wrong, but I take it this way: The real root is faster to calculate, so on a *programmable* calculator used by people mostly interested in real results, this makes a lot of sense. The 35s, along with its predecessors, is targeted at education, not engineering.

The 15C does a better job by flagging the desired mode. Maybe the 32S was on the edge in terms of memory to use this concept, and obviously the 35s is designed to follow its traces.

All just speculations, of course.

09-09-2013, 03:39 PM

Quote:

The 35s, along with its predecessors, is targeted at education, not engineering.

Now kids the real mystery about these complex numbers is that you can calculate the trigonometric functions but there's no way to find the inverse which your calculator proves returning INVALID DATA.

Cheers

Thomas

09-09-2013, 03:52 PM

Quote:

you can calculate the trigonometric functions but there's no way to find the inverse which your calculator proves returning INVALID DATA.

But of course, in that way you can't check the accuracy of the direct results :-)

That's probably because they wanted to avoid all those questions about "why the returned value doesn't match the initial" - being multi-valued.

Cheers\'AM

09-09-2013, 04:05 PM

Then why did they provide *ln z*?

09-10-2013, 12:05 AM

It's really rather bizarre. Why wouldn't they put in the inverse functions and the hyperbolics and their inverses in the HP 35s's function universe for complex arguments? I put the equations for all of this in it's equation library, and it works just fine. Doesn't make sense...

09-10-2013, 12:19 AM

Could you post your equations here, please? These sound like essential additions to the trig function set. Much appreciated.

09-10-2013, 01:21 AM

Thomas, I had "Leistungskurs Mathe" (you know what that means, dunno what that might be in english), and when there was some time left, we were asked for our interests and how to fill the remaining weeks. I asked for complex numbers, and the answer of our teacher was: "I'll see if I can find a way to explain what they are." He never did and I left school without knowledge about it.

Started functional analysis at university not before 3rd semester, introducing complex numbers to me. That was quite hard ;-).

Maybe all this was because I live in northern germany, and hopefully things are better now.

09-10-2013, 10:33 AM

Math2.org Math Tables: Hyperbolic Trigonometric Identities

(Math)

Hyperbolic Definitions

sinh(x) = ( ex - e-x )/2

csch(x) = 1/sinh(x) = 2/( ex - e-x )

cosh(x) = ( e x + e -x )/2

sech(x) = 1/cosh(x) = 2/( ex + e-x )

tanh(x) = sinh(x)/cosh(x) = ( ex - e-x )/( ex + e-x )

coth(x) = 1/tanh(x) = ( ex + e-x)/( ex - e-x )

cosh2(x) - sinh2(x) = 1

tanh2(x) + sech2(x) = 1

coth2(x) - csch2(x) = 1

Inverse Hyperbolic Defintions

arcsinh(z) = ln( z + (z2 + 1) )

arccosh(z) = ln( z (z2 - 1) )

arctanh(z) = 1/2 ln( (1+z)/(1-z) )

arccsch(z) = ln( (1+(1+z2) )/z )

arcsech(z) = ln( (1(1-z2) )/z )

arccoth(z) = 1/2 ln( (z+1)/(z-1) )

Relations to Trigonometric Functions

sinh(z) = -i sin(iz)

csch(z) = i csc(iz)

cosh(z) = cos(iz)

sech(z) = sec(iz)

tanh(z) = -i tan(iz)

coth(z) = i cot(iz)

09-10-2013, 11:29 AM

So you could program the missing ones!

09-10-2013, 01:10 PM

Here's more!

http://www.math.ethz.ch/education/bachelor/lectures/fs2012/other/ka_itet/TrigHypFunktionen.pdf

09-10-2013, 01:52 PM

That's kind of a sad story. Because it's not that difficult to explain imaginary numbers:

I had more luck with my teacher. He had an HP-41C as well and allowed me to borrow PRISMA, the fanzine of the CCD (Computerclub Deutschland e.V.). One of the programs used Bairstow's method to solve a polynomial with real coefficients. This was far from what I would understand at that time. Many years later I wrote a program for the HP-11C.

Quote:

hopefully things are better now

With the advent of the internet we have access to so much knowledge these days. The problem now is to sort the wheat from the chaff.

Kind regards

Thomas

09-10-2013, 02:01 PM

Thanks! I love Calvin & Hobbes.

09-10-2013, 05:30 PM

All math is imaginary. Hanging in my classroom:

09-10-2013, 05:56 PM

Hence:

:)

09-10-2013, 07:27 PM

Huh. I thought it was just a different integer base. English has thirteen simple names for numbers:

Zero, one, two, ... nine, ten, eleven, twelve.

After that is "thirteen" -- literally "three and ten".

So English is perfect for counting in Base 13. After twelve comes "teen".

Oneteen, twoteen, thirteen, ... nineteen, tenteen, eleventeen, twelveteen.

And after the nineties come the tenties, eleventies and twelveties.

Ninety-eleven, ninety-twelve, tenty, tenty-one, tenty-two...

And so after twelvety-twelve, comes one-hundred base 13, which equals 169 decimal (a number I'm fond of for other reasons beyond the scope of this post).

I especially like to use base 13 counting when I'm standing behind someone doing inventory... ;-)

Of course, in Spain, Latin American countries, etc., you count in hex.

Cero, uno, dos, tres, cuatro, ... once, doce, trece, catorce, quince...

... are the sixteen digits. So "veinte y catorce" is 2E hex.

I think what started me thinking about all this was when Dennis the Menace used the number eleventeen back long ago (1950s? 1960s?).

Anyway, my favorite number is eleventy-seven. It just sounds cool. But imaginary? Hardly!

Dale

p.s.: U2: 01, 02, 03, 0E ???

p.p.s: hope my spelling 'en espanol' is close. Been a while....

*Edited: 10 Sept 2013, 7:29 p.m. *

09-10-2013, 07:58 PM

A little girl once said to me: Thomas, you're so weird! I took it as a compliment and would like to pass it to you.

Cheers

Thomas

09-10-2013, 08:02 PM

Quote:

Anyway, my favorite number is eleventy-seven. It just sounds cool. But imaginary? Hardly!

Try French, where 97 is "quatre-vingts dix sept" or 4 times twenty and ten and 7 (not sure about the "s" after vingt - it's been more than 40 years since I studied this!)

After years of attending scientific meetings at which English is invariably the language of the meeting, I have decided that you can figure out what somebody's true language is by listening to him/her count/recite numbers to him/herself.

09-10-2013, 09:14 PM

Thomas,

If that was directed at me: Heard it. Often. For a long time. And thank you! Back atcha, fellow calc-nut!

Dale

09-12-2013, 11:15 PM

Dave,

Try watching engineers attempting to speak in haiku counting on their fingers!

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