Euler Identity in Home

04042014, 12:09 AM
(This post was last modified: 04042014 07:34 AM by ColinJDenman.)
Post: #1




Euler Identity in Home
Apparently, despite Euler, e^ipi + 1 = 2.07E13i, which seems quite a large difference


04042014, 12:57 AM
Post: #2




RE: Euler Identity in Home
Of course, Home view is approximate by design.
Do it in CAS, and you will get zero exactly, as Euler intended. 

04042014, 01:14 AM
Post: #3




RE: Euler Identity in Home
Hmm...
I'm surprised by the size of the approximation, and also observe that other traditional approximations (on other, ancient nonHP calculators) tend to make their presence felt with basic trig like sin(90) etc, which are caught on HP calcs. On an old (seminotorious National Semiconductor) RPN calculator sin(90) yields 0.999999999997 or thereabouts, leading to much amusement amongst rich HP owners. I wonder what other e^ix capable HPs yield  does the 50/38/39 yield a similar answer? 

04042014, 01:48 AM
(This post was last modified: 04042014 01:50 AM by Helge Gabert.)
Post: #4




RE: Euler Identity in Home
HP50G: 2.0676 . . . E13i


04042014, 05:09 AM
Post: #5




RE: Euler Identity in Home
Hmmm... part 2
Further fiddling shows, in Home mode, in degrees mode sin(180) = 0 in radians mode sin(pi) = 2.07E13 But e^i*pi in degrees mode gives the 2.07E13. I begin to feel this is less a question of Exact versus Approximate, but rather some inconsistency internally. In CAS with exact checked (with Solve as the app (just because it was set to Radians)) sin(pi) = 0 and with Parametric as the app (set to degrees) sin(180) = 0 With exact, CAS, degrees (Para) e^i*pi +1 = 0 // yippee exact, CAS, radians (Solve) e^i*pi + 1 = 0 With exact unchecked, in CAS, degrees (Parametric) e^i*pi + 1 = 1.078E14i in radians (Solve) ... exactly the same Am I wrong to be confused :) It's making me suspicious of the results I get (which on some level is probably a good thing). I'm not sure I can trust my Prime any more, which isn't a good attribute in a calculator. 

04042014, 06:05 AM
(This post was last modified: 04042014 06:06 AM by HP67.)
Post: #6




RE: Euler Identity in Home
50g CAS mode
Code: 0 It ain't OVER 'till it's 2 PICK 

04042014, 12:41 PM
Post: #7




RE: Euler Identity in Home
(04042014 05:09 AM)ColinJDenman Wrote: But e^i*pi in degrees mode gives the 2.07E13. I begin to feel this is less a question of Exact versus Approximate, but rather some inconsistency internally. In Home, π is just a floating point (BCD) number, not a symbolic constant. Thus, sin(π) simply isn't zero. I wouldn't be surprised if the result is the correctly rounded result of computing sin(3.14159265359). Marcus von Cube Wehrheim, Germany http://www.mvcsys.de http://wp34s.sf.net http://mvcsys.de/doc/basiccompare.html 

04042014, 04:18 PM
(This post was last modified: 04042014 04:18 PM by orcinus.)
Post: #8




RE: Euler Identity in Home
I honestly don't get what the confusion is.
Home = numeric CAS = symbolic Numeric pi is a numeric (!) float constant, with a limited precision. Symbolic pi is treated as a symbolic (!) entity, not a number. Therefore, you'll get an approximation limited by the precision of numeric pi on the Home screen and an exact symbolic result on the CAS screen. If you replace pi with 180 in degrees mode on Home, of course you'll get an exact result  180 is a natural number, pi is transcedental. 

04042014, 09:26 PM
(This post was last modified: 04042014 09:42 PM by Joe Horn.)
Post: #9




RE: Euler Identity in Home
(04042014 12:41 PM)Marcus von Cube Wrote: In Home, π is just a floating point (BCD) number, not a symbolic constant. Thus, sin(π) simply isn't zero. I wouldn't be surprised if the result is the correctly rounded result of computing sin(3.14159265359). You're right: it is. SIN(exactly 3.14159265359 radians) = 2.0676153735661672...E13, which rounds to the same result as given by all HP 12digit BCD calculators. Edit: Some HP 10digit BCD calculators were not as accurate. For example, the HP19C, '97, and '41 return only 2 significant digits: exactly 4.1E10 for SIN(exactly 3.141592654 radians), whereas the correct answer is 4.102067615373566167...E10. HEY... the HP65 returns 0 for SIN(3.141592654). What's up with that? <0ɸ0> Joe 

04052014, 04:08 AM
Post: #10




RE: Euler Identity in Home  
04052014, 04:18 PM
Post: #11




RE: Euler Identity in Home
(04042014 09:26 PM)Joe Horn Wrote: HEY... the HP65 returns 0 for SIN(3.141592654). What's up with that? The same result for HP25. Just tried it a few minutes ago. "f" "SCI" "3" "g" "RAD"; "g" "PI"; "ENTER"; "f" "sin"; Result: 0.000 00 "Rotate down"; "f" "cos"; Result: 1.000 00 "g" "PI"; "ENTER"; "2"; "/" "f" "sin"; Result: 1.000 00 That's what I remember from the classic calculators like the HP55 and HP67 as well. Jose Mesquita RadioMuseum.org member 

04052014, 04:28 PM
Post: #12




RE: Euler Identity in Home
(04042014 09:26 PM)Joe Horn Wrote: HEY... the HP65 returns 0 for SIN(3.141592654). What's up with that? And now with a Texas TI57, exactly the same result as well: "2nd" "Rad"; "2nd" "PI"; "2nd" "sin"; Result: 0 "2nd" "PI"; "2nd" "cos"; Result: 1 "2nd" "PI"; "/"; "2"; "=" "2nd" "sin"; Result: 1 Jose Mesquita RadioMuseum.org member 

04052014, 04:35 PM
Post: #13




RE: Euler Identity in Home
(04042014 09:26 PM)Joe Horn Wrote: HEY... the HP65 returns 0 for SIN(3.141592654). What's up with that? Also, using a Texas TI51III, got the same result again: "2nd" "Rad"; "2nd" "PI"; "sin"; Result: 0 "2nd" "PI"; "cos"; Result: 1 "2nd" "PI"; "/"; "2"; "=" "sin"; Result: 1 Jose Mesquita RadioMuseum.org member 

04052014, 04:41 PM
Post: #14




RE: Euler Identity in Home
(04042014 09:26 PM)Joe Horn Wrote: HEY... the HP65 returns 0 for SIN(3.141592654). What's up with that? And what about a CASIO FX39? Guess what... the same result again: Select "RAD" on the right side sliding switch; "INV" "PI"; "sin"; Result: 0 "INV" "PI"; "cos"; Result: 1 "INV" "PI"; "/"; "2"; "=" "sin"; Result: 1 Jose Mesquita RadioMuseum.org member 

04052014, 04:59 PM
Post: #15




RE: Euler Identity in Home
(04042014 09:26 PM)Joe Horn Wrote: HEY... the HP65 returns 0 for SIN(3.141592654). What's up with that? Sorry people for polluting this thread with so many posts in a row. That's my last one on this theme. I just tried with a CASIO FX5000F, and again, the same result as before: "mode" "Sci" "3"; "sin" "shift" "PI"; Result: 0.00 00 "cos" "shift" "PI"; Result: 1.00 00 "sin" "(" "shift" "PI" "/" "2" ")" ; Result: 1.00 00 Jose Mesquita RadioMuseum.org member 

04052014, 06:11 PM
Post: #16




RE: Euler Identity in Home
One can look to http://www.wolframalpha.com/ for another take on this:
sin(pi)=1.2246467991473532x1016 (double).sin(pi)=0 

04052014, 07:32 PM
(This post was last modified: 04052014 08:43 PM by jebem.)
Post: #17




RE: Euler Identity in Home
(04052014 06:11 PM)DrD Wrote: One can look to http://www.wolframalpha.com/ for another take on this: And again, on HP Prime in Home mode: 1) So, it is expected to not get an answer of Zero for SIN(Pi), as previously explained by others (I got a result = 2.06761537357E13); 2) However, it will give "exact" rounded values for different situations (using Scientific 4 digit format): 2.1) SIN(Pi/2) > Result = 1.0000 2.2) COS(Pi) > Result = 1.0000 To be honest, I was not expecting the results to be so rounded here, for the same reasons used to explain the SIN(Pi). I just checked my HP50G, and the results are exactly the same ones I get on my HPPrime, so I'm happy to find out that the behavior is consistent between the two calculators. There is no reason to not trust the Prime here, unless we don't trust the 50G either. Jose Mesquita RadioMuseum.org member 

04052014, 08:44 PM
Post: #18




RE: Euler Identity in Home
These older calculator results are essentially what I was expecting to see. If a 30/40 year old non CAS non Exact calculator can display the goods, why do you not expect the same from a 400Mhz ARM with 32MB of RAM?
I was tempted to ask for a poll of old HPs on the general forum, asking for the calculation of sin(pi) in radians with max Sci format display. On real hardware, since I don't know how the various emulators work. Plain and simple truth (to me): I expect better, not worse after 30 years. And no amout of telling me to use CAS or the limits of BCD (which isn't limited in the way that IEEE floats are) is likely to change my mind. I'm just a grumpy old man, reminiscing about non existent golden ages, ignore me. 

04052014, 09:18 PM
Post: #19




RE: Euler Identity in Home
(04052014 08:44 PM)ColinJDenman Wrote: These older calculator results are essentially what I was expecting to see. If a 30/40 year old non CAS non Exact calculator can display the goods, why do you not expect the same from a 400Mhz ARM with 32MB of RAM? Yap, I have made myself the same question over and over, when looking to the results from a HP50G and now from a HPPrime. Apparently, what the classic 70's calculators are doing is to round internally to Zero when the difference is below a specific value. If this is true, then they were "faking" the calculation a bit. Probably this kind of algorithms are still used in other low cost calculators, who knows. By the way, I have the real calculators with me, so the above posted results are from physical calculators from my modest collection. I am into electronics and physics as a hobby now, so mathematics are needed here and there, and I have using calculators since the 70's in school (Casio and HP). Jose Mesquita RadioMuseum.org member 

04052014, 09:54 PM
(This post was last modified: 04052014 10:25 PM by ColinJDenman.)
Post: #20




RE: Euler Identity in Home
(04052014 09:18 PM)jebem Wrote: Yap, I have made myself the same question over and over, when looking to the results from a HP50G and now from a HPPrime. I expect calculators and any digital computer to be approximate, to have cumulative errors on complex calculations. I don't expect those results to be visible on such a simple calculation. I think there is a reasonable expectation that calculation precision is better that display precision. As I mentioned in a previous post, I regard this a hallmark of older HP calcuators. I owned the notorious NatSemi I mentioned, at University in the 70s. I worked with assember on PDP11s in my first job as a programmer. I feel as though we are going backwards. It leaks out occasionally :) But let us never forget that the correct answer to sin(pi) is zero. Some bits of the Prime know this. Bits that don't, shouldn't be there. http://xkcd.com/1349/ 

« Next Oldest  Next Newest »

User(s) browsing this thread: 1 Guest(s)