Related Links (Outside this Site)

Wikipedia :   Calculators   |   HP 35s

Width :  81.8 mm
Length :  158 mm
Height:  12.7 to 17 mm
 
39 + 4  keys  
15-digit precision
2 lines of 14 characters
Two CR2032 batteries
(9 months @ 1 h/day)
 
No data link
(NCEES requirement)
 
List price :    $60
Street price:   $49

(2012-11-27)   Keyboard, Keys and Modifier Keys
Advanced calculators assign more than one function to most keys.

On the keyboards of basic calculators, every key is assigned one and only one use, digit entry, binary operation (plus, minus, multiply or divide) or unary operation (typically, only the square root function is provided on such calculators).

Scientific calculators have to provide so many functions that several uses must be assigned to a single key.  On the  HP 35s,  the additional functionalities are obtained by pressing either of two special colored keys  (blu or yellow)  before punching the main key  (whose extra functionality may be indicated by a label of the same color, either on the key itself or nearby on the faceplate).

Back in 1972, the original HP-35 only had  35  keys  (hence its name).  It featured a single modifier key, marked "arc", which was used to obtain the inverse of a trigonometric function by punching it before the key normally assigned to the direct trigonometric function.

(2015-12-09)   Reverse Polish Notation   (Jan Lukasiewicz, 1924)
RPN  is the fastest way to enter data.  Parentheses not needed.

Jan Lukasiewicz (1878-1956) Pronounciation       Video :   The Joys of RPN 

(2012-11-27)   Common unit conversions.
Just the most common ones...

(2012-11-27)   40 physical constants and a mathematical one   ( e )
Accessible via one single menu selection.

HP advertises 41 physical constants, but their 41-st is the value of the mathematical constant  e,  the base of natural logarithms  (2.718281828)...  although it's otherwise readily available in  2½ keystrokes:

[ 1 ]   [ e^x ]

This is rather unfortunate, because the symbol  e  in  any  list of physical constants always refers to the elementary charge  (the electric charge of a proton, the opposite of the charge of an electron).  Since the value of the elementary charge in coulombs  (C, the SI unit)  is unavoidable in such a list, they skirted the issue by giving the value of the electronvolt  (eV)  in joules  (J)  which is the same number  (by the very definition of the electronvolt).

The  molar volume  given in the  HP 35s  is for an ideal gas under co-called  normal temperature and pressure  (NTP, 0°C, 1 atm = 101325 Pa).  In  Casio calculators  competing directly against the  HP 35s,  standard temperature and pressure  (NTP, 0°C, 1 bar = 100000 Pa)  is used instead to define that particular constant.  Otherwise, the calculators of both manufacturers feature  exactly the same set  of  40  physical constants...

The standard convention  (discussed elsewhere on this site)  is that the digits between parentheses that follow a measured quantity indicate its experimental uncertainty  (one standard deviation)  expressed in units of the least significant digit.

The built-in physical constants of the  HP 35s  are based on  CODATA 1998 :

Description & Symbol		HP 35s  Value		CODATA 2010	Unit
Einstein's constant	c	1	299792458	299792458	m/s
Normal gravity	g	2	9.80665	9.80665	N/kg
Newton's constant	G	3	6.673 10-11	6.67384(80) 10-11	N.m2/kg2
Molar volume, 0°C, 1 bar	Vm	273.15 R / 100000		0.022710953(21)	m3/mol
Molar volume, 0°C, 1 atm		4	0.022413996	0.022413968(20)
Avogadro number	NA	5	6.02214199 1023	6.02214129(27) 1023	1/mol
Rydberg's constant	R¥	6	10973731.5685	10973731.568539(55)	1/m
Hartree energy	Eh	2hc R¥  =  a2 me c2		4.35974434(19) 10-18	J
Charge of a proton	e	7	1.602176462 10-19	1.602176565(35) 10-19	C
Electronvolt	eV			1.602176565(35) 10-19	J
Mass of the Electron	me	8	9.10938188 10-31	9.10938291(40) 10-31	kg
Mass of the Proton	mp	9	1.67262158 10-27	1.672621777(74) 10-27	kg
Mass of the Neutron	mn	10	1.67492716 10-27	1.674927351(74) 10-27	kg
Mass of the Muon	mm	11	1.88353109 10-31	1.88353475(96) 10-28	kg
Boltzmann's constant	k	12	1.3806503 10-23	1.3806488(13) 10-23	J/K
Planck's constant	h	13	6.62606876 10-34	6.62606957(29) 10-34	J/Hz
Dirac's constant		14	1.054571596 10-34	1.054571726(47) 10-34	J.s/rad
Quantum of flux	F0	15	2.067833636 10-15	2.067833758(46) 10-15	Wb
Bohr radius	a0	16	5.291772083 10-11	5.2917721092(17) 10-11	m
Electric constant	e0	17	8.854187817 10-12	8.85418781762039 10-12	F/m
Coulomb's constant	1 / 4pe0			8.9875517873681764 109	m/F
Ideal gas constant	R	18	8.314472	8.3144621(75)	J/K/mol
Faraday's constant	F	19	96485.3415	96485.3365(21)	C/mol
Atomic mass unit	u	20	1.66053873 10-27	1.660538921(73) 10-27	kg
Magnetic constant	m0	21	1.2566370614 10-6	4p 10-7	H/m
Ampere's constant	m0 / 4p			10-7	H/m
Bohr magneton	mB	22	9.27400899 10-24	9.27400968(20) 10-24	J/T
Nuclear magneton	mN	23	5.05078317 10-27	5.05078353(11) 10-27	J/T
Proton magnetic moment	mp	24	1.410606633 10-26	1.410606743(33) 10-26	J/T
Electron magn. moment	me	25	-9.28476362 10-24	-9.28476430(21) 10-24	J/T
Neutron magn. moment	mn	26	-9.662364 10-27	-9.6623647(23) 10-27	J/T
Muon magnetic moment	mm	27	-4.49044813 10-26	-4.49044807(15) 10-26	J/T
Classical electron radius	re	28	2.817940285 10-15	2.8179403267(27) 10-15	m
Z of vacuum = m0 c	Z0	29	376.730313461	376.730313461770655468...	W
Compton wavelength	lc	30	2.426310215 10-12	2.4263102389(16) 10-12	m
Compton wl for neutron	lc,n	31	1.319590898 10-15	1.3195909068(11) 10-15	m
Compton wl for proton	lc,p	32	1.321409847 10-15	1.32140985623(94) 10-15	m
Fine-structure constant	a	33	7.297352533 10-3	7.2973525698(24) 10-3
	1/a			137.035999074(44)
Stefan's constant	s	34	5.6704 10-8	5.670373(21) 10-8	W/m2/K4
Ice point  =  0°C	t	35	273.15	273.15	K
Normal Pressure	atm	36	101325	101325	Pa
Standard Pressure	bar			100000	Pa
Gyromagnetic ratio of the proton	gp	37	267522212	267522200.5(63)	rad/s/T
	gp / 2p		2 mp / h	42.5774806(10)	MHz/T
1st radiation constant	c1	38	3.74177107 10-16	3.74177153(17) 10-16	W.m2
2nd radiation constant	c2	39	1.4387752 10-2	1.4387770(13) 10-2	m.K
Conductance quantum	G0	40	7.748091696 10-5	7.7480917346(25) 10-5	S
Euler's number	e	41	2.71828182846	2.71828182845904523536...

As of 2012,  the above values have not been updated in the HP-35s since its first release, in 2007.  They are based on  on  CODATA 1998  which was already several years out of date in 2007  (CODATA 2002  was then current, since  CODATA 2006  was only officialized in 2008).  At this writing, the values listed above  (CODATA 2010)  have been current since June 2011 and are not expected to be updated until 2015 or so.

The values highlighted in yellow correspond to slight mistakes and inaccuracies in the HP 35s built-in constants.  The constants singled-out in this way are known exactly  (because of the way SI units are defined)  but have been rounded incorrectly and/or needlessly truncated below the nominal 12-digit precision of the calculator  (to say nothing of its advertised 15-digit precision for "internal computations").

---

Arnold Sommerfeld's Fine-Structure constant  (a)  is the only listed constant to be  dimensionless.  Its numerical value would be the same in any coherent system of physical units and it remains a mystery:

a   =   m₀ c e² / 2h  =   e² / 2hce₀   =   1 / 137.035999...

The following lengths form a geometric progression of  common ratio  a :

1 / 2R_¥       2p a₀       l_c       2p r_e

That's the first of many noteworthy relations between the above constants:

• 2p r_e   =   a l_c   =   a² 2p a₀   =   a³/ 2R_¥
• m_e l_c   =   m_p l_c,p   =   m_n l_c,n   =   h/c  =  2.210218901(98) 10^-42 kg.m
• m_e m_B   =   m_p m_N   =   e h / 4p   =   8.44805 10^-54 J.kg/T
• u N_A   =   0.001 kg/mol
• e N_A   =   F
• k N_A   =   R
• c₁   =   2p h c²
• c₂   =   h c / k  
• s     =   ( 2 p ⁵ k ⁴ )  /  ( 15 h ³ c ² )   =   5.6704 10^-8 W/m²/K⁴
• e₀ m₀ c²   =   1   (electromagnetic wave propagation).
• F₀   =   h / 2e   (magnetic flux quantum).
• G₀   =   2 e²/ h   =   4a / Z₀   (conducance quantum, Landauer 1957).
• Z₀   =   (m₀ / e₀ )^½  =   m₀ c   (characteristic impedance of the vacuum).

(2012-11-27)   Bug Reports
Severe problems and minor ones.

Inaccurate functions :

For the expression  tan(89.999°) = 1 / tan(0.001°)   my new  (2012)  HP 35s still gives the inaccurate value  (57295.7795401)  that users were complaining about way back in 2007.  The correct value is:

57295.7795072645567033655767369...

My first diagnosis was that it could have been due to the following beginner's mistake in the implementation of the tangent function:

The typical way to compute the tangent function quickly and with high precision is to use an optimal polynomial approximation for values of  x  whose magnitude doesn't exceed  45° = p/4  (beyond that, you compute the reciprocal of  tan(90°-|x|)  instead).

If you were to use directly a standard  Chebychev economization  of  tan(x)  for the aformentioned polynomial approximation, you'd be essentially minimizing the  absolute  error on a function that may vanish  (at x=0).  The fairly large  relative  errors in the neighborhood of  x=0  would result in  floating-point values  that would be erroneous at their nominal precision.

Instead, you obtain an acceptable polynomial approximation by multiplying  x  into a Chebychev economization of  tan(x)/x  (which is itself a poplynomial in  x² ).

Well, whatever mistake the HP engineers did does not reduce to the above.  They seem to have implemented correctly the sine function  (for which the above warning would have applied too)  and the cosine function.  Yet, the ratio  sin(x)/cos(x)  gives exactly the above erroneous value for  x = 89.999°.  This is a clue that they "cut corners" by wrongly implementing the tangent as a sin/cos ratio, which is not numerically stable...

HP Trig Accuracy?

(2012-11-27)   Complex Functions of Complex Variables
Discontinuity cliffs appear in the complex extensions of some functions.

(2012-12-09)   Programming
Turing-complete  (only marginally less versatile than the  HP 33s).

Like other calculator of its class, the HP-35s is programmed by recording the sequences of keys that would be necessary to produce a result.  Such sequences may depend on the data-entry mode that's being used  (RPN or "algebraic" infix).  Therefore a program written with RPN mode in mind will most probably not execute properly if the calculator is in algebraic mode  (and vice-versa).

My recommendation  (and the recommendation of everybody who is familiar with the HP-35s)  is to always run this calculator in RPN mode and to write programs exclusively for that mode  (the capability to use an infix entry mode is no more than a misguided marketing decision which can be safely ignored).

User-defined programs are essentially executed like predefined programs, using the "R/S" key  ("run/stop")  at the top-left corner of the keypad.  The blue-shifted function of this key  ("PRGM")  is to enter the programming mode, where you see every recorded instruction appear wih a 4-digit line number.

If all you ever wanted to do was run a single program, what ypu would do is enter the corresponding instructions staring with line 0001 and execute the sequence by making the  program counter  point to the beginning  (by pushing GTO 1)  before hitting the R/S key.  It's optional to start such a lone program with a "label" and to end it with the RTN instruction but we may as well get into the habit of doing it  (since it's mandatory if we stored more than one program, as will be discussed later).  The single-letter "label" (here A) is the name of the routine and will appear as the first character in the line numbers for that routine  (which can have at most 999 lines in it).

Example:  To compute the area of a circle of given radius:

A001 LBL A
A002 x²
A003 π
A004 *
A005 RTN

There are several ways to execute the above...

Programming the 35s  by  John  (2012).