HP-35 Reference the 1972 original · 35 keys, no shift · 49 entries

The first pocket calculator with logarithms and trigonometry: ten digits, a range of 10⁻⁹⁹ to 10⁹⁹, a four-register RPN stack, one memory register and thirty-five keys that each do exactly one thing. This sheet is the keyboard drawn from the manual’s cover, a guide to how the machine thinks — the stack, the automatic lift, the power-first x^y, the things it cannot do and how the manual does them anyway — then an index of every key and behaviour with its page in the Operating Manual, and the manual’s mortgage example traced through the stack one key at a time.

key a key on the machine · 0. what the display shows
Source: HP-35 Operating Manual, HP 00035-90008 Rev. 8/73 (Hewlett-Packard, Cupertino, 1973); the HP-35 Errata Sheet; and David S. Cochran, The HP-35 Design: A Case Study in Innovation (2010). Page numbers are the manual’s printed pages.
Colour keythe machineentry, stack & displayprogrammingmathematicsvariables & memoryreference & system

The Keyboard

35 keys · one prefix · drawn to the proportions of the machine
-1.234567809-35
OFFON
xy 
log 
ln 
ex 
CLR 
√x 
arc 
sin 
cos 
tan 
1/x 
x⇄y 
R↓ 
STO 
RCL 
 ENTER↑
CHS 
EEX 
CLx 
 −
 7
 8
 9
 +
 4
 5
 6
 ×
 1
 2
 3
 ÷
 0
 .
 π
hpHEWLETT · PACKARD35

Every function is one key

Small black keys carry the functions, with their names printed on the case above them; the blue keys are the ones you press most — ENTER↑, the four operators and the three that change or clear an entry — and the large light keys are the digits, the point and π. There is no shift key. The one prefix is arc, which makes the next trigonometric key its inverse.

ENTER↑
separates two numbers you key yourself; after an answer the lift is automatic
arc sin
the inverse; arc is pressed first, then the function
x^y
the power is keyed first and the base second — the reverse of every later HP
CLx · CLR
clear the entry, or the whole stack

Function keys are on ½-inch centres and the digits on 11/16-inch centres: the design study says the numeric keys were made larger and given the strongest contrast because they are pressed most, and the rest were shrunk so that a finger cannot hit two at once.

The power switch is a slide above the keys; ON shows 0. in the display.

blue: ENTER↑, the operators, CHS · EEX · CLx and CLRblack: a function, named on the case above the keylight: digits, the point and π

HP-35 Guide

the machine, and how to make it do things

Anatomy three things worth seeing laid out

Four registers, one of them showing

The HP-35 keeps its working numbers in a stack of four registers, X at the bottom and T at the top, and the display always shows X. ENTER↑ pushes a copy of X up into Y; an operator takes Y and X, leaves the answer in X and drops everything above. The manual’s own trace of (3 × 4) + (5 × 6):

3 ENTER↑
T—
Z—
Y3
X3
X repeated into Y
4 ×
T—
Z—
Y—
X12
product formed, stack drops
5
T—
Z—
Y12
X5
automatic ENTER↑ pushes 12 up
ENTER↑ 6
T—
Z12
Y5
X6
5 into Y, 12 into Z; 6 overwrites
×
T—
Z—
Y12
X30
30 in X, 12 drops to Y
+
T—
Z—
Y—
X42
sum

Two rules explain everything else. A number keyed after an answer lifts the stack automatically, so ENTER↑ is only needed between two numbers you key yourself. And when the stack drops, t is copied into both T and Z — a constant placed in T is reused every time an operator is pressed.

The trigonometric functions borrow T as scratch space (z replaces t), which the manual notes on page 17. Nothing else touches it.

Fifteen positions of red LED

The display is ten mantissa digits with their own sign, and a two-digit exponent with its own sign. Numbers between 10⁻² and 10¹⁰ are shown plainly with trailing zeros blanked; anything larger or smaller is shown as a mantissa and an exponent, and 9.999999999 99 is what overflow looks like.

signten significant digitssignexponent−9.754604237−12the decimal point takes a digit position×10 to this

There is no FIX key: the machine always shows every significant digit it has. When the batteries run low every decimal point lights, and because the true point owns a whole digit position the number is still readable.

Thirty-five keys and no shift

Every function has its own key, and the only prefix on the machine is arc, which turns the next sin, cos or tan into its inverse. That is why there are no hyperbolics, no x², no 10^x and no radians: what is not on the keyboard is not in the machine, and the manual shows how to build each from what is.

3 ENTER↑ ×9. — x²2 ENTER↑ 10 xy100. — 10^x, the power keyed first512 ENTER↑ 9 1/x x⇄y xy2. — a ninth root30 ENTER↑ π × 180 ÷ sinsin of 30 keyed as radians would be

The case-study explains the count: the earlier desktop HP 9100 had suggested the design, and once the keyboard was settled the calculator was named after it.

The machine

The HP-35 of 1972 is the first pocket calculator with transcendental functions: logarithms, exponentials and trigonometry in a case that fits a shirt pocket, with ten-digit accuracy and a range of 10⁻⁹⁹ to 10⁹⁹. Bill Hewlett asked for a shrunken HP 9100; the manual calls it “a high precision portable electronic slide rule”.

It works in RPN on a four-register stack, has one memory register, no shift key and no display formatting. Three nickel-cadmium cells give three to five hours; fourteen hours on the charger fills them, and it runs from the charger meanwhile.

HP planned 100,000 units and sold far more. The manual’s foreword invites you to check that 355/113 approximates π to within 8.47 millionths of one percent before reading anything else.

Keying in

Digits and the point go in left to right. CHS works before or after the digits; EEX ends the mantissa and takes a power of ten, with CHS after it for a negative exponent. A wrong entry is cleared with CLx and keyed again.

9.109 EEX CHS 319.109 -31 — the mass of an electronEEX 61. 06 — an exact power of ten needs no mantissa3 CHS-3.CLx0. — only X is cleared

Trailing zeros after the point are blanked, and answers outside 10⁻² to 10¹⁰ switch to scientific notation by themselves: 987654 ENTER↑ × shows 9.754604237 11.

It is not necessary to clear between problems. CLR clears the stack when you want a clean start.

Arithmetic

Key the first number, press ENTER↑, key the second, press the operator: the answer appears at once. Like an adding machine, + adds the last entry to what is already there and − subtracts it; × and ÷ do the same with the last entry.

12 ENTER↑ 3 +15.12 ENTER↑ 3 ÷4.1 ENTER↑ 3 + 5 + 7 + 9 +25. — no ENTER↑ after the first12 ENTER↑ 1.58 × 8 ENTER↑ 2.67 × +40.32 — a sum of products

After any answer a new number lifts the stack automatically, so a serial calculation needs ENTER↑ only once. Sums of products and products of sums fall out of the stack with no intermediate result written down.

ENTER↑ then + doubles a number; ENTER↑ then × squares it.

The stack

Four registers, X on the bottom and displayed. ENTER↑ copies X into Y and pushes the rest up, losing T. An operator combines Y and X, drops the stack and copies T into Z.

1 ENTER↑ 2 ENTER↑ 3 ENTER↑ 4x = 4, y = 3, z = 2, t = 1R↓ R↓ R↓ R↓review each register; the fourth press restores the stack512 ENTER↑ 9 x⇄ythe operands were the wrong way roundCLRall four registers to zero

Two exceptions to the automatic lift: a number keyed right after ENTER↑ or CLx overwrites X rather than pushing it. And R↓ after a string of digits terminates the number, so no more digits can be appended.

The trigonometric keys use T for scratch; keep nothing there across a sin, cos, tan or arc.

The memory register

One constant register. STO copies X into it; RCL brings a copy back, lifting the stack like a keyed number unless it directly follows ENTER↑, CLx or STO.

1 ENTER↑ 3 + 5 + 7 + 9 + STO25. stored, and still in X1 RCL ÷.04 — 4% of the total3 RCL ÷.129 RCL ÷.36

Computations never touch it. It is the only place, other than T, that a number survives a long calculation.

Powers, roots and logs

x^y raises the displayed number to the power that was keyed first: enter the power, ENTER↑, then the base. The power can be any number, and the result comes from logarithms, so the last digit may be off — 8^(2/3) shows 3.999999999.

7 ENTER↑ 2 xy128. — 2⁷17 ENTER↑ 1.05 xy2.292018319 — $1 at 5% for 17 years512 ENTER↑ 9 1/x x⇄y xy2. — the ninth root of 5122 ENTER↑ 10 xy100. — antilog₁₀ 21 ex2.718281828

ln and log take the logarithm of x to base e and base 10; e^x is the natural antilogarithm. There is no 10^x key: use x^y with 10 as the base. √x and 1/x replace x by its root or reciprocal.

The manual’s altimeter: 25000 ENTER↑ 30 ENTER↑ 9.4 ÷ ln × gives 29,012 feet from a barometer reading of 9.4 inches — “we suspect you may be on Mt. Everest”.

Trigonometry

sin, cos and tan take x in decimal degrees — there is no radians or grads mode. arc pressed first turns the next of them into its inverse, with the answer in decimal degrees.

30.5 sin.5075383628.7 CHS arc cos134.42700435 ENTER↑ 17 ENTER↑ 47 ENTER↑ 60 ÷ + 60 ÷ +35.29638889 — 35° 17′ 47″ as decimal degrees3 ENTER↑ × 4 ENTER↑ × + √x5. — a hypotenuse

Degrees-minutes-seconds and radians are converted by hand, as above. The functions use the T register while they work, so t is lost each time.

The manual’s problem 7 does polar and rectangular conversion the long way: x = R cos θ, y = x tan θ; θ = arc tan (y/x), R = y / sin θ.

Accuracy and errors

The four operators, 1/x and √x are correct to one count in the tenth digit. The transcendental functions are exact for some argument within a few counts of the one you keyed — 2 for logs and e^x, 3 for trigonometry, 4 and 7 for the two operands of x^y.

5 √x ENTER↑ ×4.999999998 — no ten-digit number squares to 55 ln1.609437912 — the log of some number in 4.999999998 … 5.0000000021 ENTER↑ 0 ÷0. blinking — press CLx

An improper operation — division by zero, √ or a logarithm of a negative, arc sin or arc cos of more than 1 — makes the display blink. Any number of magnitude 10¹⁰⁰ or more shows as 9.999999999 99; anything below 10⁻⁹⁹ shows as zero.

HP’s errata sheet lists the known wrong answers: e^x is off by up to one percent for 0.7030975114 (that is ln 2.02) and a few related arguments, and the arc functions misbehave for 0.0002 and right at 45°. Dropping the last digit of the argument works around the first; HP replaced calculators for owners who asked.

Reading more

The Operating Manual is 36 pages and worth reading whole: two sections on using the machine, sample problems with keystroke solutions, and a flowchart algorithm for evaluating any expression on a stack machine. The Errata Sheet is two pages. Both are at literature.hpcalc.org, along with the Math Pac and Surveying Pac of worked problems.

David Cochran’s The HP-35 Design: A Case Study in Innovation is the designer’s own account — the pMOS chips, the microcode, why 35 keys, and the day the first order for 10,000 arrived. The HP-45 and HP-65 that followed keep this keyboard and stack; their sheets are linked in the footer.

Keyboard drawn from the manual’s cover photograph. Every keystroke sequence and result here is either the manual’s own or was checked arithmetically.

Key & Function Index

49 entries · every key and every behaviour the manual describes, grouped by what it does · hover for details and the page

Arithmetic

9
+y + x; the stack drops
−y − x
×y × x
÷y ÷ x
ENTER↑ +double x
ENTER↑ ×square x
serial calculationa new number after an answer lifts the stack
sums of productsENTER↑ × … + … + as they occur
product of sumsENTER↑ + … × … ×

Entry, display & clearing

10
0 … 9 .digit keys and the decimal point
CHSchange sign
EEXenter exponent
CLxclear X
CLRclear the stack
displayten digits, then scientific notation
9.999999999 99overflow
blinking 0.improper operation
all decimal points littwo to five minutes of battery left
OFF – ONthe power switch

The stack

6
ENTER↑copy x into Y, lifting the stack
x⇄yexchange X and Y
R↓roll the stack down
stack dropafter any two-number operation
T and trigonometrysin, cos, tan and arc use the T register
lift after R↓a keyboard entry or RCL after R↓ raises the stack

Memory

2
STOstore x in the memory register
RCLrecall the memory register to X

Powers, roots & logs

8
x^yx to the power y — the exponent goes in first
√xsquare root of x
1/xreciprocal of x
lnnatural logarithm
logcommon logarithm
e^xnatural antilogarithm
10^xcommon antilogarithm — there is no key
πthe constant 3.141592654

Trigonometry

8
sinsine of x, in decimal degrees
coscosine of x, in decimal degrees
tantangent of x, in decimal degrees
arc sininverse sine, answer in decimal degrees
arc cosinverse cosine
arc taninverse tangent
degrees, minutes, secondsconverted by hand
radiansconverted by hand

Accuracy & limits

6
range10⁻⁹⁹ to 9.999999999 × 10⁹⁹
accuracy: arithmetic±1 count in the tenth digit
accuracy: log, ln, e^xcorrect for an argument within ±2 counts
e^x erratathe 2.02 bug
arc erratasmall arguments and 45°
temperature0 to 40 °C operating

Worked Example — a mortgage payment, with the stack shown at every key

the manual’s page 11 problem: thirteen keystrokes and no memory register
X (shown)YZT3000030000.———the principalENTER↑30000.30000——copied up.005.00530000——monthly rate: 6% ÷ 12×150.———principal × rate — the numerator, kept in the stack11.150——automatic liftENTER↑1.1150—360360.1150—writes over X after ENTER↑ENTER↑360.3601150the 150 has reached T1.0051.00536011501 + monthly ratexy6.022511501501.005³⁶⁰ — the power was keyed first; T copied into Z1/x.1660411501501 / 1.005³⁶⁰−.833951501501501 − that: the denominator÷179.86150150150150 ÷ .83395 — the monthly payment

The X column is what the display shows after each key; the manual prints the same table with the numbers rounded as here. Dashes are registers whose contents no longer matter. Numbers shown to four or five places are ten-digit values in the machine.

Find the monthly payment on a 30-year, $30,000 loan at 6% a year — 0.005 a month, 360 payments. The formula is

payment = principal × rate ÷ (1 − 1 / (1 + rate)^payments)

On a machine with one memory register and no parentheses this is a test of the stack, and the manual works it as a showpiece: the numerator is computed first and parked, the denominator built on top of it, and one ÷ finishes.

The numerator takes three keys and leaves 150 in X. Keying 1 then lifts it into Y automatically; ENTER↑ 360 ENTER↑ 1.005 push it to T. This is the whole trick of the example — the number you will need last is put in first and rides up the stack.

x^y is the key to watch. It raises the number in X (1.005) to the power in Y (360), so the power was keyed before the base. The stack drops and, as it always does on a drop, the 150 in T is duplicated into Z. 1/x and − finish the denominator, and every drop leaves another 150 behind: by the last line the stack is three copies of the numerator and the denominator.

÷ divides 150 by 0.83395 and shows the payment:

30000 ENTER↑ .005 × 1 ENTER↑ 360 ENTER↑ 1.005 xy 1/x − ÷179.8651575

The manual rounds the trace to 179.86; the full display is 179.8651575, which is what the closed-form formula gives to ten digits.

Two other page-9 figures for checking a machine: $1 at 5% for 17 years is 17 ENTER↑ 1.05 x^y = 2.292018319, and a growth from 926 to 1370 over seven years is 1.37 EEX 9 ENTER↑ 926 EEX 6 ÷ 7 1/x x^y = 1.057551118, a rate of 5.755%.