Why is x^2 shifted?



#6

I often wonder why HP (and many other brands) choosed many times to make x^2 a shifted function? In most formulas I see you´ll find a lot more squared variables than square roots. As example only one simple formula used a million times each day around the globe c = SQRT(a^2+b^2). I love the 33s for having a dedicated x^2 key. Is there some deeper going logic behind having x^2 shifted?


#7

Interestingly, the 300s has X^2, x^3, x^n and SQRT as direct keytop functions.

#8

It's designed to send us nuts!, just like in one of my recent blogs

Shifted 1/X, X^2, Log10, EXP and ENG keys are my pet hates. And don't get my started on the useless dedicated HYP key!

Dave.

Edited: 24 Sept 2009, 7:03 a.m.


#9

actually I'd like to know your issues with the dedicated HYP key - being curious about pros/cons of such implementation.

I can see cases where accessing the hyperbolics would be convenient. It's just one key to launch 6 functions (or more).

I just implemented it on the 41Z. Point in case is the user keyboard is full with "regular" functions (non-user is the real-numbers 41, user is the COMPLEX 41). HYP saves six key assignments and also presnts a more logical arrangment.

Best,
ÁM


#10

Quote:
HYP saves six key assignments and also presnts a more logical arrangment.

And both people who use hyperbolic functions regularly will appreciate the saving ;-)

- Pauli


#11

Quote:
And both people who use hyperbolic functions regularly will appreciate the saving ;-)

*grin*

Yes, it's a simple matter of need. What percentage of typical sci calc users would use Hyperbolic trig functions on a regular basis?

My guess is it wouldn't even reach a single percent.

Infinitely more sensible to make HYP a shifted function.

Dave.

#12

Don't forget the venerable HP-45, which was the first HP pocket calculator to feature the x^2 function. It has a dedicated x^2 key and the sqrt(x) function is shifted. I always have an HP-45 handy for routine calculating.

#13

Quote:
I often wonder why HP (and many other brands) choosed many times to make x^2 a shifted function? In most formulas I see you´ll find a lot more squared variables than square roots ... Is there some deeper going logic behind having x^2 shifted?

I agree that x2 is very commonly used, and that an unshifted key is convenient. However, lack of available keyboard space is a constraint. Since square root and x2 are inverse functions of each other, square root is normally unshifted, with x2 being the second function.

x2 can also be performed in RPN by [ENTER][*], which -- along with a limited keyboard -- may explain why the HP-35 didn't have it.

Quote:
As example only one simple formula used a million times each day around the globe c = SQRT(a^2+b^2).

This, of course, can be performed by a rectangular -> polar conversion, as well.

-- KS

Edited: 24 Sept 2009, 10:30 a.m.


#14

Yes, keyboard layout is a tough discussion.

Considerations:

1) Can the function be done another way? x^2 is also ENTER X, so making it a shifted function costs the same keystrokes as ENTER X. Perhaps better to save a key press and make x^2 primary then and save a keypress for the user.

2) How often is the function used? Is SQRT used more often that x^2? If so, perhaps it deserves a primary key.

#2 applies to the LOG/10^x and LN/e^x discussion. Seems as if the natural log functions occur most often and if so, these "ought" to be given a primary key position over the common functions.


#15

Quote:
#2 applies to the LOG/10^x and LN/e^x discussion. Seems as if the natural log functions occur most often and if so, these "ought" to be given a primary key position over the common functions.

Again, the Hp-45 did exactly this with ln and e^x as separate primary keys with log and 10^x as shifted keys. It should be pointed out that the much maligned HP-33s also has dedicated primary ln and e^x keys.

And while we're on the subject of primary keys, I've always liked having separate STO and RCL keys on my basic non-menu calcs. Again, both the HP-45 and HP-33s have these. Unfortunately, I have an early HP-33s with the invisible decimal point, so I don't use it very much.

#16

Quote:
Yes, keyboard layout is a tough discussion.

Oh yes! E.g. assume a small RPN calculator featuring the following primary keys:
0 ... 9 . E ± + - * / ENTER Delete
STO RCL x<>y Rv Sigma+
XEQ R/S SST BST ON/OFF
Enough space shall be left for secondary (i.e. shifted) functions. Three (3) primary keys are to be filled yet. What would you put on them?

#17

Quote:

Oh yes! E.g. assume a small RPN calculator featuring the following primary keys:

0 ... 9 . E ± + - * / ENTER Delete
STO RCL x<>y Rv Sigma+
XEQ R/S SST BST ON/OFF
Enough space shall be left for secondary (i.e. shifted) functions. Three (3) primary keys are to be filled yet. What would you put on them?

Wouldn't a "small RPN calculator" have none of that programming stuff? Like the RPN Scientifc 4-Banger dream calc some time back?

The difference really comes down to a calculator key layout optimised for programming, and one optimised for manual calculations.

As someone who rarely runs programs, I prefer a layout designed for manual calculations, so I believe all the program keys should be shifted.

But if I only had three keys left I'd put 1/x, X^2 and SQRT

And of course an ON/OFF key is the most useless thing ever invented for a low power calculator.

Dave.


#18

Quote:
The difference really comes down to a calculator key layout optimised for programming, and one optimised for manual calculations.
As someone who rarely runs programs, I prefer a layout designed for manual calculations, so I believe all the program keys should be shifted.

But if I only had three keys left I'd put 1/x, X^2 and SQRT

And of course an ON/OFF key is the most useless thing ever invented for a low power calculator.


Well, I didn't tell you SST, BST, and ON/OFF are multi-purpose keys like on the 42S. SST and BST browse catalogues as well, and ON/OFF serves also as EXIT. So there are only XEQ and R/S left being primary program keys. My ideas for the three free keys were 1/x, y^x and PI so far, so we agree on one key already. Maybe e^x and LN are a better choice than y^x and PI ?

Edited: 25 Sept 2009, 4:12 p.m.


#19

I would definately leave PI for a shifted key.

For primary keys, 1/X gets my vote as well, but it's a hard choice between Y^X, SQRT and X^2. I guess I'd go for Y^X and SQRT, leaving X^2 for a shifted function.

It almost pushes me to use the 3 keys for SIN, COS, TAN so their inverses are on the proper shifted key. It's always a matter of tradeoffs in the limited resources world.


#20

Quote:
I would definately leave PI for a shifted key.

I like the Casio method of putting PI on the EXP key.

If you haven't already entered a digit then pressing EXP gives you PI without using SHIFT. But once you enter a digit then EXP becomes the active key.

Dave.


#21

Quote:
I like the Casio method of putting PI on the EXP key.
If you haven't already entered a digit then pressing EXP gives you PI without using SHIFT. But once you enter a digit then EXP becomes the active key.

HP's traditional way was using the EEX key to enter 1*10^ directly if no input was pending. E.g. 225 SQRT E 4 * will result in 150,000. Sadly, this way was abandoned recently with the 35s.
#22

Quote:
I like the Casio method of putting PI on the EXP key.

If you haven't already entered a digit then pressing EXP gives you PI without using SHIFT. But once you enter a digit then EXP becomes the active key.


Hi, Dave --

I have a 1981 Casio fx-3600P with that feature. (Note: pi is always accessible using the shift key.) I thought it was a clever means of dual-purposing a key.

On the newer fx-115MS, EXP and pi are also on the same key, but pi always requires the shift key. A leading "1" is not necessary; EXP 6 will give "E6" representing 1,000,000 -- more HP-like in that sense.

I agree with Walter regarding the application of the old Casio method to HP's. Not only does it preclude quicker entry of power-of-10 numbers, it is inconsistent with RPN principles and could cause ambiguity in programs.

-- KS


Edited: 27 Sept 2009, 12:40 a.m.

#23

Quote:
#2 applies to the LOG/10^x and LN/e^x discussion. Seems as if the natural log functions occur most often and if so, these "ought" to be given a primary key position over the common functions.

I find it interesting that the 32s has e^x and ln as primary functions (with 10^x and log10 shifted), but the 42s has log and ln (with 10^x and e^x shifted). Which came first, and why the change?


#24

They were both introduced in 1988. I suspect the design differences are evidence that the HP42S was not related to the HP32S in any way except use of Saturn and external appearance. Other than that, these two machines seem to come from two rather different design philosophies with little in common between them. With respect to the problems they were designed to solve, there is much more in common between the HP-15C and the HP42S than there is between the HP-15C and the HP32S, or between the HP32S and the HP42S.


#25

Q:

Quote:

I find it interesting that the 32s has e^x and ln as primary functions (with 10^x and log10 shifted), but the 42s has log and ln (with 10^x and e^x shifted). Which came first, and why the change?

A:

Quote:
I suspect the design differences are evidence that the HP42S was not related to the HP32S in any way except use of Saturn and external appearance.

"not related ... in any way"? That's a rather sweeping -- and wrong -- conclusion based on just one difference in key layout.

Use of the Saturn microprocessor is a significant similarity because that means identical mathematical algorithms (for functions that are common to both). The virtually-identical appearance resulted from the use of identical packaging and basic layout, as well as a design based on one shift key and menu-based access to many functions.

Of course, the function set, display, programming paradigm, and price were quite different.

As for the minor difference in top-row keys: The HP-42S has an unshifted up-arrow key (convenient for menu and program scrolling), bumping XEQ to the top row and leaving one fewer key for transcedental math functions. This also explains why yx was shifted. (Walter B. correctly pointed out that the layout of these keys are identical to those of the HP-41.)


Back to the original poster's question: The square-root symbol (x inside a radical) can be difficult to discern in small font. That may explain why square root, instead of x2, is typically unshifted.

I suspect that aesthetics and consistency of structure played a large role in some of the layouts:

HP-15C:

UNSHIFTED: sqrt ex 10x yx 1/x
SHIFTED: x2 LN LOG % %CHG

HP-32S:

UNSHIFTED: sqrt ex LN yx 1/x
SHIFTED: x2 10x LOG % %CHG

HP-42S:

UNSHIFTED: 1/x sqrt LOG LN
SHIFTED: yx x2 10x ex

The following should be noted:

  • On the HP-15C, the three leftmost pairs are inverse functions, and the three "exponentiated to power x" functions are side-by-side, forming a group.

  • On the HP-42S, the three rightmost pairs are inverse functions. Four exponentiated functions are side-by-side, as are LOG and LN. These also form groups.

  • The HP-32S is perhaps optimized for convenience; LN and ex-- which are used more than LOG or 10x -- are unshifted on two of the five keys.

-- KS


Edited: 26 Sept 2009, 10:10 p.m.

#26

Quote:
I find it interesting that the 32s has e^x and ln as primary functions (with 10^x and log10 shifted), but the 42s has log and ln (with 10^x and e^x shifted). Which came first, and why the change?

The 42S inherited the top key sequence of the 41 (and added XEQ to this row). The 32S had the standard top key row of the scientific Pioneers (20S, 21S, 22S). So it's obvious which came first.
#27

Gene:

Quote:
Can the function be done another way? x^2 is also ENTER X, so making it a shifted function costs the same keystrokes as ENTER X. Perhaps better to save a key press and make x^2 primary then and save a keypress for the user.

I sort of remember that some earlier machines did x^2 using logarithms as a special case of y^x with the result that

1. x^2 was less accurate than x X x, and

2. x^2 was slower than x X x

Do you remember anything like that?

Palmer

#28

Using ENTER^, X will lose the contents of the T register... which may be an important consideration for some (programs, of course).

#29

That's a good point, Frank. I think you're suggesting that they put x^2 and sqrt(x) on the same key, make sqrt(x) shifted and x^2 unshifted, rather than the other way around.

I wrote a long note on HP's support site shortly after getting my 50g giving some thoughts on the keyboard. I recognize the difficulty in changing such things, but it's still helpful to speculate I think.

More specifically, I noticed that several unshifted keys (TOOL, VAR, NXT, and SYMB) and more than a dozen shifted keys do nothing more than change the soft-menu context. Many others (like APPS, MODE and shifted functions CHARS, EQW and CAT) just start applications. I'd put nearly all of these functions behind a single "Menu" key. Some functions like EQW are really key from a marketting perspective so they might still be on the keyboard.

I'd also change it so that pressing a shift key would change the softkey labels to indicate the new function (shifted soft keys already PERFORM different functions, but the labels don't change to indicate what the new function is). Finally, I'd do two rows of 5 soft keys instead of 1 row of 6 (and two rows of labels on the screen to match). Note that this would allow an extra character in the softkey label that appears on the screen

By adding this emphasis on soft keys, you could now remove a ton of clutter. You'd probably need just one shift key, which would make room for a large ENTER key. With 20 functions available in each menu (10 unshifted and 10 shifted) you wouldn't need the NXT key at all. In the unlikely event that a menu set required more than 20 functions, you could add NXT and PREV soft functions.

Dave

#30

The 27s gives "equal time" to everybody: SQRTx,x^2,e^x,LNx,10^x,LOGx,y^x,1/x are ALL shifted. Only the trig functions are arranged in unshifted/shifted groupings.


#31

Hi Martin,

Actually that isn't completely true. If you are working from the MAIN menu position, then all of the blue "shifted" functions in the top row are direct press (UNshifted) as there are no softkeys active!

Maybe you already knew this but if not, it is useful to know.


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