Why do even the simplest calculators feature a square root key?
05-05-2018, 06:39 AM
Post: #1
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
Why do even the simplest calculators feature a square root key?
While this forum is clearly focused on midrange and high end calculators, there are also the ubiquitous simple four function devices that offer nothing but the four basic arithmetic operations. Even HP offers these, e.g. the EasyCalc 100, so I take the liberty of posting this in the General Forum. ;-)

While it is clear that the typical user of such a device does not care about logarithms, powers or trigonometric functions, and adding a few numbers or calculating a markup essentially is all that it is used for, I always wondered why most of these simple devices still have a √ key. I have never seen anyone calculating a square root on such a calculator, so I wonder why this function exists at all. Where and why does the typical user of a calculator that does not feature anything beyond +, –, × and ÷ need a √ key? Or is it simply because this function is so easy to implement that it is included only because of that?

Dieter
05-05-2018, 07:52 AM
Post: #2
 pier4r Senior Member Posts: 2,111 Joined: Nov 2014
RE: Why do even the simplest calculators feature a square root key?
Throwing ideas: could be due to the Pythagorean theorem that needs square roots to compute distances? The need to compute distances may always arise. Although as Dieter I never do anyone using the square root on such calculators.

Wikis are great, Contribute :)
05-05-2018, 10:52 AM
Post: #3
 Gamo Senior Member Posts: 713 Joined: Dec 2016
RE: Why do even the simplest calculators feature a square root key?
This is my Citizen FS-50WH II it came with Metric Conversion Table also have square root. Maybe someday in a life time user might need to calculate Triangle or area that need to use this function.

Gamo

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05-05-2018, 11:35 AM (This post was last modified: 05-05-2018 11:55 AM by sasa.)
Post: #4
 sasa Member Posts: 116 Joined: Apr 2018
RE: Why do even the simplest calculators feature a square root key?
(05-05-2018 06:39 AM)Dieter Wrote:  Where and why does the typical user of a calculator that does not feature anything beyond +, –, × and ÷ need a √ key? Or is it simply because this function is so easy to implement that it is included only because of that?

The answer may lie in todays dedicated IC standardization. While in latest 60's and early 70's, especially at the dawn of electronic calculators in early '60s making simple calculator was quite demanding and expensive task, today is required only low powered COB with appropriated dedicated LCD driver, keyboard scanner and few registers to produce a result. Probably minimum except elementary operation and percentage calculation is calculating square root and having one additional memory storage.

For instance, one of these dedicated ICs is EMPCD081A:
http://www.emc.com.tw/eng/database/Data_...CD081A.pdf

On calculator manufacturer is only to make a design and choose what functionality to support.
05-05-2018, 12:06 PM
Post: #5
 SlideRule Senior Member Posts: 1,325 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
Perhaps an artificial partition relating to business, scientific, etc category by inclusion/exclusion of mathematical functions, operations, etc. It has the appearance of marketing as much as technology. Just a thought.

SlideRule
05-05-2018, 09:17 PM (This post was last modified: 05-05-2018 10:12 PM by Vtile.)
Post: #6
 Vtile Senior Member Posts: 406 Joined: Oct 2015
RE: Why do even the simplest calculators feature a square root key?
Because you can simplify great many advanced and even complex problems of of engineering and mathematics to the simple geometrical problems and most simple geometrical problems can be easily solved by Pythagorean formula of c^2=a^2+b^2. If the problem is really bad you can then solve the determinant discriminant of the parabola.

That said I still prefer my 50g.
05-05-2018, 10:19 PM
Post: #7
 Gene Moderator Posts: 1,282 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
At this year's HHC in September, I plan to cover the Unisonic brand of calculators. They eventually had four "scientific models" but a hoard of them that had basic functions plus square root or square or 1/x. More to come.
05-05-2018, 11:31 PM (This post was last modified: 05-06-2018 12:06 AM by hibiki.)
Post: #8
 hibiki Junior Member Posts: 24 Joined: Sep 2016
RE: Why do even the simplest calculators feature a square root key?
Sometimes I use a simple calculator with square root function to calculate the absolute value of electrical impedances (polar form) when in rectangular a+jb format. It uses the pythagorean formula. Also it is suitable to calculate the ressonance frequency of a LC filter f=1/(2.PI. SQR(LC)) PI is typed 3.14159.
If you wish to have an idea of the length of sides of an squared area, the square root function is always welcome. I use a lot to calculate the diameter of copper wire from its section circular area.
Another use of square root function useful for me is to calculate the impedance ratios of output audio transformers and the voltage ratios n = SQR (Zp/Zs).
Some basic electrical formulas use square root function: I = SQR (P / R) V = SQR (P x R)
In electronics, the most of formulas do not need a scientific calculator, but a simple calculator with square root is enough.

I can think also for calculation of quadratic functions roots, as is needed to calculate the SQR (b^2-4ac) as a part of the formula.
Finally, with the square root function is possible to do the cubic root calculation of a number, using few iterations steps.

My calcs: HP12C, HP15C, HP17BII+, HP20S, HP20B, HP32SII, HP34C, HP35S, HP42S, HP48GX.
05-06-2018, 12:51 AM
Post: #9
 Carsen Member Posts: 204 Joined: Jan 2017
RE: Why do even the simplest calculators feature a square root key?
Woah. A key with a + and = sign? Very interesting!
05-06-2018, 03:20 PM
Post: #10
 Maximilian Hohmann Senior Member Posts: 1,018 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
(05-05-2018 11:50 PM)Mike (Stgt) Wrote:  Do they?

And regarding the original question my guess is that a lot of these simple calculators are used to calculate sales prices. Stuff which is either sold by area units (e.g. wallpaper or floor panels) or length and width needs to be converted one way or the other, for which a square root key can be useful.
05-06-2018, 07:18 PM
Post: #11
 Carsen Member Posts: 204 Joined: Jan 2017
RE: Why do even the simplest calculators feature a square root key?
(05-06-2018 02:42 PM)Mike (Stgt) Wrote:  If you don't mind that it is not a RPN machine you may give it a try.

I will try it. I'm rather curious.

In regards to the topic, I am currently required (sadly) to use a basic calculator for my accounting classes so far. So if I run into a use for the square root key there, I'll let you know.
05-06-2018, 07:26 PM
Post: #12
 AndiGer Senior Member Posts: 438 Joined: Oct 2015
RE: Why do even the simplest calculators feature a square root key?
HP-10 is not RPN as well if you don't mind
05-06-2018, 07:30 PM
Post: #13
 Maximilian Hohmann Senior Member Posts: 1,018 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
(05-06-2018 07:26 PM)AndiGer Wrote:  HP-10 is not RPN as well if you don't mind

05-06-2018, 08:10 PM
Post: #14
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
(05-05-2018 06:39 AM)Dieter Wrote:  I have never seen anyone calculating a square root on such a calculator, so I wonder why this function exists at all.

First of all thank you very much for your replies so far.

Yes, it is obvious that there are useful applications for a √x key on a calculator. But for me the essential question is this:

Why is it the √ function and not another one that might be even more useful? For instance, if I were to design a simple calculator and I had to choose one single additional function, it would be the power function yx. This is far more useful and it can even be used for calculating roots. Or, looking at the typical user of such calculators, why not add a percent function instead of a rarely used √x ? Is there a technical reason? I don't think that the typical user of such a calculator has to deal with the Pythagorean theorem very often. ;-)

Dieter
05-06-2018, 09:07 PM
Post: #15
 rprosperi Super Moderator Posts: 5,378 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
(05-06-2018 08:10 PM)Dieter Wrote:  ...if I were to design a simple calculator and I had to choose one single additional function, it would be the power function yx. This is far more useful and it can even be used for calculating roots. Or, looking at the typical user of such calculators, why not add a percent function instead of a rarely used √x ?

It seems lots of folks believe that since they've never seen a low-end calculator user actually use the square root button, this somehow means that it wasn't used. Personally, I don't watch folks using low-end machines closely enough to discern if they use that function or not.

I believe it's included on such low-end machines simply to make them appear more sophisticated (and therefore justifiably more expensive) than the nominal 4-banger. Also, it takes very minimal efforts to test and document this function.

Though technically you are right that the power function is more useful and flexible, these aren't key concerns for such a machine, and only the tiniest fraction of such machine's users even know what it's for. And again, explaining and testing this takes much more effort. And as for using power function for roots, this is something even many high-end machine users don't know/do.

Finally, I agree that percent seems to be far more generally useful, and have often wondered why it isn't included on every calculator. I imagine any person that purchases any calculator would need, and be able to use percent sooner or later, but in any case more often than a square root.

--Bob Prosperi
05-07-2018, 01:06 PM
Post: #16
 Accutron Member Posts: 111 Joined: Aug 2014
RE: Why do even the simplest calculators feature a square root key?
In modern times, a basic electronic calculator is a piece of dollar store trash that the tech-illiterate use to balance their checkbook, but in the 1960s they were serious business. Early adopters of calculators included scientists and engineers, which had come to expect a square root function from their slide rules. While implementing square (and cube) roots on a slide rule is extremely simple, it was quite complicated to do it with discrete diodes and transistors. In the late 1960s, the inclusion of a square root function represented a cost increase measured in hundreds of dollars. The square root function became associated with advanced and costly scientific functionality before true scientific calculators became prolific.

To give you some actual numbers, the earliest machine I have which includes a square root function is a Busicom 162. It was manufactured in 1967, and retailed for 298,000 yen, or about $830 in 1967 dollars (the yen/dollar exchange rate was fixed at 360:1 throughout the 1960s). The price difference between the 162 and the root-less model 162C was 38,000 yen or$105, approximately a \$500 price difference when adjusted for inflation.

When the price of hardware imploded in the early 1970s, square root was implemented in some commodity calculator chips from companies like TI and Mostek, and often became a differentiator between high and low end four-bangers in the same series. Companies would offer multiple iterations of the same machine, with and without a square root function. Casio in particular was fond of including it right in the calculator's model name, with models like the ROOT-121A (which actually used the square root symbol in its model name) and ROOT-8S. By this point, the cost of square root was measured in tens of dollars. The majority of four-bangers still did not have a square root function, and the inclusion of square root was still a marketable feature, even as true scientific calculators were transitioning from the HP-35 price point to the TI-30 price point in the mid-70s.

By the early 1980s, square root had become a value-added feature with a negligible production cost increase, and was simply included to avoid losing a sale to another calculator with square root. Forty years later, they have become a tradition.
05-07-2018, 02:07 PM
Post: #17
 Egan Ford Member Posts: 172 Joined: Dec 2013
RE: Why do even the simplest calculators feature a square root key?
(05-07-2018 01:06 PM)Accutron Wrote:  By the early 1980s, square root had become a value-added feature with a negligible production cost increase, and was simply included to avoid losing a sale to another calculator with square root.

This was my thought as well. If your competitors have it, then you need it. If your competitors do not have it, you still need it to differentiate (recognize or ascertain what makes (someone or something) different, not that other kind :-).
05-07-2018, 04:19 PM
Post: #18
 Vtile Senior Member Posts: 406 Joined: Oct 2015
RE: Why do even the simplest calculators feature a square root key?
... But it had a reason to be there at the first place, so the relative monetary value doesn't matter in that example. Another point is that if you only have general x^a function and when you do need sqrt(9) you must you need to type: [9][^][0][.][5][accept] or something on those lines, while if you have dedicated square root in a key then that procedure contains two and much more economical calculation.

Lets take an example. I will calculate sqrt(9) in HP50g in RPN mode, I do how ever write the unnecessary zero at the front of decimal point.

Case 1 - No dedicated square root is used.

Key Travel No.Press
[9] 0 mm 1
[Ent] 35mm 2
[0] 77mm 3
[.] 91mm 4
[5] 113mm 5
[y^x] 155mm 6

Case 2 - Dedicated square root is used.

Key Travel No.Press
[9] 0 mm 1
[y^x] 45mm 2

The case shows ~70% reduction in finger travel and ~66.5% reduction in key presses. I would say that is pretty significant. This also underlines the importance of the real estate analysis of the keyboard map when engineering such as a calculator keyboard.
05-07-2018, 05:01 PM (This post was last modified: 05-08-2018 05:59 AM by pier4r.)
Post: #19
 pier4r Senior Member Posts: 2,111 Joined: Nov 2014
RE: Why do even the simplest calculators feature a square root key?
(05-06-2018 09:07 PM)rprosperi Wrote:  I believe it's included on such low-end machines simply to make them appear more sophisticated (and therefore justifiably more expensive) than the nominal 4-banger. Also, it takes very minimal efforts to test and document this function.

This is actually a very good hypothesis. As even the late mechanical 4 function calculators started to implement the square root function. (And there one can also see that the work done is a lot)

Wikis are great, Contribute :)
05-07-2018, 11:16 PM (This post was last modified: 05-07-2018 11:18 PM by hibiki.)
Post: #20
 hibiki Junior Member Posts: 24 Joined: Sep 2016
RE: Why do even the simplest calculators feature a square root key?
Dieter, aside the marketing aspects explained above, I think I found another clue for square root function present in simple calculators.
I made a research in the book "Inside Your Calculators - Gerald R. Rising - Wiley 2007". This books presents the algorithms used in calculators.
In synthesis, once you have implemented the four arithmetic basic operations and have some free registers memories, to implement the square root function is very easy using Newton´s Method (sucessive aproximation) that is a short routine and fast to find the root. The implementation of Integer Power is easy too, but Rational powers is not so easy to implement. The algorithm described in the book for rational powers uses the square root function, and is a bit longer routine. So in my understanding based on this book, the square root function precedes the power function, and is not dependent of nothing more than the basic operations division, addition and compare.

My calcs: HP12C, HP15C, HP17BII+, HP20S, HP20B, HP32SII, HP34C, HP35S, HP42S, HP48GX.
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