
Would you notice if your calculator was lying? Research says probably not - EndXA
https://cosmosmagazine.com/technology/would-you-notice-if-your-calculator-was-lying-to-you
======
jbarham
I've personally witnessed scammers using people's trust in calculators being
correct to cheat them.

It was at the Zambia-Zimbabwe border at Victoria Falls in 1999. We needed to
change USD into Zimbabwe dollars and were approached by an informal money
changer who ostentatiously did the calculation on his calculator in full view.
We knew roughly what the exchange rate should be and confirmed that the money
changer was using close to that on the calculator. However, the final result
of the multiplication on the calculator was fraudulently low. I thought the
result was "off" but given the relatively high exchange rate (this was pre-
hyperinflation), by the time I'd done the math in my head he'd taken our USD
and disappeared. Not a huge deal as we were only out by maybe tens of USD, and
we had to grudgingly admit that it was a very effective scam.

~~~
jlg23
> I've personally witnessed scammers using people's trust in calculators being
> correct to cheat them.

Whenever people shove a calculator into my face, I expect a scam. One of the
oldest, used a lot in restaurants: 50+0 (showing customer the zero) + one
coffee + ...

~~~
achow
Without calculator how would one calculate things in places like Zimbabwe or
any Mom-Pop place where there are no computers.

Apart from building 'trust', calculator plays important job of communication -
people in places like Zambia probably do not know how to write in numbers in
English (or are not very fluent) and also probably do not know how to say the
numbers in English.

Being traders however they would _recognize_ numbers readily, so they
communicate via calculator by punching out the numbers. That is, calculator is
used even when there is no calculation involved.

[Edit - Corrected the Roman numerals to English numbers]

~~~
username90
> Without calculator how would one calculate things in places like Zimbabwe or
> any Mom-Pop place where there are no computers.

People can calculate things in their heads, have you never bought something
where there wasn't a calculator?

~~~
carlmr
Also I feel like most people now own smartphones?

~~~
vectorEQ
nah, there's no computers or anything in africa :'D

------
perl4ever
On a related note, I was listening to a talk on information security the other
day when the speaker gave "balancing your checkbook" as an example of a
control/check on a process. And the funny thing was he didn't ask the audience
" _do_ you balance your checkbook?", he asked "how _often_ do you balance your
checkbook?"

Anyway, this is why Fermi estimation is worthwhile; I never understood why
people hate it or think it's pointless. Anything you don't check on for not
being absurd will eventually get absurd enough for you to notice.

Also, when it comes to phishing, the paragraph on why we can't train people to
recognize phishing ignores (as usual) the elephant in the room - corporate and
government environments are plagued with _legitimate_ emails that are full of
red flags, thus training people to ignore things like misspellings, weird
source addresses, and so on.

You'll never make any progress on a problem when you're studiously ignoring
the important factors...and assuming people balance their checkbooks.

~~~
foxyv
To add to this, even if you don't own a checkbook, keeping a tracking budget
with something like YNAB, Mint Wallet, or Personal Capital makes a huge
difference. I've caught so many weird charges on my credit card that way. It
also let me know when my bank decided to start charging a "Checking Fee" on
their "Free" checking account >_<

~~~
perl4ever
I have a bad habit of reading the agreements, and I can't stomach the ones for
services like that. Lots of stuff about power of attorney...

~~~
foxyv
Yeah that's part of why I don't use the account imports and do everything
manually. Plus it's pretty limited. For example, from YNAB:

"FOR PURPOSES OF THIS AGREEMENT AND SOLELY TO OBTAIN AND PROVIDE THE ACCOUNT
INFORMATION TO YOU AS PART OF THE SERVICES, YOU GRANT THE COMPANY A LIMITED
POWER OF ATTORNEY,"

------
maaaats
Reminds me of when I was TA for calculus at university. Lots of people got one
answer wrong, having done the calculations correctly. Found out roughly half
of the calculators gave the answer in a different than expected quadrant.

Found the picture: [https://imgur.com/a/Ve8DVXu](https://imgur.com/a/Ve8DVXu)

~~~
roelschroeven
Unless I'm missing something the right half of that picture is simply wrong,
not just another quadrant.

tan(-1/3 pi) = -sqrt(3), so arctan(sqrt(3)/-1) = arctan(-sqrt(3)) = -1/3 pi !=
1/3 pi

Why do they use sqrt(3)/-1 instead of simply -sqrt(3)? I have the impression
something else is going on. Are we seeing the last line of a multi-line
expression? Why do we see the bottom curls of the parentheses but not the top
curls?

~~~
papln
Despite convention, it is reasonable to consider sqrt(3) to have ambiguous
sign since inverse(square) is a multi-valued function (as is arctan). So you
can have arctan(-sqrt(3)) = arctan(sqrt(3)) = pi/3 (allowing for arbitrary
selection from multi-valued functions)

This is a problem in general with the design of calculators and single-result
algorithms in general.

The -1 might be needed to trigger the bug.

The Citizen calculator has a reputation for bad math:

[https://forum.kerbalspaceprogram.com/index.php?/topic/119681...](https://forum.kerbalspaceprogram.com/index.php?/topic/119681-stupid-
calculators/)

------
EndXA
The original study that's mentioned here:
[https://journals.plos.org/plosone/article?id=10.1371/journal...](https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0223736)

Abstract:

> Calculators are often unnecessary to solve routine problems, though they are
> convenient for offloading cognitively effortful processes. However, errors
> can arise if incorrect procedures are used or when users fail to monitor the
> output for keystroke mistakes. To investigate the conditions under which
> people’s attention are captured by errant calculator outputs (i.e., from
> incorrectly chosen procedures or keystroke errors), we programmed an
> onscreen calculator to “lie” by changing the answers displayed on certain
> problems. We measured suspicion by tracking whether users explicitly
> reported suspicion, overrode calculator “lies”, or re-checked their
> calculations after a “lie” was presented. In Study 1, we manipulated the
> concreteness of problem presentation and calculator delay between subjects
> to test how these affect suspicion towards “lies” (15% added to answers). We
> found that numeracy had no effect on whether people opted-in or out of using
> the calculator but did predict whether they would become suspicious. Very
> few people showed suspicion overall, however. For study 2, we increased the
> “lies” to 120% on certain answers and included questions with “conceptual
> lies” shown (e.g., a negative sign that should have been positive). We again
> found that numeracy had no effect on calculator usage, but, along with
> concrete formatting, did predict suspicion behavior. This was found
> regardless of “lie” type. For study 3, we reproduced these effects after
> offering students an incentive for good performance, which did raise their
> accuracy across the math problems overall but did not increase suspicion
> behavior. We conclude that framing problems within a concrete domain and
> being higher in numeracy increases the likelihood of spotting errant
> calculator outputs, regardless of incentive.

------
kevinmchugh
My cousin lived in rural Moldova from 2002-2004. She said at many markets,
vendors had both calculators and abaci. The vendors mostly totalled bills with
an abacus, because customers assumed the calculators were built to cheat them,
and they could follow the summing when performed on an abacus.

I wonder how it goes now, with more penetration of technology. What would a
calculation interface designed for verification look like? How could you build
a calculator app that's as trusted as an abacus?

~~~
CamelCaseName
If people are unwilling to do the math themselves, then it becomes a question
of what can you trust?

Something physical (an abacus, say) could potentially be manipulated by slight
of hand or optical illusion.

Something digital (an iPhone app) could silently do the same.

But what if you make it really easy for the user to do the math themselves, by
explicitly breaking down the calculation into a video of smaller steps where
the numbers move to show the calculation.

E.g. $19 + $25 + $28 on such an app:

19

25

+28

\---

10

20

14

+28

\---

10

20

10

20

+12

\---

30

10

20

+12

\---

40

20

+12

\---

60

+12

\---

72

~~~
catalogia
> _" Something physical (an abacus, say) could potentially be manipulated by
> slight of hand or optical illusion."_

That's true, but people often overestimate their ability to see through such
ploys. That's why the shell game
([https://en.wikipedia.org/wiki/Shell_game](https://en.wikipedia.org/wiki/Shell_game))
is thousands of years old and still going strong, across the world.

------
codeulike
_Would you notice if your calculator was lying?_

Well no, because thats the point of calculators, right? You trust them to get
it right but if its something important you run it through twice just in-case
you made a mistake.

Its like saying " _Would you notice if your coffee machine was adding a bit of
alcohol to your coffee?_ " . We're obviously not expecting mundane appliances
to do weird shit to us, and it would mostly be a waste of time and energy to
worry about things like that.

~~~
mkl
Calculators generally don't lie [1], but people do frequently make mistakes
pressing their buttons. Calculators' accuracy cannot be trusted, because
people's accuracy cannot be trusted. I think this experiment is really
simulating things like entering mistakes, and you _should_ pay attention and
notice when the numbers seem off. I just finished marking an exam question
that depended heavily on calculator work, and quite a few people need to work
on their calculator mistrust...

[1] Well, they do! But only things like small rounding errors that _usually_
don't matter.

~~~
codeulike
Dealing with operator error is different from having doubts about the accuracy
of an appliance. For quite a lot of things done on a calculator, trying to
judge whether the numbers seem 'right' is going to be difficult. A more
reasonable way to reduce operator error is to key the calculation through the
device a couple of times and make sure you get the same answer.

~~~
mkl
For purely keying errors, yes, but at least as big a problem are errors in
translating formulas to the calculator, like leaving out parentheses, and
these errors will probably just be repeated.

------
orionblastar
I had a calculator from 1985 a TI Scientific SLR solar powered. I did not know
it had a bug in the square root function. It was enough to flunk Algebra and
get a C in Statistics. I bought a more modern calculator later on when I went
back to college and learned the older one had a square root bug. I never
learned to do the math in my head, that would have been a neat trick.

~~~
nullc
Arbitrary precision sqrt in your head:

There are two ways to do it, one is via a process that looks like long
division, the other is via newtons method. I find the latter more useful
because I for me its easier to remember and even a single step of it gives a
pretty good result.

First guess get an initial guess. If you're good with base-2, you can crop off
half the number's binary digits. Otherwise, think of the number in scientific
notation and take the sqrt of the mantissa and halve the exponent.
Alternately, guess and check some nearby squares.

Now we apply newton's method using the derivative of sqrt() to refine your
guess:

guess_n+1 (x) = 0.5 __* (guess_n + x /guess_n)

That division is a PITA mentally, so don't be a chump-- keep your intermediate
results in rational form and keep multiplying up the denominator (the rapidly
bloating denominator also gives you an idea of how fast the precision of this
process increases-- very fast...).

Sometimes I get the sign wrong, but that or any other mistake I've encountered
obvious really fast.

~~~
SamReidHughes
That algorithm should be called square rootus horribilis.

Let's say we're square rooting a 3-4 digit number (or you can multiply/divide
out the exponent accordingly).

I recommend starting with the (x + .5)^2 = x(x+1) + .25 shortcut, multiplied
by 100, to be able to compute all multiple-of-five squares 5, 10, 15, 20, ...,
95. Then, from some multiple of 5 that is x, with k in {-2,-1,0,1,2}, use
(x+k)^2 = x^2 + 2xk + k^2 to tweak that to a particular perfect square. E.g.
42^2 = 1764 = 1600+160+4. Now, we're square rooting a number, so find the two
nearest perfect squares. Depending on how close you are to halfway between
them, and depending on if you're in the mood for that sort of thing, poke it
up by <0.25 to account for the quadratic factor of whatever you're square
rooting (because (x+.5)^2 is 0.25 less than the average of x^2 and (x+1)^2),
and then linearly interpolate between the two perfect squares. (You don't
really have to calculate both neighboring squares to interpolate, because the
difference is about 2x, and you probably already calculated that when you did
the 2xk calculation.)

So for example, what's the square root of (types randomly) 5719? Why, 56 is
7*8, so 5625 is 75^2. We're like, 95 more than that, out of 150? So, 75.6 and
change. Or more exactly, 94/151? That's like a +4% -.6% adjustment of .6 [1],
so maybe 75.62. Or even closer, you might use 94.2/151, sure.

[1] (a+n%)/(b+m%) can be approximated (a/b) + n% - m%, for small percentages.

------
perilunar
I bought a QAMA calculator to improve my estimating abilities, but then put it
in a drawer and forgot about it.

[http://qamacalculator.com/index.html](http://qamacalculator.com/index.html)

------
mindB
At the bottom of the article, it links to [1] as the original article. Could
we please update the link?

[1] [https://theconversation.com/would-you-notice-if-your-
calcula...](https://theconversation.com/would-you-notice-if-your-calculator-
was-lying-to-you-the-research-says-probably-not-126027)

------
pimmen
I work in data engineering, and we have multiple systems built by other people
talking to our pipeline. Just one of them being off by a little might send the
KPIs, the machine learning models and much more into a spiral where they end
up completely wrong.

This has unfortunately made me paranoid and uncertain of anything data or
calculation related. I cross check all my day to day calculations and,
annoyingly, my friends' calculations too. Any known remedies from people with
similiar problems would be appreciated.

------
flybrand
In 1998 my first job involved manually checking very large xl sheets for
government cases to confirm that the math was right.

If there are $s involved, someone will notice.

~~~
jmkni
Was the maths ever wrong on the Excel spreadsheets? What was the cause when
that happened?

~~~
nwallin
The fdiv bug in the Pentium x87 unit was discovered because someone was
manually checking their Excel spreadsheets.

[https://en.wikipedia.org/wiki/Pentium_FDIV_bug](https://en.wikipedia.org/wiki/Pentium_FDIV_bug)

~~~
jmkni
Waow, nuts/interesting.

Thanks

------
WalterBright
I recall this being researched in the 1970s. The calculators had to be off by
a factor of 2 before people became suspicious.

------
goatinaboat
Back in the day it was not uncommon to do important calculations twice, once
on HP and once on TI, just in case of bugs.

------
cellular
Once my friend wrote a program for an hp48sx that would add 1 to any
calculation. He tried to let people borrow his calculator.

~~~
copperx
Did he teach them how to use RPN before letting them borrow it?

~~~
cellular
Most of us geeks already knew rpn. He was an uber geek. :-)

