Would you notice if your calculator was lying? Research says probably not
cosmosmagazine.com
cosmosmagazine.com
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.
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 + ...
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]
People can calculate things in their heads, have you never bought something where there wasn't a calculator?
And as mentioned, calculator I have seen being used as a communication mechanism (You ask what's price? Seller taps on the calculator and shoves it under your nose without uttering a single word : 72.50)
The scam I talked about only works when adding up larger number of items (like a table of 4 at a restaurant) but there it is pervasive.
With honest traders I never had a problem, neither in Africa nor in South America: Calculator on the table, reset in front of me, slowly, step by step, add up the items so I have time to punch it into my own calculator/phone.
Don't know where you get this impression from; based on my first-hand experience, it is very likely wrong. If you are referring to people who are illiterate in English, fair enough (although even then they most likely can say the numbers in English). But then you would need to check the rate of illiteracy in Zambia (and their language of business) before making a blanket statement like above
Imagine Germany, Russia, Indonesia, China, S.Korea...
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.
"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,"
Found the picture: https://imgur.com/a/Ve8DVXu
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?
The parantheses are just how it looks on that model, I tried inputing the exact same sequence on multiple calculators.
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...
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.
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?
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
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) is thousands of years old and still going strong, across the world.
20 + 25 + 30 - 3 = 72
Works well with multiplication too.
12 * 13 = 10 * 13 + 2 * 13 = 130 + 26 = 156.
or
19 * 15 = 20 * 15 - 15 = (10 * 15) * 2 - 15 = 285
There's no simple rule I know for multiplying by 19, but for 20 you just add a zero and double it. It's all about transforming the numbers into ones that are easier to work with in your head, and just accounting for your transformations at the end. Even if you forget to account for your padding, you're still in the right ballpark and have a rough idea of what the answer should be near.
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.
[1] Well, they do! But only things like small rounding errors that usually don't matter.
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.
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.
[1] https://theconversation.com/would-you-notice-if-your-calcula...
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.
If there are $s involved, someone will notice.
Easier items - where rounding resulted in paper errors were common.
Thanks