Should You Trust Your Calculator?
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- Опубліковано 6 січ 2024
- Calculators disproving Fermat's Last Theorem? Surely not? Let's take a look at examples of strange calculator behaviour!
Harvard's Near Misses:
people.math.harvard.edu/~elki...
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For those of you that are new here, hi there 🌞 my name is Ellie and I'm a Part III Mathematics Graduate from the University of Cambridge and current Astrodynamics Software Engineer! This channel is where I nerd out about maths, physics, space and coding so if that sounds like something you're interested in, click the subscribe button to follow along ☺️
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I actually had one time my calculator gave me a wrong number for a simple addition which was pretty nuts.
In computer science, this kind of behaviour has been known for a while one reason being finite memory in the calculators
Love this! I think you first point illustrates that pocket calculators can only display 10 significant figures, hence the missing fraction part after a 10-digit integer part.
I would be curious to know how accurate (in terms of significant figures) pocket calculators are these days.
Pocket calculators that show 10 significant figures have certainly been widely available, since the 1980s.
Not sure if you have heard of me, but I am one of the UKs Mental Calculations World Cup representatives. Most multiplication we do in the World Cup is 8-digits by 8 digits, for a 15 or 16-digit answer. When my friend was trying to break a multiplication world record, he came up with a different answer from the calculator system that was being used. It later turned out that his answer was correct, and the computer was wrong.
After you mentioned 5^14, and were about to mention a famous theorem, I thought it was going to be Fermat's LITTLE theorem, and we were going to have some disproof (of the calculator's result) to do with remainders.
On the Fermat's Last Theorem piece, I am going to take a look and see how different one 12th power is from the sum of the other two.
If this disproof were true, then of course we'd have a disproof for cubes and 4th powers.
Will also take a look at the 'near miss' page - am keen to see how close some of them get.
In the states, we often use Texas Instruments (TI) calculators. I personally use a TI-84 Plus CE, and it showed me the subtle differences for each of these calculations. I cannot speak to a Casio, but for my graphing calculator, it never displayed this strange calculator behavior
I was looking at this calculator today. The Maths Sorcerer uses one so I thought i'd check it out. Impressive, if expensive, piece of kit!
I was in Bletchley Park yesterday and saw a paper on display by Alan Turing from 1948 “rounding off errors matrix processes” I thought of this video by you when I saw that paper.
I want you to make videos on Feynman 's technique for solving mathematics
Like,Dirichlet integral by Feynman trick etc...
Hey cool video!
So I work in science testing labs, and per the standards (ISO) you are required to check lab calculations periodically to make sure the calculations (say dilution factors or Z scores) aren’t being distorted by the computer or calculator. You’d be surprised how often these calculations on excel can change over time.
I watched a video on this earlier today. IIRC it was a fairly involved calculation involving radicals and exponents. Some calculators turned out a result of '1' and others returned a result of '16'
In Excel youe get a 3 as last number. But if you get one decimal then you git 2.5 in the end of the answer
Great video! Thanks.
Thanks PROF this was pretty interesting and useful VIDEO.
There is an excellent treatment of problems with rounding etc called "Lies My Calculator and Computer Told me" by James Stewart (the "Stewart's Calculus" Stewart).
FWIW my HP35S actually works correctly on the 5^14 example.
You are super dedicated toward your video. I like it🤩🤗
Can you upload something Related to Physics? Waiting for it also...
Calculators, like any other modern gadgets are considered to be just tools
It's not what calculator you got
(ei. TI-84)
It's how you use it
(Garbage In-Garbage out) wrong input, wrong process, wrong output
Wrong process? perhaps, its the power of the processor, 1024 bit multi bus chips inside, but not for a pocket calculator.
the more wrong INPUT, the more time it will take exponentially (ei.PT) to process it
Trustworthyness pertains to reliability and accuracy to support something considered as valuable and as a modern conveniences as opposed to pen and paper approach, which is also a tool.
I know this video is probably about more complex calculators, but my wife and I just had to buy a calculator for our son in elementary school because we realized that the calculator on our iphones was not actually correct all the time. Not exactly sure what it was doing but sometimes it was just off by a little bit in certain more advanced calculations... really couldn't believe how often it was wrong though... enough so it was affecting his grade, since we were incorrectly correcting his answers on homework and study guides.
Hi, Ellie, the sum of last digits is 7 hance its not divisible by 2 but if sum were 6 then its divisible by 2. I wish you where my neighbors, you inspire me with some of your tips to study. I really appreciated your videos thank a lot!!
Ellie, i am wrong by sum of two being even then it's divisible by 2, I think I forgot the method am talking about.
I also use this calculator. I'm an Engineering student and my department is INFORMATION AND COMMUNICATION TECHNOLOGY (#ICT)Love from Bangladesh 🇧🇩 and love from #CSTU (#Chandpur_Science_and_Technology_University)
I always use a TI-83 Plus calculator, that’s my go to.
Thanks mam for making video.
I was wondering if you would react to an exam from Math 55 at Harvard. It's supposed to be one of the most difficult undergraduate math classes in the USA.
Strange Calculator Behaviour is my mental state in a non-calculator exam
calcuate the sum from n=1 to infinity of 1/n (underflow)
try to multipy with results far above hardware register width or variable length (overflow)
devide by very small values (mantissa not accurate)
store a big int to sint (sign bug)
in most cases assembler stepping explains the miracles.
symbolical calculation is better, but you can not do that in complicated designs.
you can use simple cases calculated symbolically and general mathematic knowlege to validate your numeric solution for a complicated design.
In fact, the Simpson's example can be quite quickly shown to be incorrect on several counts.
Both numbers on the left-hand-side are divisible by 9 (from 3987, 3+9+8+7=27, and from 4365, 4+3+6+5 = 18). However, the 4472 is not even divisible by 3 (let alone 9).
We could also disprove it by considering the last 2 digits of each number. We could raise the last 2-digits of each number to the power of 12, and ignore anything before.
Raising to the power of 12, we square twice, and cube once.
From 3987, 87^2 = 7569, and 69^2= 4761. To work out 61^3, we will square it first, discard anything infront of the last two digits, and multiply by 61 again. So 61^2 = 3721, and 21*61 = 1281. This means the last 2 digits of 3987^12 are 81.
For 4365, not so much effort needed. Any power of 4365 will end in 5, and any square number with a factor of 5 will end in 25. So 4365^12 ends in 25.
This means 4472^12 should end in (81+25)(mod100) = 06. However, any square number that does end in 6 must have an odd number in the 10s column. In this case 4472^12 ends in 16.
I guess the bottom line of this is that sadly calculators are geared up to the business world, and so the later digits of a number are not considered to be as important as the earlier ones.
Look like this example deviates in the 11th significant figure - doh!
This is simply a rounding or truncation error due to the underlying architecture; most calculators implement floating-point arithmetic using 10, 13 or 15 binary-coded decimal digits, with a 10-digit display, while programming languages generally use IEEE 754 64-bit binary floating point. If you perform the Fermat's Last Theorem example using an arbitrary-precision math library like GP/PARI, you get much better results:
(10:41) gp > 3487^12 + 4365^12 - 4472^12
%1 = -12902831680537941692248968175840903150415310
The moral of the story is to always be aware of the underlying number representations in use - which is really a question of computer science and engineering, rather than mathematics. ;)
I have a collection of both Texas Instruments (TI) and Casio calculators, though I will admit my TI collection is larger as I am from the United States where we almost exclusively use TI calculators. Are you aware of whether or not this problem persists in TI as well? Also do you know the actual cause of the issue? I am aware that there seems to be an issue, however, you never actually discussed what could be causing it, which intrigued me more than just the existence of the problem itself. Also are calculator errors like this one ever large enough that they have resulted in serious error or miscalculation because of it? i.e. is there any record in history of this causing something unexpected to happen?
I am first year student
Wow
Just wow!!
ellie good evenung ....as a maths lover i also want to be a mathematician like you ...and be cambridge graduate ..can you plzz make a video how you enterd cambridge means which exam we have to clear to be a student at cambridge, and what are the fees structure thereplzzzzzzzzzzzzzz didi
🥰🥰
Hello, I just seen your channel and Subscribed it...
Can you advise me which best Calculator to use it in Complex math?
I have the same one in the video, But I have problem with it ...
I waiting your replay on the message
GCSE Teacher: "Machines do not lie."
Numerical Analysis: "Hold my beer!"
I trust my CG50 with my whole life
From INDIA country,ಇಂಡಿಯಾ🎉🎉
I was thinking about these calculator weirdnesses… could it be that these large numbers fill up the calculator’s buffer so it can’t show a precise enough value to show that the numbers are different? Maybe some kind of buffer overflow type thing where it is forced to round the number due to its weak processing power
Not sure 😂
There are also other peculiar errors in calculators (CASIO-fx991CEX makes them). Several of them are discussed in the video "Why do calculators get this wrong? (We don't know!)" by Matt Parker. For instance, 11^6 ÷ 13 ≈ (156158413/3600)π. And many more in the comments like 11^6/16.999999999995. Certainly, the example with 5^14/2 is intriguing! It should indeed be added to the list of CASIO quirks. However, on the other hand, it doesn't seem to me that the example with a fulfilled Fermat's theorem is in the same category. It would be better to check the difference between these numbers (which is not equal to zero).
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Hey there! I like your content.. I am from India.. I love maths and physics ❤ but in ohr country.. Like for example if u love just physics ( or math) then too you have to compoulsary study all the subjects... I am very disturbed😢
Cám ơn bạn đã chia sẻ video ❤ đồng hành cùng bạn ❤ OK bạn rồi ❤❤❤❤❤
Thanks for sharing Ellie. I hope you can respond to my comments on your last convex hexagon video when you can. Thanks very much.
Thank you so much! Could you post your comment here? I can’t seem to find it! ☺️
@EllieSleightholm Sure if someone doesn't think of the trick of extending the triangles tomorrow the other triangles, can't younsolve just algebraically I think? Or I thoight of surrounding the hexagon with a rectangle and solving thst way..maybe by drawing triangles then within the rectangles or just with the rectangles...wouldn't thst be possible? Thanks
@@EllieSleightholm Hope you can respond agsin whenever you can.
❤maths ❤
I don't get it, what is this "strange calculator behaviour"?
why does it give an answer ending in 3 and also what did you mean by near misses?
I'm sorry if this is ignorant/dumb of me to ask but I really don't get it.
It's just a result of floating-point arithmetic. The behavior comes from a truncation error where the machine doesn't have enough bits to represent the number and instead rounds the number to one it can represent (in this case, one order of magnitude larger than the true value).
The "near-misses" refer to two integers that, raised to some power, ALMOST equal another integer raised to that power. An example from the linked from Noam Elkies' page on near misses (linked in video description): 3472073^7 + 4627011^7 = 1.00000000000000000000036... · 4710868^7
i see, thanks@@nimzodragonlord
Small angles, roots, natural logs, exponents, multiplication of 4digit numbers etc.., and even matrices and almost all engineering problems need calc. You are doomed without it my professor used to kick students out of class without it in my 1st year of engg as you will need to get used to it in earlier years of engg, if you are pursuing STEM its highly recommended to get used to it as it'll play crucial part through out the course.
Nice meet you sister love from India I want to become like you 😊😊
first person?
May be you calculator rounded 2.5 to 3
Second ❤
Can you Marry me
Calculate the odds and send her a mathematical proof of why she should marry you...duh..
@@leif1075
So, I will spend all my life in vain lol
@@assa1843 whybdo you say that?
I guess fermat's last theorem proof was a billion dollars problem..
I think it's pathetic that calculators are even allowed 🤮