EEVblog 1479 - Is Your Calculator WRONG?

Is your calculator WRONG? It could be!
Looking at the issue of implied multiplication and how it can affect your calculations. Why do some Casio and TI calculators give a different result to others? And why do they differ from your phone calculator, google calculator, or Wolfram Alpha?
How Order of Operations matters.

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[KarlAdamsAudio]

The Casio adding extra parentheses for clarity, it's also a programming practice I highly recommend - you may know your chosen language's operator precedence rules like the back of your hand, but parentheses are free (in a compiled language, at least) and you don't want your code's intent to be apparent only to a select few.

[mattb]

I recall making many mistakes with a calculator due to having trust in the parenthesis logic..that was some 25 years ago. Same with excel. I basically trained myself to use lots of parenthesis to avoid order of operations errors.

[Gameboygenius]

My impression has always been that the reason for giving implicit multiplication a different priority is not for use with parentheses, but for use with variables. If you type 1/2x it's more likely that you meant 1/(2x) than (1/2)x.

[koharaisevo]

This is why I always use exra parentheses to make it absolutely clear how the order of operation goes.

[markc]

If you look at it as 6 / 2(x+y), you would not do the 6 / 2 before the 2(x+y) . The same rules apply whether you are using numbers or variables.

[best]

Funnily enough in Germany there are both variants: my casio fx-991ES, which I used in the exams for my high-school diploma gives out 9 as a result (as shown in the video) and the fx-991ES PLUS, which my girlfriend used, gives out 1. Both are allowed to use in the exams for the high-school diploma and while studying computerscience

[hjups]

Even including the explicit multiplication could be ambiguous, so I would opt to include parenthesis around the division as well. And if it's really that critical of a calculation, you want to be as precise as possible, where the result shouldn't depend on how one calculator might work over another. The other thing you could do to be explicit while entering into a calculator, is treat division like multiplication, where you explicitly do 2^-1 (via the inverse function).

[Pongo]

The ambiguity is from the inline typesetting of the division sign. It is unclear whether the intention is 6/2 x (2+1) or 6/ ( 2(2+1) ). The Casio resolves this with putting in extra ()'s to clarify the operation. I think that it is a great feature.

[Grantherum]

When I was a kid, in the USA, we were taught the 6/2(2+1) would have been 1. Sometime in the 90's or so, they changed over to the =9 version. They wanted to make things difficult for some reason. I was always taught that you do parenthesis first, including the implied multiplication... then do the rest of the multiplication in order of precidence.

[LoneTech]

As usual, my favourite calculator Qalculate! does a little helpful step: like Wolfram Alpha, it displays its interpretation, and when you hit Enter/= it actually explains the ambiguity and asks which variant you want (which is then set under Mode - Parsing Mode). It's not an issue in RPN mode, of course (6 2 2 1 + * /). You can even get the chain syntax (1+2*3=(1+2)*3, strict left to right) if you want.

[adriansdigitalbasement]

HP 48G: as you type it in, it adds a period between the 2 and ( basically explicitly saying there is a multiplication there. The result is of course 9 on this calculator.

[skunkmyrddyn]

These kinds of errors with algebraic input is why I prefer rpn input on my hp. It is much more difficult to get order of operation incorrect when dealing with a single operation at a time with values on a stack.

[mosfet]

Thanks Dave.
I use the HP's but in RPN mode which, as an engineer, I prefer. So I get to decide the order, not the calculator. My answer is "1". I live in the USA, by the way. I think the system I learned in school is PEMDAS (Parenthesis, exponents, multiply, divide, add, subtract.) The system doesn't seem to differentiate between implied or not implied, however, when I see equations like this it is implied to me.

[jimwolsiffer]

I am in the US and I was taught in School and College math to use the system that results in the answer of 1 for this equstion.

[Scouarn]

Please just use fractions, this calculator has a fraction key. The meaning of the division symbols is not always the same.
But the fact it adds brackets is really cool.

[darkforce]

One more reason I prefer RPN style calculators :).

RealCalc that you used on the phone at the end in RPN mode has mostly replaced my hardware HP calculators though.

[usagi_hase]

The fx-991DEX sold in the german market also has a higher priority for the implied multiplication.

[chrisose]

The removal of priority of implied multipliers in the US is a "new math" thing. When I went through algebra and higher maths in the 70's, implied/distributed properties were handled before base multiplication/division. Granted, this was a time when we wrote everything out and the only calculation tools we had access to were the tables in the textbook and a slide rule.

Something I noticed on the Wolfram page you showed. It reconfigured the equation to completely remove the distributed property by making it a discrete fraction times the result of the parenthetical expression. Yet the inputted equation was clearly written as a fraction with "6" as the numerator and "2(2+1)" the denominator.

[skonaslp]

First of all, you should always use fractions (and such) for clarity, but it's not always possible. Juxtaposition precedence rules are more a scientific thing, for example when formulas get in-lined (meaning only one line is there for utilization), they can't use stuff like fractions. Extra brackets could cause confusion when reading, as it's a bit messy, and many are too lazy to add them everywhere. That's where juxtaposition comes into play. 1/2x=1/(2x) and examples shown in the video would normally be interpreted like the Casio did, but for clarity it should always be made clear (say at the start that some paper uses juxtaposition rules). The Casio is doing a super job at this, even adding brackets for clarity, but it's not something you'd likely encounter in school. Not even schools in Europe (in my experience) teach kids about implied multiplication and its precedence rules, as mentioned before it's just a "style"-thing and it's not likely to be encountered anywhere for the average student.

[billcrowell]

It is concerning that the results are region dependent. Math should be universal.

My takeaway is that one should always use the operator as this appears to be an implied operator quirk. 6/2x(2+1).

Of course, some of us use RPN and our calculations are always correct. <grin>