Gotcha. Cool.
I find it interesting so many people are using the term “PEMDAS”, while I learned order of operations in school, we didn’t learn it through a mnemonic. In fact now that I think of it I can’t actually think of anything I learned in school in which the teacher used a mnemonic as an aid in their lesson, even though I hear people use the mnemonics in casual conversation often enough to know that many people learned things that way.
Personally I prefer that I was just taught that you evaluate inside parentheses (and if they are nested you do the most interior and work your way out), then exponents, then multiplication/division and then addition/subtraction.
I will say that I don’t think excluding “R” from the mnemonic is a failure per se, as far back as I can remember doing basic Algebra I remember knowing that you exponents can express roots (x ^ (2/3) is the same as the cube root of x squared.)
I was taught the same way. Never remember any sort of mnemonic necessary for order of operations. That said, in trig, we did learn soh-cah-toa for sine, cosine, tangent, and it’s proven being useful. However, that’s the only mnemonic I recall being taught.
Oh, and ROY G BIV for the colors in the spectrum, but that’s not math.
Ah, I do remember ROY G BIV, but that was very early in school so I had no remembered it until now. It may have been 1st or 2nd grade even, and I do remember the teacher using it.
With trig relationships I remember the first day we started on them we basically just had to write down a little chart and part of the required homework was to memorize that chart and to reference it throughout until you basically just knew it.
So instead of SOH-CAH-TOA I just remember a little chart scrawled in pencil like:
Sin = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Looking at it now I do see how easily it lends itself to a mnemonic, so I’m not surprised they were used heavily.
I didn’t even learn this in school. (Well, the mnemonic; I certainly learned the definition of the sine, cosine and tangent of angles in right triangles.) I think I was a grad student before a teacher friend of mine mentioned this word and I had no idea what she was talking about. And it just sounds like a Native American warrior to me.
Note that 1 + 2 x 3 + 4 is not equal to 3 + 4 x 1 + 2, on the conventional way of reading these expressions.
Does this mean multiplication is not commutative, and that the order of multiplying 1 + 2 by 3 + 4 matters?
No. It just means the conventional way of reading expressions makes expression juxtaposition (with a “x” in the middle) is not commutative. But no one ever claimed expression juxtaposition was commutative.
When people speak of the commutativity of addition or multiplication, they aren’t talking about commutativity of expression juxtaposition; they aren’t talking about notation at all. They’re talking about the actual operations of addition or multiplication, which mean the same thing regardless of how you choose to write them down.
If you really nitpick it to a fine point, multiplication does not take precedence (as the previous thread tries to explain.)
It’s just that it’s a common convention to interpret the sequence as multiplication first. However, there’s no immutable law of the universe that the symbols will be parsed that way.
Consider the following “wrong” answer you mentioned:
You punch this into a calculator from left to right as stated and the answer you will get is 8. Is the calculator “wrong” or is the human who uses PEMDAS “wrong”? Neither. It’s just 2 different interpretations of a particular set of symbols.
Most of the time, (the “convention” in other words), we evaluate it as 6.
The PEMDAS folks would have you believe that written math is unambiguous but that’s not true. When you write down an equation, you’re really just depending on a set of probabilities that others will interpret it the way you think they will. PEMDAS is not an inescapable law – it’s just a guide about probabilities of interpretation (you hope others have learned and remembered PEMDAS and therefore they will interpret your symbols via PEMDAS.)
Depending on the calculator. Most scientific calculators will give 6 for that, but most non-scientific calculators will give 8.
The reason given in the other thread that the answer was not wrong was not that you can interpret it however you want, but that many mathematicians would still interpret it as ambiguous.
This example is far less ambiguous. I’m pretty sure that, if you asked any mathematician for the “correct” answer, they would give 6. In the previous example, some would say that implied multiplication takes precedence.
Cuz I’m an old fart;)
The way almost any mathematician/physicist/engineer would read the OP is indeed, “solve 2X2 + 2 = 0”. The word “solve” implies an algebraic, not arithmetic question, and if both sides are not given, the default equation is pretty much always “= 0”.
Given the OP says both “solve” and “equation” it is very hard to make a case that the question is arithmetic. Use of the words if an arithmetic question was intended is simply wrong.
Then the slight confusion about what 2X2 means. Since it was an uppercase X not lowercase, it is hard to make the case that 'x" was intended to mean multiplication anyway. So this further suggests an algebraic rather than arithmetic interpretation. Then, what is the “2” doing on the right of the “X”? Most people would interpret that as being a poor shorthand for x squared. Thus the OP is read as a conventional quadratic equation. 2x^2 + 2 = 0
In this case the solutions are i and -i aka the square root of minus one and its negative value.
I self admit to being absolutely fucked at math, but I was looking for the parens for a clue as to which way to solve it. I wanted to read it as 2+(2X2)=6. mrAru wanted to go the other way and get 8.
I think the correct answer is 1 . If you want it to be 9, you’d have to write it : (6/2)(1+2).
At best it could be either, but 9 isn’t the only right answer.
I’m sorry I didn’t understand anything you wrote: I interpreted:
When = pie
You = hot
Write = penguin
and at that point I just gave up.
Yes, you have the correct idea. What I wrote also applies to English words (written or auditory phonemes) as you have aptly demonstrated.
When you write down an English sentence it is impossible to make it 100% unambiguous, just like mathematics symbols. Writing a sentence is working with an expectation of probabilities for interpretation.
Your point is pretty much the opposite of truth. If it met a true statement it would blow up and create a supernova. The point of an equation is that it is unambiguous. Operators have precedence, they are defined and accepted, and they are unambiguous (certainly for one as simple as in the OP).
There is no “probablity” aside from that there is a chance that someone reading the equation may not know the rules or will imply them incorrectly. That does not mean that the equation is ambiguous.
The very existence of someone “maybe not knowing the rules” is exactly why it’s a probability. The very existence of 2 threads about this subject on SDMB in the last month with differing answers proves that it’s a probability. Hopefully, you will write the equation in such a way that the probability is virtually guaranteed (99.9% interpreted the way you intend) but it’s never 100%. It’s impossible for it to be 100%.
PEMDAS is not “truth” and it doesn’t make things “unambiguous”.
Here’s a question for you:
1 + 1 = ?
The above expression is ambiguous. (Yes, for practical everyday purposes one could say it is not ambiguous and the answer is 2 but if you really nitpick with rigor, it is ambiguous.) You also cannot make it unambiguous no matter what qualifications or meta symbols you try to put around it. It is impossible to do so. Godel’s Incompleteness Theorem proved that it was impossible.
OK, tell that to all the mathematicians in the previous thread on this subject which is linked to above.