-5=5

[Count Arioch the 28th]

I gotta say, you're one of the more skilled trolls of the day, Doom.

[Doom]

Uh, yeah.

Anyway, anyone who's actually curious and still not satisfied can do what Count and his ilk would never do: pick up a book on the subject. Read.

[jadagul]

You know, if I were smart I'd just ignore Doom because he's proven himself crazy. But I did want to jump in and suggest that his dogged use of "arabic notation" is extremely odd. Wikipedia says there is something called arabic notation but it's the notation that people use when they're writing in Arabic and has nothing to do with anything we're talking about.

We do use Arabic numerals. They are the symbols on the following list:

Code: Select all

0
1
2
3
4
5
6
7
8
9

Things not on the above list: the plus sign. The minus sign. Any division sign. Any exponentiation sign. Any operation at all. Nothing except the ten numerals we use in our modern decimal system. (Incidentally, a large fraction of higher math only uses "0" and "1" from that list. Sometimes "2" or "3". Anything else is unusual).

Now, we do have conventional systems of order of operations, but they're really arbitrary and totally depend on what group of people you're usually communicating with. For instance, I'm pretty comfortable with "a/bc" representing the same thing as "a/(bc)", but one of the physicists I talked to said she'd be really uncomfortable doing even that and would always want the parentheses in.

Of course, there are other notation systems, like Reverse Polish Notation. Which really has a well-defined order of operations and is totally unambiguous and doesn't need parentheses. But most people are really uncomfortable working with it, because it looks like this:

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  5 3 7 2 9 1 4 + - / * + -
= 5 3 7 2 9 5 - / * + -
= 5 3 7 2 4 / * + -
= 5 3 7 .5 * + -
= 5 3 3.5 + -
= 5 6.5 -
= -1.5

That system is actually unambiguous. Which is why certain fields use it instead of infix notation--they actually need something that's unambiguous without parentheses.

[Count Arioch the 28th]

I have an ilk now? Cool, I've always wanted an ilk. Does an ilk do any tricks or taste good?

[Doom]

Not sure what the personal attacks are all about, but you're correct, I'm using "arabic notation" with less than absolutely perfect precision. I simply wanted to distinguish the notation we use in the schools (which is based on Arabic) from the mathematical languages used in various computer and software programs. They are all versions of 'mathematical notation', I saw little need to split hairs to such a level.

Only in MPSIMS can it take three pages of my referencing every textbook in existence regarding basic, fundamental definitions...and still there a few here that just don't believe it.

[shadzar]

I have an ilk now? Cool, I've always wanted an ilk. Does an ilk do any tricks or taste good?

i think it comes in hocolate lavor

[Count Arioch the 28th]

I have an ilk now? Cool, I've always wanted an ilk. Does an ilk do any tricks or taste good?

i think it comes in hocolate lavor

I prefer Shockolate. It's made with lightning. REAL LIGHTNING!

[jadagul]

We don't believe it because as far as I know, I have never in my life met a working mathematician, scientist, or engineer who cares about any of this stuff, actually uses it, or would rely on it to clarify ambiguous notation. I don't think I've met anyone who would consistently interpret "a/bc" as the same thing as "ac/b", which is one of your core claims.

Let me let you in on a secret: the people who write high-school or middle-school or elementary-school level math textbooks either are absolute blithering idiots, or are forced to pretend to be by the committees choosing textbooks (that last isn't just a sop; it's highly probable). They tend to have an impressively low level of mathematical sophistication and a minimal grasp on how people who use math in their actual careers interact with it. They focus on rote memorization and pointless rules (like the one you're trying to foist on everyone) over actually having the barest goddamn understanding of what's going on.

Bonus question for you: how would you interpret

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a \sum_{n=1}^{10} b^n

and why?

[Doom]

We don't believe it because as far as I know, I have never in my life met a working mathematician, scientist, or engineer who cares about any of this stuff, actually uses it, or would rely on it to clarify ambiguous notation.

I don't think I've met anyone who would consistently interpret "a/bc" as the same thing as "ac/b", which is one of your core claims.

Um, above, you yourself claimed to have talked to a physicist who had issues with writing a/bc if she meant "a/(bc)". This isn't my 'core claim'...it's how the notation works. You're basically arguing that "B" is pronounced "C" in the English language, and rejecting any information to the contrary.

Do the rules for mathematical notation have 'issues'? Sure...so do the rules for spelling and grammar in every other language.

Let me let you in on a secret: the people who write high-school or middle-school or elementary-school level math textbooks either are absolute blithering idiots,

Fair enough, but that is the level of mathematics that some people here are struggling with. It's worth noting that even (barely) college level textbooks say the same thing (more advanced texts often don't address the topic, since it's so basic).

They focus on rote memorization and pointless rules (like the one you're trying to foist on everyone)

Ask any master of any discipline, he'll tell you the basics are important.

You really seem to be confused here. We're addressing arbitrary rules. For example, "B" is pronounced the way it because, well, that's how it's pronounced. "Right" is spelled the way it is because, well, that's how it's spelled (yes, I know there are historical reasons, but as far as the modern world is concerned, that's how it goes). "Lute" is pronounced the same way as "loot" because, that's just the way it is. Languages tend not to be perfectly consistent. The order of operations rules? Yeah, they're arbitrary, just so everyone can agree exactly how to express calculations and concepts. I'm not 'foist'ing anything on anyone, they are the rules the world uses for reading and writing mathematics.

So, yeah, when it comes to order of operations, there's nothing for it but to learn the arbitrary rules, any more than a small child is forced to simply memorize the alphabet, or how to pronounce or spell certain words.

For your bonus question, you'd have to explain why whatever that notation is is relevant to the questions of 'order of operations' and 'square root properties'...but I acknowledge I don't know that notation; as I've never claimed to know all languages your question is pretty pointless. I guess I could counter by asking you something in Mandarin?

On the other hand, you keep claiming to be a mathematician, so bonus irrelevant question for you: Find the area of intersection between R = Cos (T) and R = Sin (T), polar coordinate system, exact answer; seeing as you claim to be such an incredible mathematician, I'm sure this will take you no more than it did for me a few days ago (i.e., 2 minutes). This comes from a first-year calculus course, so I'm sure it won't be that big a deal...you just keep writing such non-mathematician things that I'd like to see some evidence that you've had at least a year of training.

Seriously, a mathematician that neither understands order of operations, nor comprehends why it'd be nice to have a reliable precise method for clearly defining computations is...odd, to say the least.

[Username17]

We don't believe it because as far as I know, I have never in my life met a working mathematician, scientist, or engineer who cares about any of this stuff, actually uses it, or would rely on it to clarify ambiguous notation. I don't think I've met anyone who would consistently interpret "a/bc" as the same thing as "ac/b", which is one of your core claims.

Uh... no, he's totally right about that. Multiplication and division are read left to right. I don't know why you're spazzing out about this.

Now, Doom's tirade about "Arabic Notation" is nonsense, because no such thing exists. But I am unaware of any system wherein a/bc would be anything other than ac/b.

a/bc

=/=

a
bc

-Username17

[DSMatticus]

I simply wanted to distinguish the notation we use in the schools (which is based on Arabic)

This is totally, completely wrong. Everything you're referring to came about in the 1500's and later of Europe. The only thing we stole from the Arabians specifically was numerals. It is not based on Arabic mathematics of the time or prior.

But I am unaware of any system wherein a/bc would be anything other than ac/b.

I believe the main confusion is nobody handwrites a/bc, they write...

Code: Select all

a   /               a   /               a            ab
  /   c     or        /       or      ----     or   ----
/  b                 /  bc             bc            c

Except not that ugly. And when people see a/bc, they wonder "do they deliberately mean a/(bc), or is this a technical limitation of using text to write a mathematical expression and they mean number 1?" Not that that makes them right, it's just that the notations people actually use don't work well in text medium and it makes people doubt the intent of the writer.

The order of operations rules? Yeah, they're arbitrary, just so everyone can agree exactly how to express calculations and concepts.

You're missing the point. The rules? They're incomplete. They don't describe what to do with the unary negation operator, the summation operator, or any other of a dozen operators. The conventions for those things are not widespread, if any convention exists at all. The language of math is not a complete, unambiguous grammar. It's unambiguous where it's complete, but it's incomplete and that's important.

Seriously: -1^2. Parse this, explaining what rule you're applying at each step of the parsing.

[jadagul]

Frank--no, I do. DSM sort of pointed towards this, but I often write "a/bc" when I mean "a/(bc)", at least for private notes. The physicist I was talking to was saying she'd be uncomfortable writing "a/bc" ever under any circumstances because it's so ambiguous. Just like I asked her to evaluate "-1^2" and she looked at it for a minute and said, "well, that depends where you put the parentheses."

There are various rulesets that can give those expressions interpretations, but since no one uses them consistently the correct answer to being handed something like that is to hand it back and ask for clarification.

Doom: \sum is the summation sign--you know, the one with the capital sigma? So \sum_{n=0}^3 n is 0+1+2+3. I ask because that's a fairly common thing to write down and involves nothing other than addition, multiplication, and exponentiation, and I'm actually not sure how you're going to tell me to interpret it.

As for the rest...I really shouldn't let myself get trolled like this. My original point was that you don't have any idea what real math looks like, just basic arithmetic and calculus and crap. Real mathematicians don't computer fucking polar integrals because that's what mathematica is for; we do stuff like this. The only reason I've done any calculus at all in the past three years is TAing a freshman class. But I'm still tempted to do the problem if you'll explain why you're using polar coordinates with a negative R value--do you want absolute values or do you actually want the curves intersecting the origin all over the place, in which case the "area of intersection" isn't really neatly defined?

[Doom]

Real mathematicians don't computer fucking polar integrals because that's what mathematica is for;

LOL. Real mathematicians actually know how to do things. It's curious that you link to very specialized topics in abstract algebra; are you aware that there are other fields?

But I'm still tempted to do the problem if you'll explain why you're using polar coordinates with a negative R value--do you want absolute values or do you actually want the curves intersecting the origin all over the place, in which case the "area of intersection" isn't really neatly defined?

Um, R = Cos t and R = Sin t are circles, a very basic concept learned in high school trig, R possibly being negative doesn't matter (do you even know what a negative R means?). The intersection is very neatly defined, but there's no reason to discuss this further; you've already failed, so you need not continue.

[Starmaker]

Now, Doom's tirade about "Arabic Notation" is nonsense, because no such thing exists. But I am unaware of any system wherein a/bc would be anything other than ac/b.

But yeah, if someone says "-1^2=-1" then you are perfectly within your rights to punch them in the mouth. Because while the computer text does literally map to the pen and paper text that says that, in actuality that computer text is supposed to stand in for the pen and paper text where a positive one should be on the right side of the equals sign.

It stands for the pen and paper text where the "/" would be loooong. "bc" in code is most often interpreted as a single variable. a/b*c = a*c/b.
Also:

But yeah, if someone says "-1^2=-1" then you are perfectly within your rights to punch them in the mouth.

Even though they're from another country? This from a person who created a meme to mock grammar nazis who abuse non-native speakers? I am disappoint.

*reports to the barrel*

[Doom]

Look, I'm sorry I wanted to use a shorter expression than "the type of written mathematics that everyone (should be/is) taught as a child.", the shorter expression is certainly wrong if taken in a very literal sense, but I still want something to distinguish from the many other forms of mathematical notation. I won't bother citing anything explaining the shorter reference, and instead just use this longer and more clear expression (abbreviated to TTOWMTESB/ITAAC) if it happens to come up again.

You're missing the point. The rules? They're incomplete. They don't describe what to do with the unary negation operator, the summation operator, or any other of a dozen operators. The conventions for those things are not widespread, if any convention exists at all. The language of math is not a complete, unambiguous grammar. It's unambiguous where it's complete, but it's incomplete and that's important.

Seriously: -1^2. Parse this, explaining what rule you're applying at each step of the parsing.

I feel your pain. You've taken an expression like "-2", converted the (binary operator, called 'subtraction', with an assumed zero) into a (unary negation operator--technically still have addition, but now with another operator of transforming into additive inverse), and now you're afraid that the entire world will collapse because all calculations are now using undefined terms.

Now, either

A) the world collapses because nobody can calculate anything, or

B) you're confused about something.

As I said, I feel your pain, so allow me to help. I'm sure by now you've noticed the world hasn't collapsed into an innumerate mess, so let me try to explain to you why it hasn't.

Here's what you do. Instead of converting subtraction into something undefined in order of operations, don't do that. In fact, what you want to do, is take undefined operators and convert into something defined.

Let's go over that carefully. When you see and expression like "5 - x", don't transform that into "5 + (-x)", and get confused because your unary operator is undefined in order of operations. Just leave it as subtraction.

Let's explain that some more: your "unary negation" operator can be written in a form mathematically equivalent to the subtraction operator. So, just use subtraction, and now the order of operations applies.

There are, certainly, other operators in mathematics; you can, of course, put them in terms defined in order of operations as well. The summation operator, for example, can be defined in terms of addition. You can even, if you want, write exponents as a form of multiplication, or multiplication as a form of addition, if you feel the need to go further down in order of operations.

From past experience, I know you don't learn in the forum format, so if you have any further questions on how this works, PM me your university, and I'll put you in contact with someone in the mathematics department (or developmental math department) who can explain it to you in person.

I've taken the liberty of assembling 3 different textbooks that all assert things along the line of "-4^2" = -16 and made a short video; I'll upload it to YouTube late, to show that indeed TTOWMTESB/ITAAC really is read this way.

Frank still has a point. Online communications often don't easily allow for exactly perfect representation of concepts in all languages, whether that language is Mandarin or mathematics or numerous others. It should be expected that when someone writes in the 'forum language', that occasionally some things will get lost in translation.

[DSMatticus]

LOL. Real mathematicians actually know how to do things. It's curious that you link to very specialized topics in abstract algebra; are you aware that there are other fields?

Yeah, I haven't touched polar coordinates in four years, albeit I'm not pure math major. It's kind of funny that you think they're relevant. If I recall, you teach physics courses? Or is it just introductory mathematics courses? Either one explains it, I suppose.

You've taken an expression like "-2", converted the (binary operator, called 'subtraction', with an assumed zero) into a (unary negation operator--technically still have addition

So your position is that the unary negation operator doesn't exist. At which point, okay, we get it; you have no idea what you're talking about.

Let's explain that some more: your "unary negation" operator can be written in a form mathematically equivalent to the subtraction operator. So, just use subtraction, and now the order of operations applies.

P.S., multiplication 'can be written' in a way that is mathematically equivalent to iterative addition, but multiplication is not resolved at the same precedence level of addition. As someone already said in this thread, and you failed to understand or respond to them.

Also, this is circular reasoning. You can only write the unary negation operator as being equivalent with subtraction if you give it an order of precedence that makes it equivalent with subtraction. You've got a nice circular loop here where your premise leads to your conclusion and your conclusion leads to your premise and you have demonstrated exactly neither of them using a wide-spread convention.

So, yeah. Knock yourself out; find the convention about implied zero that actually supports what you're saying. That'll be fun.

[Doom]

From past experience, I know you don't learn in the forum format, so if you have any further questions on how this works, PM me your university, and I'll put you in contact with someone in the mathematics department (or developmental math department) who can explain it to you in person.

[fectin]

Fectin: the easiest example where order of operations would make a difference is in the expression

Code: Select all

1 - 1 + 1

which under usual conventions (and the way I'd probably read it) equals one. You can, however, rewrite it with parentheses

Code: Select all

1 - (1 + 1)

to get negative one instead.

Derp. I am dumb.
I was only think of distribution. You're (obviously) correct.

[Psychic Robot]

-5 = -1(5)
-5^2 = -1(5)^2
-25 = -25
therefore -5 = 5

[Username17]

Even though they're from another country? This from a person who created a meme to mock grammar nazis who abuse non-native speakers? I am disappoint.

I actually consider people who say "-1^2=-1" to be doing exactly the same thing as grammar nazis. They are using purposefully opaque communication and misapplying rules in order to make fun of people. As Josh said when this first came up, the entire purpose of that line was "to make fun of the mathematically illiterate".

Now the fact that unary negation exists and it isn't even correct doesn't really matter. The point is that the people who make that argument are purposefully confusing people with bad communication in order to make fun of them. That means that you should punch them in the mouth.

-Username17

[Doom]

It won't make any difference, of course, but here is a quick video of me showing the same type of calculation in three different textbooks:

http://www.youtube.com/watch?v=UMDbAdSCnVA

[Username17]

It won't make any difference, of course, but here is a quick video of me showing the same type of calculation in three different textbooks:

http://www.youtube.com/watch?v=UMDbAdSCnVA

But in forum text you don't have minus X squared. You can't write "minus X squared" in symbolic notation. You can write an X, and you can use the "^" symbol to indicate an exponent. But you don't have superscript and you don't have a symbol that is unambiguously minus. You have a single symbol that is both the minus and the negation. It's both symbols.

So when you write "-X^2" in computer talk, you don't have the notation in the book you just filmed. You have something that is most plausibly "negative X squared". And if you still don't get that at this stage, I really do pine for your students.

-Username17

[Doom]

Frank, I agree with you (like I've said several times in this thread), and have never disagreed with you on this point...forum text is NOT the same thing, and allowances should be made for poor translations. As I said before, you have a point, and by this I mean a valid point, not sure why you thought I was disagreeing with you. It's someone else disagreeing on this issue, I think.

On the other hand, taking a course under a "mathematician" with shaky grasps of order of operations, trigonometry, and calculus....yikes, just yikes, I would very much feel for such students.

[shadzar]

this is all because qwerty keyboards were made for typewriters, not typesetters, so you cant just type in:

6 ÷ xy = 3x

and have to use the customary computer equivalent of:
6/xy=3x

but then most people dont format it right WITH THE PARENTHESIS, or space it out to indicate what should be done when...

symbols used least often in communication by the majority, just dont appear on them. such wonderful advances in technology, yet complex method must be gone through because a lack of foresight, and lack of intention to add things into easy access use for people.

[jadagul]

Huh, that's cute. Don't think I've seen that before.

solution:

Then it's two circles, you have heights varying from sqrt(1-x^2) to (1 - sqrt(1-x^2)) so you integrate

Code: Select all

\int_0^{sqrt{2}/2} 2 sqrt{1-x^2} - 1

and at this point we decide that doing it in cartesian is a pain in the ass. So shifting to polar, we see that we can split it into two integrals, where the first is

Code: Select all

\int_0^{\pi/4} \sin^2 t dt 

and the figure is symmetrical so we can just double that. We check the half angle formula and the power reduction formula and get that 2 \sin^2 t = 1- \cos 2t which has antiderivative t - (\sin 2t)/2, and evaluating from 0 to \pi/4 we get

Code: Select all

\pi/4 - 1/2

which sounds about right.

Of course, this all misses the larger point that the people who do calculus are engineers, not mathematicians. Seriously, apparently you hang around LSU? Go find a research mathematician and ask him how often he computes an integral. I'm an algebraic number theorist/arithmetic geometer, so go ask one of them. It doesn't happen.

Now do you want to tell me how you read

Code: Select all

a Σ b^n

where the sum is as n varies from 1 to 5?


Now, for the idiotic "-1^2" thing you keep harping on--yes, there are ways to handwrite it unambiguously. What's written down in ASCII has nothing to do with that, and really is ambiguous, though I'd probably wind up reading it the way you do if I had to choose (because that's the way TeX would read it, and that's usually the context in which I see ASCII math). But you're missing the larger point, which is that people who use math in real life don't worry about that. I'm ignoring you for the same reason linguists ignore people who say that "every child should take out their book" is wrong--that's how actual people actually speak, and what's written in the textbook is basically irrelevant.

Now, I freely tell my students in calc classes that some of them are probably better at computing integrals than I am; unfortunately, around here the pure math department has to teach the calc classes because the applied mathematicians have money and we don't. But I still understand what's going on better than any of them do, and can explain it and do it better, because rather than knowing a whole bunch of formulas and solution techniques I actually understand what's going on under the hood.