Mathematics and precedence rules

[noparlpf]

If you posted an equation with an integral on facebook and forced 100,000 people to respond to it with what they think the answer is, I would bet that a similar number of people would not be able to. This does not prove that integrals are misleading or that all calculus equations should be represented with a shaded graph - it just means that those people do not understand calculus.

If you were to ask 100,000 people what the capital of Zaire is, most of them would likely not know the answer. That does not mean that the capital of Zaire is somehow ambiguous - merely, that most people do not know the capital of Zaire for whatever reason. It does not mean that whenever you write, say, "The President went to Kinshasa" you have to write "The President went to Kinshasa, Capital of Zaire, a nation in Africa located at the middle of the continent bordering, intersecting the equator, etc. etc." even if a number of people on facebook cannot pinpoint the capital or the nation on a map.

...Is Zaire a place?
See, I can do math, but nobody ever taught me geography.

On a completely random note, I actually think that mathematicians are more likely to disregard trivial stuff and write in absolutely ambiguous manners. I can't count how many times I wrote

log x+2

to mean log (x+2) and

log x + 2

to mean (log x) + 2 on my exams for courses like probability theory.

I make sure to be super careful about things like that, mostly so I don't confuse myself.

[Coidzor]

...Is Zaire a place?
See, I can do math, but nobody ever taught me geography.

From about the 70s to the late 90s it was what the Democratic Republic of the Congo was called, near as Google and Wikipedia can figure.

[Heliomance]

I quite agree. Nonetheless, an equation can deliberately be written to be more obscure than strictly necessary.

1+1 = 2
is the same as
(3^0) + (natural log of e) = ((10/10) * (1000) ) / 500

but the first is much more clear. As I stated, there is an art to writing an equation such that it is as clear as possible to as many readers as possible. The equation in the OP could be immeasurably improved for general consumption by an audience of high school graduates by the addition of brackets.


Respectfully,

Brian P.

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[pendell]

Heliomance: You win.

Respectfully,

Brian P.

[Rawhide]

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Calculators tend to evaluate X<op>Y left to right.

I know, see my earlier post - you should always use Scientific mode, which does it correctly. Windows Calculator in Standard mode will do it incorrectly regardless of brackets, Windows Calculator in Scientific mode will do it correctly regardless of brackets.

If you posted an equation with an integral on facebook and forced 100,000 people to respond to it with what they think the answer is, I would bet that a similar number of people would not be able to. This does not prove that integrals are misleading or that all calculus equations should be represented with a shaded graph - it just means that those people do not understand calculus.

If you were to ask 100,000 people what the capital of Zaire is, most of them would likely not know the answer. That does not mean that the capital of Zaire is somehow ambiguous - merely, that most people do not know the capital of Zaire for whatever reason. It does not mean that whenever you write, say, "The President went to Kinshasa" you have to write "The President went to Kinshasa, Capital of Zaire, a nation in Africa located at the middle of the continent bordering, intersecting the equator, etc. etc." even if a number of people on facebook cannot pinpoint the capital or the nation on a map.

The equation is not at all ambiguous simply because there is a very clearly defined, internationally-agreed upon procedure for the order of operations; one that is taught within elementary school. The fact that 30,000 out of 100,000 people on Facebook does not know PEMDAS does not negate the fact that it exists and is one of the basics of mathematics in the same way that x number of people not knowing the capital of Zaire doesn't make Kinshasa "obscure and unclear in communication."

This. Exactly this.

The equation is not in any ambiguous, and no amount of claiming it is so will change that. If you know and follow the very clearly defined, internationally-agreed upon procedure for the order of operations, you will always get the same answer. There is absolutely no point where something could be one thing or the other, by following the rules correctly, you will always know what to do and always get the same answer.

The original poster on Facebook observed that some people were not following the rules taught in basic maths to complete this equation and released it into the wild to trap them, but that does not make it in any way ambiguous!

It's not like the sentence "Bob met Fred at the train station, he had recently had a haircut." where either Bob or Fred could have had the haircut. Correct rules of maths dictates that it must be one particular way.

[AsteriskAmp]

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Doesn't the right side end up as 2 while the left side ends up as 1?
I vaguely suspect, if my previous statement isn't wrong, that it should be a z and not a 2 in the exponent inside the first limit so that it ends up as e, whose Neperian Logarithm is 1, making the left side 1+1.

[pendell]

I know, see my earlier post - you should always use Scientific mode, which does it correctly. Windows Calculator in Standard mode will do it incorrectly regardless of brackets, Windows Calculator in Scientific mode will do it correctly regardless of brackets.

Now I'm puzzled -- why does it behave differently depending on standard mode and scientific mode? I'd have thought that a copy-pasted formula would always be evaluated the same way, and the difference between standard and scientific would simply make more functions available to the user. But that's obviously not the case.

If you know and follow the very clearly defined, internationally-agreed upon procedure for the order of operations, you will always get the same answer.

Emphasis mine. I agree with your statement. I will, however, add that it seems obvious to me that most respondants to that page did not know and follow the procedure. If my interest is for ordinary people to correctly solve the problem, it is in my interest to add brackets to make the order more obvious. Because even if it is clear and unambiguous , it's still not going to be correctly interpreted and applied by its audience.

Respectfully,

Brian P.

[Heliomance]

Doesn't the right side end up as 2 while the left side ends up as 1?
I vaguely suspect, if my previous statement isn't wrong, that it should be a z and not a 2 in the exponent inside the first limit so that it ends up as e, whose Neperian Logarithm is 1, making the left side 1+1.

On rechecking, you are in fact correct, due to ln(1) = 0. I don't think changing the 2 to a z would do anything useful, the easiest way to fix it is to get rid of the ln.

The other problem is that it's mixing matrices and real numbers - it really needs to take the determinant of (X^-1)^T-(X^T)^-1.

[AsteriskAmp]

On rechecking, you are in fact correct, due to ln(1) = 0. I don't think changing the 2 to a z would do anything useful, the easiest way to fix it is to get rid of the ln.

The other problem is that it's mixing matrices and real numbers - it really needs to take the determinant of (X^-1)^T-(X^T)^-1.

Changing the 2 to a z would produce the limit's definition of e [limit of ((1+1/z)^z) as z tends towards infinity], which, inside ln, would produce 1. The only issue with the matrices is if they can in fact be inverted, otherwise it does work since the inverse of the transposed is the transposed of the inverse if the original was invertible.

So if you take that X is an invertible matrix as a hypothesis and replace the 2 in "limit of ((1+1/z)^2) as z tends towards infinity" with a z it does add up.

I'll also note Wolfram Alpha can't solve this, due to being unable to reason out through properties that the most complex calculations cancel out without needing to be performed.

[snoopy13a]

More to the point, a 514 out of 800 on the SAT math section is an indication of mathematical deficiency given the current difficulty of the SAT math section*. It's a mixture of fairly basic arithmetic, fairly basic geometry, and extremely trivial algebra. It also briefly touches upon trigonometric functions and matrices, though there were few enough of them as recently as 2010 to reduce a score more than 40 or 50 points were every single one of them failed. That doesn't explain the loss of 286 points on average, let alone the loss of 386-401 points for approximately 15.9 percent of the test takers, assuming a roughly normal distribution. This is consistent both with a grasp of elementary math and utter inability in higher math, or with a pattern of knowledge gaps throughout mathematical levels, the latter of which would also explain the results gathered from Facebook**, particularly once one takes into account how neither the SAT nor Facebook are measures that accurately evaluate the population, due to heavy selection biases in both.

*I found data that indicate a 514 mean in 2011, with a standard deviation of approximately 100 to 115, with different sources varying. I will thus be using these numbers.

**I have not personally confirmed these, so this is a very provisional statement.

The average math SAT score is not the average score of the population. It is the average score of those who take the SAT--high school students who plan on attending college. A substantial minority of people do not take the SAT, such as high school dropouts and high school students with absolutely no interest in college. Therefore, the actual average is likely lower.

Second, most people get SAT math questions wrong because of careless mistakes in a time sensitive test. The actual concepts are designed to be accessible because it is an aptitude test, not a test of one's math education.
The math section tests for speed--especially with problems that aren't hard conceptionally but require time to solve--and attention to detail more than actual math concepts.

[Rawhide]

Emphasis mine. I agree with your statement. I will, however, add that it seems obvious to me that most respondants to that page did not know and follow the procedure. If my interest is for ordinary people to correctly solve the problem, it is in my interest to add brackets to make the order more obvious. Because even if it is clear and unambiguous , it's still not going to be correctly interpreted and applied by its audience.

If you wrote that equation for people who know basic maths, then there is absolutely no reason to change it. If you wrote that equation as a test to see if people know basic maths, then there is absolutely no reason to change it. If you wrote that equation for people you know don't know basic maths, then you should not have written that equation at all, not without at least a paragraph explaining where Zaire is, what its capital is, and why it's important.

[Heliomance]

Changing the 2 to a z would produce the limit's definition of e [limit of ((1+1/z)^z) as z tends towards infinity], which, inside ln, would produce 1. The only issue with the matrices is if they can in fact be inverted, otherwise it does work since the inverse of the transposed is the transposed of the inverse if the original was invertible.

Yes, absolutely. But ((X^T)^-1)-((X^-1)^T) is not zero, it's the zero matrix, and taking the factorial of the zero matrix is nonsensical.

[AsteriskAmp]

Yes, absolutely. But ((X^T)^-1)-((X^-1)^T) is not zero, it's the zero matrix, and taking the factorial of the zero matrix is nonsensical.

Oh, forgot...

Just stick a "det" to the right of the parenthesis, add another layer of parenthesis to indicate factorial of determinant and it does work out. Which is what you originally suggested.

And this is why parenthesis are really needed, in order of operations they are just trivial (and well, for playing around with a negative sign in front of parenthesis), for slightly more complex things it's to make things make sense (and exponentiation of grouped terms)

[nedz]

I know, see my earlier post - you should always use Scientific mode, which does it correctly. Windows Calculator in Standard mode will do it incorrectly regardless of brackets, Windows Calculator in Scientific mode will do it correctly regardless of brackets.

I have seen Windows Calculator produce errors
In fact pasting "6-1x0+2/2=" produces the answer 6. Hitting '=' does give 7, then 8, ...

Try 48/2(9+3)

Oh and nice example (well almost) Heliomance

[Kd7sov]

Just posting here to say how much I love you guys. It's been enough years since I last took a math class that I don't understand all of Heliomance's equation, but watching you all discuss it is such a joy.

Try 48/2(9+3)

Error. Is that 48/(2*(9+3)) or (48/2)(9+3)?

[AsteriskAmp]

Error. Is that 48/(2*(9+3)) or (48/2)(9+3)?

Implicit multiplication is prioritized.

[noparlpf]

Just posting here to say how much I love you guys. It's been enough years since I last took a math class that I don't understand all of Heliomance's equation, but watching you all discuss it is such a joy.

I only don't get the bit with X and T or whatever. The rest I could work out if I cared, or just recognise.

Error. Is that 48/(2*(9+3)) or (48/2)(9+3)?

Neither, it's 48/2(9+3).
=48/2(12) and now just left to right
=24(12)
=288 (all in my head, this is basic stuff)
WolframAlpha agrees with me, and so does my TI-84.

[AsteriskAmp]

I only don't get the bit with X and T or whatever. The rest I could work out if I cared, or just recognise.

X is a matrix, transposing and inverting it and doing that on the reverse order are the same and give a null matrix.

Neither, it's 48/2(9+3).
=48/2(12) and now just left to right
=24(12)
=288 (all in my head, this is basic stuff)
WolframAlpha agrees with me, and so does my TI-84.

The way WolframAlpha calculates implied multiplication is odd, as well as the TI series.

Implied multiplication in general is odd in terms of order of operations because it is considered above division and explicit multiplication.

It should be: 48/(2*(9+3))
Which gives off 2.

[nedz]

2, yes, but not on Windows 7

Ed: it ignores the 2 unless you add the extra parenthesis.

[noparlpf]

X is a matrix, transposing and inverting it and doing that on the reverse order are the same and give a null matrix.

The way WolframAlpha calculates implied multiplication is odd, as well as the TI series.

Implied multiplication in general is odd in terms of order of operations because it is considered above division and explicit multiplication.

It should be: 48/(2*(9+3))
Which gives off 2.

Ah, haven't managed to learn matrices yet.

The way I was taught it, implied multiplication is just multiplication, and in PEMDAS, M and D are equivalent, it's the order from left to right that takes precedence.

[nedz]

The way I was taught it, implied multiplication is just multiplication, and in PEMDAS, M and D are equivalent, it's the order from left to right that takes precedence.

The 2 should bind to the (), but windows just throws the 2 away.

[Rawhide]

I have seen Windows Calculator produce errors
In fact pasting "6-1x0+2/2=" produces the answer 6. Hitting '=' does give 7, then 8, ...

Try 48/2(9+3)

Oh and nice example (well almost) Heliomance

That's because you're entering the wrong formula. The correct one is 6-1*0+2/2=, not 6-1x0+2/2=.

Windows Calculator does not handle implied multiplication at all. So that other example is pointless. As above, you must enter the equation into the calculator in an accepted format, or it will not produce expected results.

[ForzaFiori]

Emphasis mine. I agree with your statement. I will, however, add that it seems obvious to me that most respondants to that page did not know and follow the procedure. If my interest is for ordinary people to correctly solve the problem, it is in my interest to add brackets to make the order more obvious. Because even if it is clear and unambiguous , it's still not going to be correctly interpreted and applied by its audience.

Respectfully,

Brian P.

When something is taught in elementary school, and then used in EVERY subsequent class in that field, it should be considered common knowledge. Just like you can assume that if you ask someone where you use a period or a question mark, they should know, because they've been doing it their entire life probably, and certainly their entire school career. People should know basic grammar and punctuation, and they should know the basic rules of math.

The main problem is that sadly, in the US (and probably other nations, but not having gone through their school systems, I can't make statements about them) math is taught in an absolutely horrid manner, and being proficient in math (or any subject, really) is seen as uncool by the majority of the youth. The over dependence on calculators (which are often wrong if you don't know how to use them, and most people don't) just complicates the problem, until you have the vast majority of the population essentially illiterate in math. (and nearly illiterate in English, but that's a different problem)

basically, it's not the equation's fault. it's either the students, for not caring enough to learn the main rule in all of math, or the teachers for not stressing it's importance or teaching it well enough.

[AsteriskAmp]

Ah, haven't managed to learn matrices yet.

The way I was taught it, implied multiplication is just multiplication, and in PEMDAS, M and D are equivalent, it's the order from left to right that takes precedence.

They are rather simple, much of them is intuitive except the products which are annoying to both operate and conceptualize.

Implied multiplication is above multiplication and division. The main argument is because its main use is alongside variables and constants.

So the idea is that 1/2x is not x/2 but 1/(2x). The T series and Wolfram Alpha consider it of higher precedence when no parenthesis is present between the implied multiplication terms, but disregard the priority when a parenthesis is involved in the conjoining.

Implied multiplication is an odd component of Order of Operations because it isn't used until much later, when in algebra, by that time the division sign is no longer in use due to the fact that writing fractions is infinitely more comfortable for working around with. The majority of authors give it higher precedence by citing the way it works with variables and constants.

In this case:
48/2q
Where q=9+3
So after replacement

48/(2*(9+3))
48/24
2

[Felyndiira]

Oh, forgot...

Just stick a "det" to the right of the parenthesis, add another layer of parenthesis to indicate factorial of determinant and it does work out. Which is what you originally suggested.

And this is why parenthesis are really needed, in order of operations they are just trivial (and well, for playing around with a negative sign in front of parenthesis), for slightly more complex things it's to make things make sense (and exponentiation of grouped terms)

It would be much easier to just define X as a scalar. Since a scalar is a 1x1 matrix, the transpose of a scalar is itself, and the invert is just 1/itself. This gives us (0)! without needing a scalar transform on a matrix.

Coincidentally, I am curious what the X_bar is supposed to represent. We usually denote matrices with an uppercase letter or a bold letter, and I do not recall any operations that is denoted by a bar (other than mean, although that's a different field of math).

[noparlpf]

They are rather simple, much of them is intuitive except the products which are annoying to both operate and conceptualize.

Implied multiplication is above multiplication and division. The main argument is because its main use is alongside variables and constants.

So the idea is that 1/2x is not x/2 but 1/(2x). The T series and Wolfram Alpha consider it of higher precedence when no parenthesis is present between the implied multiplication terms, but disregard the priority when a parenthesis is involved in the conjoining.

Implied multiplication is an odd component of Order of Operations because it isn't used until much later, when in algebra, by that time the division sign is no longer in use due to the fact that writing fractions is infinitely more comfortable for working around with. The majority of authors give it higher precedence by citing the way it works with variables and constants.

In this case:
48/2q
Where q=9+3
So after replacement

48/(2*(9+3))
48/24
2

Yeah, I looked it up later and realised I was remembering it wrong. So yeah, should be 2.

[Asta Kask]

When something is taught in elementary school, and then used in EVERY subsequent class in that field, it should be considered common knowledge. Just like you can assume that if you ask someone where you use a period or a question mark, they should know, because they've been doing it their entire life probably, and certainly their entire school career. People should know basic grammar and punctuation, and they should know the basic rules of math.

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

[nedz]

Coincidentally, I am curious what the X_bar is supposed to represent. We usually denote matrices with an uppercase letter or a bold letter, and I do not recall any operations that is denoted by a bar (other than mean, although that's a different field of math).

Conjugate perhaps ?

[DeusMortuusEst]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

One would think that something that you're supposed to use frequently for several years (from about age 12 to 17, at least) is something that would stay with you...

[noparlpf]

People know and use very advanced grammar every day. People do not use what you call the basic rules of math every day. People forget what they rarely use.

XDDD
Heh, haha, wow. No, they don't. Maybe where you live people can manage grammar, but hardly anybody around here knows even some of the pretty basic stuff. The subjunctive, who vs whom, &c.

One would think that something that you're supposed to use frequently for several years (from about age 12 to 17, at least) is something that would stay with you...

Well, that's the idea of school, but...no.