The Language of Math - Order of Operations, Notation, and other subjects

[Felius]

To give room for a discussion that got out of hand in a story thread ( https://forum.questionablequesting....-spider-man-marvel-chaos-gacha.39705/page-192 )

This is not about modern politics. Keep the discussion about math and about the language of math, the notation of it! Arguments that enter the subject of modern politics are likely to infringe rule 8, and should be better avoided.

And back on topic, to kick of the thread:

I think I know where the misunderstanding is coming from. The column I talk about in division is a vertical stack with a line in between the expressions. That's one way to express the same thing, and it's better that way and not the linear one. It's easier to solve the equation with it.

AND! Collumns are NOT only meant for kids. You're confusing operational columns, the so called "math columns" with the actual structure in maths. It's a structure used on all levels of maths, sine. As additional edu program in med institute in my country we had to have broader understanding of maths, cuz physics are part of the reason why biology works in the first place, that's biophysics for a reason, so we were taught about differential equations on the 1st year and I do remember that matrices with their collumns are in fact needed to solve some of the problems.

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Just to clarify one small point, the big issue with this is mostly that Latex is a pain, specially as it starts to go for formulas more and more complex. Or rather, it's not strictly a pain, but learning it is basically like learning a whole programming language just for the purpose of writing math formulas; Unless you're a mathematician (or an academic in a field that massively uses complex math), or a massive math nerd, it's generally not something people find worth their time to learn.

\[ \frac{2a}{3b} \]

 

[Talerax]

\[ \frac{2a}{3b} \]

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Today I learn something new.

 

[JuanJuan]

Just to clarify one small point, the big issue with this is mostly that Latex is a pain, specially as it starts to go for formulas more and more complex. Or rather, it's not strictly a pain, but learning it is basically like learning a whole programming language just for the purpose of writing math formulas; Unless you're a mathematician (or an academic in a field that massively uses complex math), or a massive math nerd, it's generally not something people find worth their time to learn.

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See the glory of column, my eyes shine with its blessed light! Jokes aside tho, I didn't even knew that xeno supports it, heel yeah it's be goddamn good to write it the way it is supposed to be!

\[ V_m = \frac{RT}{F} \ln \left( \frac{P_{\text{Na}}[\text{Na}^+]_{\text{ECF}} + P_{\text{K}}[\text{K}^+]_{\text{ECF}}}{P_{\text{Na}}[\text{Na}^+]_{\text{ICF}} + P_{\text{K}}[\text{K}^+]_{\text{ICF}}} \right) \]

Today I learn something new.

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And it's great, I got to know something actually cool today

 

[Caelleh]

As a Texan educated bachelors holding 10year practicing computer engineer degree holder, when I read

8/2(4)

I see it as 8 divided by 2(4), which then becomes 8 divided by (2x4), or 8 divided by 8.

I don't see it as 8/2 times 4. The parentheses/brackets always matter and need to be addressed. I don't see ambiguity here, that's the language I read it in.

 

[Als]

As a Texan educated bachelors holding 10year practicing computer engineer degree holder, when I read

8/2(4)

I see it as 8 divided by 2(4), which then becomes 8 divided by (2x4), or 8 divided by 8.

I don't see it as 8/2 times 4. The parentheses/brackets always matter and need to be addressed. I don't see ambiguity here, that's the language I read it in.

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The problem is as an italian bachelor holder of a 2 year computer engineer degree when i see:
8/2(4) i read it as 8÷2×4 = 4×4 = 16 , the problem is that for you multiplication came first of division, for me no, it's not inherently wrong, but if in my calculus class i had wrot the result as 1, it would be marked as wrong.

 

[Felius]

See the glory of column, my eyes shine with its blessed light! Jokes aside tho, I didn't even knew that xeno supports it, heel yeah it's be goddamn good to write it the way it is supposed to be!

\[ V_m = \frac{RT}{F} \ln \left( \frac{P_{\text{Na}}[\text{Na}^+]_{\text{ECF}} + P_{\text{K}}[\text{K}^+]_{\text{ECF}}}{P_{\text{Na}}[\text{Na}^+]_{\text{ICF}} + P_{\text{K}}[\text{K}^+]_{\text{ICF}}} \right) \]

And it's great, I got to know something actually cool today

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Which does help to prove my point in a way. While the output of the latex form is quite readable, learning to write this bloody thing:

V_m = \frac{RT}{F} \ln \left( \frac{P_{\text{Na}}[\text{Na}^+]_{\text{ECF}} + P_{\text{K}}[\text{K}^+]_{\text{ECF}}}{P_{\text{Na}}[\text{Na}^+]_{\text{ICF}} + P_{\text{K}}[\text{K}^+]_{\text{ICF}}} \right)

Is not exactly trivial. And this is not even a particularly complicated equation.

As a Texan educated bachelors holding 10year practicing computer engineer degree holder, when I read

8/2(4)

I see it as 8 divided by 2(4), which then becomes 8 divided by (2x4), or 8 divided by 8.

I don't see it as 8/2 times 4. The parentheses/brackets always matter and need to be addressed. I don't see ambiguity here, that's the language I read it in.

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That might be true within your experience, within your school and your university (that, while heavily using math, is not a math degree, specially not one focused on academics; It's not a demerit, but it is a degree more focused on using math as opposed to explaining math and the premises used).

The ambiguity is not really within the scope of "two people from the same school and same teachers", but in the greater academic context, specially taking into account places outside the US of A. Math is math, but the language that ones uses to write down the math changes, sometimes in rather subtle ways that can lead to very wrong results.

People in fields that are users of math often don't hear about that sort of discussion and controversy, but it's very real in fields of (Pure and Applied) Math, specially in Academia.

 

[RomanQrr]

Like I said in the thread and like I tell my students I tutor: division symbols are a sin, use fraction lines instead.

I've been schooled in a school attached to a science university, so we stopped using division symbols somewhere around 3rd year of middle school if not earlier. And that carried all the way into my math classes in uni.

Let's just say I have been very unpleasantly surprised when my highschool students suddenly started not only using division symbols, but actively converting fractions into in-line division (while also forgetting to put down the implied brackets and thus making a huge mess).

Also, this entire conversation revealed to me that I actually follow what sinereal calls "Old Math". I just don't realise that because I generally don't touch division until multiplication is done. Again, because of the fraction lines.

 

[Splattered Cloak]

There is an issue where implicit multiplication appears to bind more tightly than explicit multiplication, but nobody gets taught that.

Now I wish I'd kept my old college physics homework to see what I really did, but I suspect it contains many instances of multiplication after a slash as part of the denominator. But anything too ambiguous, I'd have parenthesized. Including any instances of division within the denominator; I simply cannot abide the machine interpretation of x/y/z.

For that matter, I once wrote a simple calculator that refused to parse that notation. I can understand why early Fortran allowed it, though; it's annoying enough to debug a program when you get instant results; having to submit punchcards to a queue just to be rejected for a syntax issue would be agonizing.

Honestly though, ISO should just make a proper international standard convention for order of operations.

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ISO can't even get people to agree on which of 5.000 or 5,000 is greater.

 

[JuanJuan]

As a Texan educated bachelors holding 10year practicing computer engineer degree holder, when I read

8/2(4)

I see it as 8 divided by 2(4), which then becomes 8 divided by (2x4), or 8 divided by 8.

I don't see it as 8/2 times 4. The parentheses/brackets always matter and need to be addressed. I don't see ambiguity here, that's the language I read it in.

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The whole equation is a goddamn bait. Its meant to be confusing. The real answer for me is 16.

\[ \frac{8}{2}(2 + 2) = 16 \]

 

[JuanJuan]

Like I said in the thread and like I tell my students I tutor: division symbols are a sin, use fraction lines instead.

I've been schooled in a school attached to a science university, so we stopped using division symbols somewhere around 3rd year of middle school if not earlier. And that carried all the way into my math classes in uni.

Let's just say I have been very unpleasantly surprised when my highschool students suddenly started not only using division symbols, but actively converting fractions into in-line division (while also forgetting to put down the implied brackets and thus making a huge mess).

Also, this entire conversation revealed to me that I actually follow what sinereal calls "Old Math". I just don't realise that because I generally don't touch division until multiplication is done. Again, because of the fraction lines.

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Yeah, mate, you get it! I totally forgot about how they are called in English, FRACTION SIGNS, THANK YOU!

 

[Felius]

The whole equation is a goddamn bait. Its meant to be confusing. The real answer for me is 16.

\[ \frac{8}{2}(2 + 2) = 16 \]

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You did. The problem is that a lot of people parse that, specially because of the implied multiplication, as

\[ \frac{8}{2(2 + 2)} = 1 \]

 

[Lewd11]

There never should be anything implicit in math.

 

[JuanJuan]

You did. The problem is that a lot of people parse that, specially because of the implied multiplication, as

\[ \frac{8}{2(2 + 2)} = 1 \]

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Yeah, that's why the whole thing is a bait. The in line equation divide people in two groups. People think of different things thinking they are right. They are, but the moment they try to explain why...

 

[Felius]

There never should be anything implicit in math.

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In that case it's not really a "implicit" as in "unwritten rules", but just a matter of notation and omiting a symbol, so you don't write

\[ 2*a+b \]

but instead just

\[ 2a+b \]

The problem comes not from the difference in notation, but that people tend to attribute different "weights" to it, thus

\[ {2}\times{a}\div{3}\times{b} \]

gets parsed differently than by some people

\[ 2a\div3b \]

 

[Talerax]

As a Texan educated bachelors holding 10year practicing computer engineer degree holder, when I read

8/2(4)

I see it as 8 divided by 2(4), which then becomes 8 divided by (2x4), or 8 divided by 8.

I don't see it as 8/2 times 4. The parentheses/brackets always matter and need to be addressed. I don't see ambiguity here, that's the language I read it in.

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I believe you see that as 8 divided by 2(4) because it resembles the form of X/YZ, as in

\[ \frac{X}{YZ} \]

The problem is, the in-line equation can also be read as 8/2*(4) too, and with an explicit multiplication like this instead of the implicit one in 8/2(4), the answer can be different, depend on how you interpreted it now. Which means it is bad math.

 

[TheWave]

This 8/2(2+2) is one of those troll social media posts that baits engagement becasue it is an ambiguous shorthand, used deliberately to confuse. Use one more set of brackets to remove ambiguity, and it either becomes (8/2)(2+2)=16 or 8/(2(2+2))=1 , or use fraction lines

\[ \frac{8}{2}(2+2) = 16 \]

\[ \frac{8}{2(2+2)} = 1 \]

like a normal person and there is no more ambiguity.
Maths is a language, and if you speak unclearly and ask a vague question, then you should not be surprised at getting multiple answers that each belive they are correct.
Hell, there was an entire thing about multiplication always happening before divison, like division is not simply multiplication flipped on its head:

\[ X÷Y = X*\frac{1}{Y}\ \]

Edit: LaTeX is weird to write in, but seems interesting to learn.

 

[JuanJuan]

This 8/2(2+2) is one of those troll social media posts that baits engagement becasue it is an ambiguous shorthand, used deliberately to confuse. Use one more set of brackets to remove ambiguity, and it either becomes (8/2)(2+2)=16 or 8/(2(2+2))=1 , or use fraction lines

\[ \frac{8}{2}(2+2) = 16 \]

\[ \frac{8}{2(2+2)} = 1 \]

like a normal person and there is no more ambiguity.
Maths is a language, and if you speak unclearly and ask a vague question, then you should not be surprised at getting multiple answers that each belive they are correct.
Hell, there was an entire thing about multiplication always happening before divison, like division is not simply multiplication flipped on its head, like X÷Y = X*(1/Y)

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It's because of PEDMAS and that BODMAS thing I think. Sine was so hard on because of it.

 

[kalifianto]

https://en.wikipedia.org/wiki/Reverse_Polish_notation RPN is an alternative mathematical notation where we can ignore order of operations and parentheses, but we need to care about the stack (the current result of operations) and how many numbers do each operands take, which is weird.
e.g.
1 2 - 3 x
= -1 3 x
= -3

while the equivalent in infix notation is (1 - 2) x 3

It's simple to implement and use, which is why HP calculators used to use it, and Forth, one of the earliest low-to-high-level language, is based on it (stack-based language, with Words and side-effectful functions applied to the stack and additional data as necessary)

A combination of Forth & APL (array programming language) yield https://www.uiua.org/ which is really fun to play with, and allow you to do complex algorithms in a few lines.

e.g. the following program

[1 5 8 2]
⟜/+ # Sum
⧻ # Length
÷ # Divide

lead to:
[1 5 8 2] in the stack
[1 5 8 2], 16 -> apply copy, sum
4, 16 -> apply length on first array
4 -> apply divide

Another alternative notation is "prefix" notation, where you put your operators in the leftmost position, but you'll still need parentheses to make sure you know which operators are going to be applied first.
This is also natural in another programming language family -> Lisp family, which, for the above example:
(/ (sum arr) (len arr))
which means you don't need to memorize the order of operations too, without the need for implicit stack, but with the need to use parentheses there

Sorry that my examples both use programming languages rather than mathematics itself, but it's fun to see why you'd like to use alternative notations, no?

 

[zecorezecron]

I was taught in the Old Ways and the New. My teacher, with a masters degree, would have absolutely never allowed anyone to treat multiplication and division as separate steps in order of operations.

N/2 is equal to .5×N, and your order of operations should allow perfect substitution between them without changing the answer. Same with 2N and N/.5

The real issue, as said earlier, is how everyone and their mom has different ideas about inline implicit equations.

As far as i am concerned, as an engineer that uses math,

___8___
2×(2+2)

should be written inline as

8/(2×(2+2))
Or
(8/2)/(2+2).
Or
(8/2)×(1/(2+2))
Or
8×.5×.25 after simplification.

Not 8/2(2+2)

You put the wholeass denominator in parenthesis. Otherwise it is dividing by the first term only.

N/AB is (N/A)B, not N/(AB)
Thesecondoneislikewritingwithoutspacesbetweenwords. It's the difference between a badass hole and a bad asshole.

It's not P>E>M>D>A>S, it is P>E>M/D>A/S.

Division is multiplying a fraction.
Subtraction is adding a negative.
Those steps are concurrent, not sequential.

TLDR, use parentheses motherfuckers, you ain't gonna run out.

 

[IlAg]

Stupid question from 16 follower) 6-3+2=1, if addition is before subtraction?
Just to understand logic of strict PEMDAS

 

[aladdin47]

It's because of PEDMAS and that BODMAS thing I think. Sine was so hard on because of it.

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The dumbest thing about this whole discussion is that PEDMAS is just a way for kids in USA to easily remember the order, not an actual rule as to only be used as written.

Even arithmetic book written as far as 1932 by Kiselev says that division and multiplication are the same class in importance order and should go in order from left to right, no ambiguity possible.

 

[joethelesser]

"the error is being made primarily not by those who give "wrong" answers, but by those who post the problem in the first place (or pass it on)."

The ambiguous way the equasions are shown in line is just bait.
For ease and correctness, you'd add parenthases to show better order, OR, you'd correctly line up the fraction lines.

```
6
--------
2(2 + 1)

or

6
--- (2 + 1)
2
```

You put the wholeass denominator in parenthesis. Otherwise it is dividing by the first term only.

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This is also my take. In biology we used less math than physics and chemistry, but specificity is always key.