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Geek Culture / Maths gurus, are you genius enough to explain...

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Seditious
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Posted: 5th Jun 2014 10:37 Edited at: 5th Jun 2014 10:38
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... this simple algebra.

Having never studied algebra I've decided to give it a go now, on the site mathsisfun.com. All is going well except I'm stuck on one specific part of an equations/formulas tutorial. Either I've misunderstood order of operations or the example is wrong.

http://www.mathsisfun.com/algebra/equation-formula.html "Changing the Subject"

This is specifically what I don't understand:

Quote: "divide both sides by l: V / hl = w"


Since I'm under the impression that dividing by l means: V / h / l as opposed to V / (h * l) which is equivalent to what is written.

Am I just being stupid and not understanding the explanation?

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easter bunny
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Posted: 5th Jun 2014 10:47 Edited at: 5th Jun 2014 10:53
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Try this:

l=2, w=3, h=4
V=24

V = l*w*h

V/h = l*w
24/4 = 2*3

V/hw = l
24/(4*3) = 2
V/h/w ALSO = l
(24/4)/3 = 2


Basically (x/y)/z is the same as x/yz. Just the second way is better because it's impossible to get the order of precedence confused.


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TheComet
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Posted: 5th Jun 2014 11:01 Edited at: 5th Jun 2014 11:06
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Quote: "Basically (x/y)/z is the same as x/yz"


No it's not, because multiplication does not precede division. x/yz is equivalent to xz/y.

(x/y)/z is equivalent to x/y/z.
x/y/z is equivalent to x/(y*z).
x/y*z is equivalent to x*z/y.

It's easier if you do division with fractions:



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Seditious
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Posted: 5th Jun 2014 11:16
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Thanks for the replies guys, though neither of you have answered my (implicit) question: is v / hl = w equivalent to v / h / l = w or v / (h * l) = w ?

As it happens I e-mailed the guy who runs the site and he's just replied with:

Quote: "I should really put () around hl to be clear: w = V/(hl)"


And that still doesn't really answer my question.

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TheComet
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Posted: 5th Jun 2014 11:21 Edited at: 5th Jun 2014 11:24
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Quote: "is v / hl = w equivalent to v / h / l = w or v / (h * l) = w"


v/hl is ambiguous, that's why he replied with that e-mail. If I go back to the website, I see there are now brackets around it: v / (hl).

v/h/l=w is equivalent to v/(h*l)=w.

v/hl could mean v/(h*l), but it could also mean l*v/h, if you consider the "/" not to be a fraction but a division operator.

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Seditious
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Posted: 5th Jun 2014 11:23 Edited at: 5th Jun 2014 11:24
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I thought there was a standard way of writing equations and formulas though, along with a standard order of operations. so for example "v/hl" would always mean "v/h/l" or whatever.

[edit]

I guess that would be written as v / h / l if so, right?

Anyway thanks for help clearing it up. *brofist*

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TheComet
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Posted: 5th Jun 2014 11:25
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Sorry for always editing my posts, you're replying too quickly. See above post for answer.

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Seditious
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Posted: 5th Jun 2014 11:31
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I think what I'm misunderstanding is whether or not joined numbers (ie. 1x, xy, etc.) are done before other operators?

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TheComet
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Posted: 5th Jun 2014 11:34 Edited at: 5th Jun 2014 11:36
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In primary school, we first learned the order of operators was:

* and ÷ comes before + and -, making 2+3*4=14 instead of 20.

We also learned that division is NOT interchangeable, i.e. 8÷2÷2 is NOT the same as 8÷(2÷2), but rather 8÷(2*2). Multiplication IS interchangeable, i.e. 2*4*8 = 8*2*4. This is still perfectly valid when dealing with algebra.

The ambiguity is coming from the fact that programmers are using the symbol "/" instead of "÷" for division, and it has become common to think that "/" actually denotes a fraction rather than a division operator.

This is wrong. / still means ÷ (both are division operators!).

If you were to write v/(hl)=w as a fraction:



Note how you don't need the brackets around h*l any more because the fraction line already implies that everything below it is encased in brackets.

People will commonly incorrectly write that as v/hl where "/" is supposed to be a fraction, where really they meant v/(hl).

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Seditious
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Posted: 5th Jun 2014 12:04
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I think I might just be understanding it now. I already understand about operator precedence (mul/div before add/sub etc) so maybe...

The way we (programmers) write equations involving division wouldn't produce the same results when written in the correct mathematical way (as in your code snippet)?

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easter bunny
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Posted: 5th Jun 2014 12:11
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That's why I always use brackets (or parentheses if you rather) in my code


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Posted: 5th Jun 2014 13:17
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Quote: "The way we (programmers) write equations involving division wouldn't produce the same results when written in the correct mathematical way (as in your code snippet)?"


Programmers don't have fraction notation, so whenever they encounter a fraction (such as in my code snippet), they have to convert it to a "one liner", which means everything underneath the fraction must be enclosed in parentheses. My code snippet would have to be converted to w=v/(h*l) or w=v/h/l and not w=v/h*l.

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Green Gandalf
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Posted: 5th Jun 2014 19:57
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Quote: "v/hl is ambiguous"


No it isn't.

The standard rule in algebra is that implied multiplication, such as hl, is done before both multiplication and division. It's just a convenient way of avoiding excessive use of brackets so, yes, v/h/l is algebraically the same as v/hl.



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TheComet
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Posted: 5th Jun 2014 20:19 Edited at: 5th Jun 2014 20:21
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Quote: "The standard rule in algebra is that implied multiplication, such as hl, is done before both multiplication and division. It's just a convenient way of avoiding excessive use of brackets so, yes, v/h/l is algebraically the same as v/hl."


Really? Well that's confusing

I mean, yeah that's an easy to understand example, but tell me please how on earth I'm supposed to know if multiplication is to be executed first in this:

5/4*x + 2/3y = 2x - 1/4y

Is that now
x*(5÷4) + 2÷(3*y) = 2*x - 1÷(4*y)
or is it
5÷(4*x) + 2÷(3*y) = 2*x - 1÷(4*y)
or maybe it's
5÷(4*x) + y*2÷3 = 2*x - y÷4

[EDIT] Or what about fractions within fractions?



Is that written
a/b/c
or
a/(b/c)

?

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Indicium
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Posted: 5th Jun 2014 20:20
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Quote: "v/h/l is algebraically the same as v/hl"


Agreed, sub in numbers if it helps ease your mind.


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TheComet
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Posted: 5th Jun 2014 20:25 Edited at: 5th Jun 2014 20:26
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Huh, interesting.

http://en.wikipedia.org/wiki/Order_of_operations
Quote: "Hence, with this interpretation we have that 1/2x is equal to (1/2)x, and not 1/(2x). However, there are examples, including in published literature, where implied multiplication is interpreted as having higher precedence than division, so that 1/2x equals 1/(2x), not (1/2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[5] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifsh*tz and the Feynman Lectures on Physics.[6] Wolfram Alpha changed in early 2013 to treat implied multiplication the same as explicit multiplication (formerly, implied multiplication without parentheses was assumed to bind stronger than explicit multiplication). 2x/2x, 2*x/2*x, and 2(x)/2(x) now all yield x2.[7] The TI 89 and TI 86 calculators also yield x2 in all three cases."


Well we all know that mathematicians are lazy, so it's understandable they've come up with implicit parentheses. I still think it's ambiguous, especially from a programmer's point of view.

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Indicium
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Posted: 5th Jun 2014 20:30
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Quote: "I still think it's ambiguous, especially from a programmer's point of view."


I do agree with you, but normally when you see equations they will be written in a non-ambiguous form. 3/2x will either be '3' on one line, and '2x' on the bottom side of the fraction, or '3x' on one line and '2' on the bottom of the fraction. This completely removes the ambiguity - it's only when it's written onto a computer it gains this ambiguity which can be fixed with parentheses quite easily.


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Green Gandalf
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Posted: 6th Jun 2014 01:06
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Quote: "I still think it's ambiguous, especially from a programmer's point of view."


Quote: "it's only when it's written onto a computer it gains this ambiguity which can be fixed with parentheses quite easily."



You don't have implied multiplication in any programming language I've come across so it doesn't arise.

I agree that adding brackets is unambiguous for everyone though.

Quote: "Well we all know that mathematicians are lazy, so it's understandable they've come up with implicit parentheses."


It's the driving force of mathematics - making things simpler. Why waste time, paper and ink by writing "area of a triangle is half the base times the height" when you could simply write "A = bh/2"? [Programmers are slightly less concerned with such things and would probably write "A = 0.5*b*h" - unless they are writing in COBOL I suppose. ]



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Neuro Fuzzy
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Posted: 6th Jun 2014 07:07 Edited at: 6th Jun 2014 07:09
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@green gandalf/everyone, it's not "real" math, it's just convention. But for what it's worth:


At the same time, in books that I've read, if there's a small sentence where someone just wants to use inline text and they write "a=b/hl" they do mean "a=b/(hl)".


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Green Gandalf
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Posted: 6th Jun 2014 14:35
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Quote: "At the same time, in books that I've read, if there's a small sentence where someone just wants to use inline text and they write "a=b/hl" they do mean "a=b/(hl)"."


Which of course contradicts your previous statement. I guess this is an oblique way for you to canvassing for votes in the forum election.

Quote: "it's not "real" math, it's just convention"


True - but the rules are there so we can all understand each other.



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TheComet
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Posted: 6th Jun 2014 14:47
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My Voyage 200 says:
> v/hl=vl/h
> true

I guess bottom line is: If it's not obvious, put some damn brackets around it!

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Green Gandalf
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Posted: 6th Jun 2014 15:34
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Quote: "My Voyage 200 "


Ask for your money back.

Many old calculators simply processed operations as they were entered so you got nonsense like "1+2*3=9". Not surprising there's so much confusion around - and a whole generation lost with maths.

The bottom line is to make sure the software you're using is written by someone who knows what they are doing.



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BiggAdd
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Posted: 6th Jun 2014 23:19
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Isn't it just because of this rule:

y = x/z = x * 1/z

Therefore:

y = x/z/a = x/z * 1/a = x/za

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Green Gandalf
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Posted: 7th Jun 2014 00:28
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Quote: "Isn't it just because of this rule:

y = x/z = x * 1/z

Therefore:

y = x/z/a = x/z * 1/a = x/za"


Where's the rule? Or, in other words why choose the second / rather than the first?



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