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charliealphabravo (Structural) (OP)
21 Aug 12 14:05
OK, so my architect boss is trying to stump me with this one. He says he saw it on some discussion group that was saying that engineers are poor in math because they answer 7. That was my answer as well, and since I am an engineer I must ergo be bad at math.

I am pretty sure there is no planet where the answer to this is 5, but what do I know?

Discuss.
structSU10 (Structural)
21 Aug 12 14:14
PEMDAS... so unless there are parenthesis missing, 7 is what i get. The only way to get 5 i see is (6-1)*(0+(2/2)).
dik (Structural)
21 Aug 12 14:45
It's the mathematical order of operations... which simplistically is as follows:

1. terms inside parentheses or brackets
2. exponents and roots
3. multiplication and division
4. addition and subtraction

Do your mulitplication first and end up with -0, then the division 2/2 = 1, and then you have a simple 6+1 = 7

Dik
stevenal (Electrical)
21 Aug 12 14:57
The question was asked in the spreadsheet forum. Microsoft says 7.
cowski (Mechanical)
21 Aug 12 14:59
If you have excel on your computer, enter it as a formula and see what you get (the programmers and mathematicians at MS agree the correct answer is 7). If he disagrees, he should file a bug report with MS and any/every other calculation software (mathcad, maple, mathematica, etc).

www.nxjournaling.com

cowski (Mechanical)
21 Aug 12 15:02
Anyone have a link to the original "discussion"? I'd love to see some of the logic (or lack thereof).

www.nxjournaling.com

Helpful Member!  BigInch (Petroleum)
21 Aug 12 15:05
If you type that expression in Excel using Architectural font, the answer is 1.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

Denial (Structural)
21 Aug 12 17:28
What answer do you want?
If I apply the operations in left-to-right order, I get 1.
If I apply them right-to-left, I get 5.
If I also read the numbers right-to-left, I get -5.
If I apply the operations alphabetically, in English, I also get 5.
Spanish, anyone?.
Martian?

However if I use the long-established and universally-observed rules of (earth-bound) mathematics, I get the same as everyone else on this post (except BigInch's "architectural font", apparently). That's one of the beauties of mathematics, and allows it to be described as the universal language.

But only among those with rudimentary mathematical literacy.
BigInch (Petroleum)
21 Aug 12 17:38
Spanish could be as high as 8.
The time between same days of one week to the next is often described as 8 days.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

IDS (Civil/Environmental)
21 Aug 12 19:43
I haven't got Architectural font, but I think Wingdings 3 gives the most appropriate response.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

fel3 (Civil/Environmental)
21 Aug 12 20:19
The only unambiguous approach to this problem is to invoke the standard order of operations rules that dik posted. We all agree that this results in 7. Mathcad also agrees.

The architect is full of himself. In my experience, engineers are generally better at mathematics than everyone except mathematicians and probably physicists. I have worked with dozens of very smart architects and none of them have the math skills of a typical engineer, except for two who also have an engineering degree on top of their architecture degree.

==========
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill

msquared48 (Structural)
22 Aug 12 6:36
Mathematics is supposed to be the universal language.

This string is universally confusing to me. My logic says 1.

Mike McCann
MMC Engineering
http://mmcengineering.tripod.com

electricpete (Electrical)
22 Aug 12 8:12
msquared - typical convention: Multiplication/subraction has higher precedence than addition.
6 - 1 x 0 + 2 / 2 = 6 - (1 x 0) + (2 / 2) = 6 - 0 + 1 = 7

I apologize if I missed the meaning of your post

=====================================
(2B)+(2B)' ?

electricpete (Electrical)
22 Aug 12 8:20
No coffee yet this morning...
meant to say "Multiplication/division has higher precedence than addition/subtraction"

=====================================
(2B)+(2B)' ?

IRstuff (Aerospace)
22 Aug 12 10:35
There should be no confusion about this. Order of operations is mandatory, not optional. This is not right to left convention; this is not described as a processor stack operation. Left to right has lower precedence than PEMDAS.

This is a fundamental convention that all math programs and textbooks have to adhere to, minus signs in Excel not withstanding.

TTFN
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stevenal (Electrical)
22 Aug 12 14:02

Quote (Irstuff)

Left to right has lower precedence than PEMDAS

It would seem the PEMDAS acronym is the problem here, puting the A before the S. In fact, order of operations states these two operations are at the same level, and should be performed left to right.

http://www.facebook.com/permalink.php?story_fbid=4...

PEMDAS as interpreted by some where addtion is performed before subtraction: 6-[(1*0)+(2/2)]=5

Actual order of operations performs left subtraction prior to the addition: [6-(1*0)]+(2/2)=7
IRstuff (Aerospace)
22 Aug 12 14:31
"6-[(1*0)+(2/2)]=5"

PEMDAS cannot assume this first case, because there are NO PARENTHESES in the original statement; you cannot arbitrarily add parentheses where none were given. Any such "assumptions" are grossly incorrect.

TTFN
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rb1957 (Aerospace)
22 Aug 12 14:46
i don't know about that ...

addition and subtraction are the same level operators, it doesn't matter which order that the sums are done in.
6-0 = 6, 6+1 = 7
-0+1 = 1, 6+1 = 7

i think to interpret 6-0+1 as 5 is Wrong, it implies brackets around 0+1 which would change the order of operations.

it is obvious (no?) that adding or subtracting zero has no effect on the result.
handleman (Automotive)
22 Aug 12 15:46
PEMDAS is an acronym to help remember a rule. It is not the rule itself, and is indeed an imperfect representation. PEMODAOS (Parens, Exp, Mult Or Div, Add Or Sub) is more correct but not as easy to remember.

There are lots of people out there who are just smart enough memorize acronyms, but completely incapable of really understanding the underlying concepts. Apparently, your architect fellow falls into that category.

-handleman, CSWP (The new, easy test)

cowski (Mechanical)
22 Aug 12 15:52
The negative sign is a unary operator:
6 - 1 is shorthand for: 6 + (-1)
I think the problem is some people are trying to separate the negative sign from the 1 then reusing it (incorrectly) as a later operation.
6 - 1 * 0 + 2 / 2
1 * 0 = 0
6 - + 2 / 2 =
6 - 1 =
5


The negative sign stays with the 1
-1 * 0 = 0
6 + 2 / 2 =
6 + 1 =
7

www.nxjournaling.com

handleman (Automotive)
22 Aug 12 16:03
cowski,

In this expression, the - is not unary. If it were unary, the expression would be meaningless due to lack of an operator between the 6 and the -1.

Neither of your scenarios is correct. Just because the second yields the correct answer in this case doesn't make it less wrong.

1*0=0. -1*0=0. Zero is different from nothing. You can't just take it out and make it disappear. You have to keep the zero in there. The subtraction operator then acts on the zero.

6-1*0+2/2 becomes 6-0+2/2

-handleman, CSWP (The new, easy test)

cowski (Mechanical)
22 Aug 12 16:56
The first scenario is incorrect, that is my point. I was trying to figure out how they were coming up with 5.

I stand by my assertion that 6 - 1 is shorthand for 6 + (-1).

www.nxjournaling.com

MiketheEngineer (Structural)
22 Aug 12 17:12
Going left to right I get 1!!!
DRWeig (Electrical)
22 Aug 12 18:59
Stevenal and handleman have the right interpretation of PEMDAS.

P = Parentheses
E = Exponent
MD = Multiply or divide
AS = Add or subtract

Mathematicians who set the standard to begin with follow this.

Good on ya,

Goober Dave

Haven't see the forum policies? Do so now: Forum Policies

Helpful Member!  IRstuff (Aerospace)
22 Aug 12 19:10
This is no other interpretation to be had. EVERY textbook follows these rules; if you've passed 5th grade, you learned these rules.

If you haven't passed 5th grade, you have no business arguing about this.

TTFN
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poli60 (Chemical)
23 Aug 12 1:35
Very interesting thread since it's a sign of the time.
When I was young anyone more than 5 years old knew the right answer: 7!
Maybe the only exception were the architects...
Now we have computers!!!
IRstuff (Aerospace)
23 Aug 12 2:24
Computers are programmed by people; Nonetheless, Mathcad, Excel, TI nSpire, Casio fx-115ES, Wolfram Alpha all give the same answer, 7

I don't get why this is even an issue of any sort.

TTFN
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MiketheEngineer (Structural)
23 Aug 12 10:39
Are you smarter than a 5th grader ----- cute TV show!!
IRstuff (Aerospace)
23 Aug 12 10:49
That depends greatly on where that 5th grader is from. I recall that when I was in school, transfer students from Hong Kong were about 2 years ahead of us in math, i.e., they had finished algebra by 7th grade, when we were just getting started.

TTFN
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IRstuff (Aerospace)
23 Aug 12 10:54
One possible interpretation of the problem is in the usage of 1x0 rather than 1*0, which is the keyboard convention in the US.

If the "0" is actually a letter "o", then it might represent 1 in octal representation, resulting in 6-1+2/2 = 6.

TTFN
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stevenal (Electrical)
23 Aug 12 11:32
Irstuff,

The parenthesis were not added arbitrarily, I added them to show operation order; both incorrectly and then correctly. The wrong way performs addition prior to subtraction as advocated by many on the Facebook forum and evidently by at least one architect.

All,
I think most ( but not all; see Facebook)everyone can agree the expression simplifies to 6-0+1. Left to right works. 6+(-0)+1 works. Addition prior to subtraction as in 6-(0+1) does not.
IRstuff (Aerospace)
23 Aug 12 13:11
Since when are Facebook entries even remotely relevant? Input from people who can't remember how to do math is not worth dissecting.

TTFN
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BigInch (Petroleum)
23 Aug 12 13:38
IRstuff,
"I don't get why this is even an issue of any sort. "
Don't worry. It's only an issue to those that don't get 7.

All the rest of us have better things to do than argue about this. I'm moving on. I suggest you and the rest of us 7's do too.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

stevenal (Electrical)
23 Aug 12 15:11
I added the link as a likely answer to Cowski's question 21 Aug 12 15:02.
In this thread the question began with an architect who can't remember math. Entirely relevant to the topic at hand.
Denial (Structural)
23 Aug 12 17:04
Nicely put, IRstuff.
I would expect nothing better from Facebook.
But I find it hugely depressing that such a "controversy" can run on a site supposedly for numerate professionals.
Denial (Structural)
23 Aug 12 17:10
BTW. My post above was not aimed at Stevenal, but at the general notion that Facebook might have anything useful to say about anything. Reading that Facebook page gave me my first smile for this entire thread, even if it was a slightly perverted & convoluted smile.
Hoaokapohaku (Geotechnical)
23 Aug 12 21:11
The discussion on Facebook shows why the USA is in trouble. That the discussion even exists here (except for laughs) shows why the USA is in DEEP trouble.
IDS (Civil/Environmental)
23 Aug 12 21:41
I think that looking at how simple unambiguous rules can be misinterpreted is both interesting and instructive.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

IRstuff (Aerospace)
24 Aug 12 1:33
OK, I couldn't see the Facebook page at work, and my worst fears were confirmed, it's another version of Yahoo!Answers, which is nearly worthless

TTFN
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Drew08 (Civil/Environmental)
24 Aug 12 8:53
I got 2.228169pi....ummmm pie.
dgallup (Automotive)
24 Aug 12 11:42
7

----------------------------------------

The Help for this program was created in Windows Help format, which depends on a feature that isn't included in this version of Windows.

xwb (Computer)
24 Aug 12 18:09
I've never heard of PEMDAS: had to look it up. I learnt it as BODMAS.

What's interesting is that I was taught that O stood for of, an alternative to multiplication: not order (equivalent of exponents). Guess it is a bit difficult teaching a 6 year old what exponent means.
stevenal (Electrical)
24 Aug 12 18:46
I never heard of either of them til now, and managed to learn order of operations without them. Perhaps misleading acronyms should be retired.

Is there really any controversy here? Except for a few wisecracks, did anyone here really think the answer was not 7?

If you like controversy, I offer:

-3^2=?
IDS (Civil/Environmental)
24 Aug 12 19:03

Quote:

Is there really any controversy here?

The only controversy seems to be whether the whole discussion is a complete waste of time, or whether it is something worth talking about.

In my opinion it is worth talking about, if only as a reminder that we should not assume our non-engineer colleagues are numerate.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

IDS (Civil/Environmental)
24 Aug 12 19:14

Quote:

If you like controversy, I offer:

-3^2=?

Interestingly, if you enter -3^2 in Lotus 123 it tells you the answer is -9. If you save the file as xls and open in Excel it still says -9, but it now displays in the edit box as =-(3^2).

1-3^2 = -8 in both programs, so Excel doesn't need to add any brackets in this case.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

Denial (Structural)
24 Aug 12 23:21
XWB. I also learned it as BODMAS, which I suspect shows both our ages. The O does stand for "of", and I never understood why "of" should have any priority over other ways of expressing a multiplication operation.

As for the absence of exponentiation in that aide-memoire, I suspect it comes about because way back then (at school level at least) exponentiation was denoted without any operator at all (32 rather than 3^2) so the issue did not arise. Also, as has already been pointed out above, we are talking here about a cryptic aide-memoire to (parts of) the universal convention, not the formal definition of the full convention.
IRstuff (Aerospace)
25 Aug 12 0:54
The minus sign unary operator subject has been beat to death a couple of times in the spreadsheet forum already. PEMDAS resolves both expressions, differently, but resulting in the same end result.

TTFN
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IDS (Civil/Environmental)
25 Aug 12 4:32

Quote:

The minus sign unary operator subject has been beat to death a couple of times in the spreadsheet forum already. PEMDAS resolves both expressions, differently, but resulting in the same end result.

I'm not sure how one spreadsheet giving 9 and another one giving -9 for the same input is "resulting in the same end result".

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

BigInch (Petroleum)
25 Aug 12 9:11
They're both wrong.

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

IRstuff (Aerospace)
25 Aug 12 12:34
"=-(3^2)"
"1-3^2 = -8"

TTFN
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desrod (Electrical)
29 Aug 12 7:57
The 5 is the result of the architect's quote maths
The 7 is actual maths

We now know why architectural projects are over budget most of the time!

Daniel D
Adamshive (Mechanical)
8 Sep 12 21:43
I joined the forum simply because of this question.
I had originally said the answer was 5.
I work in CNC Machining and lines of code and be interpreted differently by different machine drivers.

A very basic yet extreme example with something like 6-0+5 could be interpreted by the computer saying "ok, well everything after the - is negative". Ok, bear with me I know there's a zero there. Some lines of code will literally drop a zero or "uncheck" it as if each character in a line were on a checklist.

Some lines of code won't be interpreted until after each character is accounted for, the driver processes whatever algorithm and creates a result.

A CNC operator may look at this as "blocks" of code. Some drivers will group long lines of code into smaller ones.

One of my biggest arguments is that a machine cannot do "0" or nothing. It could very easily read this and say "subtract zero? I can't do that...hmm but I have to subtract something so I'll go to the next character set".

The obvious answer is 7, but I think there is a valid argument for 5 under certain circumstances and even 1 which would be strictly left-to-right.

By the way- the #1 answer on Facebook in USA was 1. Go figure.

Does any of this make sense or am I trying to validate something that isn't there?
Denial (Structural)
8 Sep 12 22:45
That is an interesting new perspective, Adamshive.

Also very worrying. It would seem that the CNC industry wants to re-write the long-standing conventions of mathematics, but does not even have a consistent convention to offer as an alternative.

How is this problem currently dealt with, out there at the coal face?
Adamshive (Mechanical)
8 Sep 12 23:53
Haha, well it's not rewriting math it's just turning into code that a post-processor can understand.

In the early days on NC machines a single line of text was punched into a ticker-tape which would then be optically processed by the machine. The roots of these unique codes could well be in those single lines but that was way before my time.

I guess it really is comparing apples to oranges since its essentially and industry-specific language but I thought it was funny that I dropped the 0 without batting an eyelash.

Something else interesting that we see almost every day is how we're taught to round up so many decimal spaces while some processors simply drop decimal spaces. So 9.9969 is 9.996 to some computers when we'd just round up to 9.997. Since our tolerances are ridiculously small- single digit microns- you can see where it can be a pain.

Also the "order of execution" in g-code rings a bell and I believe that is what overrides the conventions of mathematics. It's been awhile since I've used pure g-code though so don't quote me.
IRstuff (Aerospace)
9 Sep 12 0:03
I get that; I used to program a machine that read paper tape one byte at a time (don't ask how long ago).

It's incumbent on the programmer to parse the expression correctly to ensure that they get what they want. Therefore, any decent machine language or assembly programmer knows to what to put on the stack and in what order. That's one thing that using RPN instills in you, since that's the only way to get the correct answer in RPN.

TTFN
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Adamshive (Mechanical)
9 Sep 12 18:02
I just found out that this has less (or nothing, mind you) to do with cnc code and more to do with an improper redundant zero "rule".

A friend at Harvard said that using the order of operations would get you 7. Dropping the redundant 0 gives you 5. He also mentioned a whole lot of stuff about axioms and things I didn't understand.
Helpful Member!  IRstuff (Aerospace)
9 Sep 12 18:25
The problem, of course, is that it's not a "redundant zero", but the improper handling of the " - 0 + " sequence. You cannot eliminate the "0" without eliminating its associated "-" sign. That's just BAD MATH. Replace the "0" with "x", and you wouldn't be tempted to do that sort of nonsense, i.e., "6 - 1 * x + 2 / 2 must equal 7 - x. That clearly demonstrates that if x=0, the answer is 7, and absolutely nothing else. The rules of math must be consistent with whatever is in the "0" slot. You cannot arbitrarily change the way the equation works if x=0; that's just plain nonsensical, BAD MATH.

TTFN
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Adamshive (Mechanical)
9 Sep 12 19:33
^reminds me of the time I took some BAD METH. Woke up three days later at a Tijuana donkey show.
charliealphabravo (Structural) (OP)
9 Sep 12 21:01
yikes, isn't that from a movie?

...but to your point. As long as you know the rules that the CNC code interpreter is using then a programmer can get the correct answer from it. It is not really bad math and it doesn't change the order of operations required to get the correct answer.

msquared48 (Structural)
10 Sep 12 0:00
It is interesting to note that a similar question on the net yielded a 73% wrong result. If math is the universal language, it should be more universally understood, and correctly.

The problem here is that the average person does not remember the "heirarchy" of the operations as it were. That is why the use of parenthesis and brackets. That nomenclature leaves absolutely no doubt of the sequence of the operations required. That's why I use them all the time - accurate communications without having to remember mankind's confusing "rules".

Mike McCann
MMC Engineering
http://mmcengineering.tripod.com

IRstuff (Aerospace)
10 Sep 12 1:17
"s long as you know the rules that the CNC code interpreter is using then a programmer can get the correct answer from it. It is not really bad math and it doesn't change the order of operations required to get the correct answer. "

It is bad math if the resulting answer is wrong, unless they're willing to cop to plain old incompetence. But, that's most likely because modern programmers are actually CS majors, as opposed to math majors that had to find a job after graduation, like in the old days.

TTFN
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boottmills (Mechanical)
10 Sep 12 8:58
The one thing that scare me most about this, is how many of the people on the facebook thread (which I believe is where this originated from) are getting the answer incorrect. That's the future of our world folks...

Boottmills

jhardy1 (Structural)
13 Sep 12 18:22
It's scary enough that people relay so heavily on search engines like Google etc to answer just about any question. (At least Google agrees that "6 - 1 x 0 + 2 / 2" is 7: https://www.google.com.au/search?q=6+-+1+x+0+%2B+2... ) But try asking Yahoo: http://answers.yahoo.com/question/index?qid=201208... - the "Top Answer" there at the moment is 1!

But accessing social media sites like Facebook to resolve your maths problems? Now that's REALLY scary!

http://julianh72.blogspot.com

jhardy1 (Structural)
13 Sep 12 18:29
But try asking Yahoo: http://answers.yahoo.com/question/index?qid=201208... - the "Top Answer" there at the moment is 1!

(Has anyone EVER had a useful result to ANY question they have found on "Yahoo! Answers"? surprise )

And I love the fact that the following poster quite rightly corrects the "Best Answer", saying that you have to use PEMDAS - but then goes on to "prove" that the "correct" answer using PEMDAS is "5", and signs off with the following tag-line:

"Years of schooling, and a COMPLETE understanding of the PEMDAS method."

Priceless!

http://julianh72.blogspot.com

IRstuff (Aerospace)
13 Sep 12 18:56
I've GIVEN useful results to questions there, and racked up about 13,000 points before I gave up. The current leader has over 1 million points. After a while there, I realized that it was too pitiful to spend much time there.

It seems like the question has been deleted from the site.

TTFN
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IFRs (Petroleum)
26 Sep 12 16:55
This is just too hard a problem! I've been using 6 - 3 x 2 which the correct answer is zero but half of those I have asked said six. When they say six, I ask them to check it on their smartphone which tend to give the correct answer. Many calculators give the wrong answer. I'd love to give this to the folks in Congress!
Helpful Member!  charliealphabravo (Structural) (OP)
26 Sep 12 17:02
I need to kill this thread before it proves my boss right :8\
jimstructures (Structural)
27 Sep 12 10:38
HP 48GX - '6-3*2' eval = 0
KENAT (Mechanical)
27 Sep 12 10:45
What's this PEMDAS nonsense, it's BODMAS thank you and still gets 7.

Posting guidelines FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm? (probably not aimed specifically at you)
What is Engineering anyway: FAQ1088-1484: In layman terms, what is "engineering"?

CorBlimeyLimey (Mechanical)
27 Sep 12 13:00

Quote (Denial)

XWB. I also learned it as BODMAS, which I suspect shows both our ages. The O does stand for "of", and I never understood why "of" should have any priority over other ways of expressing a multiplication operation.
As for the absence of exponentiation in that aide-memoire...
If I remember correctly, the "O" stands for "Order", not "of" ... and "Order" (UK) equates to "Exponent" (US)
KENAT (Mechanical)
27 Sep 12 13:28
I was taught "O" as 'of' in the UK in the early 90's but this was before I'd done much with exponents other than squared.

Posting guidelines FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm? (probably not aimed specifically at you)
What is Engineering anyway: FAQ1088-1484: In layman terms, what is "engineering"?

dgallup (Automotive)
27 Sep 12 14:27
I think either of those acronyms is too complex and obscure to be useful as all these posts about their meaning have demonstrated. Just learn the proper order of precedence and use it.

----------------------------------------

The Help for this program was created in Windows Help format, which depends on a feature that isn't included in this version of Windows.

IRstuff (Aerospace)
27 Sep 12 16:03
Of????

supposedly, it's "order" which makes some sense relative to exponentials.

TTFN
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KENAT (Mechanical)
27 Sep 12 16:07
Just saying what I was taught.

Posting guidelines FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm? (probably not aimed specifically at you)
What is Engineering anyway: FAQ1088-1484: In layman terms, what is "engineering"?

Denial (Structural)
27 Sep 12 17:01
The O most definitely stood for "of" in the Australian (Victorian) education system in the early 1960s. ("Of" as in "a quarter of a gallon".)
IRstuff (Aerospace)
27 Sep 12 17:18
Still sounds weird, but these guys are hedging by allowing both: http://www.cimt.plymouth.ac.uk/projects/mepres/boo...

BBC claims it's BIDMAS: http://www.bbc.co.uk/schools/gcsebitesize/maths/nu...

This says it's both, i.e.,

TTFN
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IDS (Civil/Environmental)
27 Sep 12 19:33
It's sort of interesting that even though we all know how to do it, there are such a variety of different statements of the rules, several of which give different results.

For instance two of those posted above don't mention the equal precedence of DM and AS, and the one that does gives a version in which "Of" has the same precedence as "Order".

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

BigInch (Petroleum)
5 Oct 12 5:52
IT SEEMS THAT ROMNEY MATH ALSO GETS A 0

"People will work for you with blood and sweat and tears if they work for what they believe in......" - Simon Sinek

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