sbk
http://mathforum.org/library/drmath/view/52582.html
Date: 11/22/2000 at 10:56:37
From: Brian Huffine
Subject: History of Order of Operations
I was teaching a computer class and the history of order of operations
came up. Where, when and with whom did the order of operations first
originate? Was it the Greeks or Romans?
Thank you! There is a whole class waiting to hear the answer.
Brian Huffine
Date: 11/22/2000 at 12:12:26 From: Doctor Peterson Subject: Re: History of Order of Operations Hi, Brian. The Order of Operations rules as we know them could not have existed before algebraic notation existed; but I strongly suspect that they existed in some form from the beginning - in the grammar of how people talked about arithmetic when they had only words, and not symbols, to describe operations. It would be interesting to study that grammar in Greek and Latin writings and see how clearly it can be detected. At the other end, I think that computers have influenced the subject, so that it is taught more rigidly now than it used to be, since programming languages have had to define how every expression is to be interpreted. Before then, it was more acceptable to simply recognize some forms, like x/yz, as ambiguous and ignore them - something I think we should do more often today, considering some of the questions we get on such issues.
[snip]
-----Original Message-----
From: Vinaire <vinaire@yahoo.com>
To: Math4u@yahoogroups.com
Sent: Tue, 11 Dec 2007 5:31 pm
Subject: Re: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on Evaluation Order
What do you mean?
All I am saying is that confusion exists in the mind of those who do not understand the reason for the order of operations in the first place.
Why is multiplication senior to addition? Or, why is division senior to subtraction? Other similar questions may be asked.
Someone earlier implied that this was arbitrarily set by convention. Is that so ?
Vinaire
All I am saying is that confusion exists in the mind of those who do not understand the reason for the order of operations in the first place.
Why is multiplication senior to addition? Or, why is division senior to subtraction? Other similar questions may be asked.
Someone earlier implied that this was arbitrarily set by convention. Is that so ?
Vinaire
----- Original Message ----
From: Michael S. <M.Suesserott@gmx.net>
To: Math4u@yahoogroups.com
Sent: Tuesday, December 11, 2007 4:33:48 PM
Subject: Re: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on Evaluation Order
From: Michael S. <M.Suesserott@
To: Math4u@yahoogroups.
Sent: Tuesday, December 11, 2007 4:33:48 PM
Subject: Re: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on Evaluation Order
Vinaire,
the question I am dealing with here is why 16 / 2 * (8 – 3 * (4 – 2)) +
1 evaluates to 17, not 5, as some have suggested, and why this is a de
facto convention nowadays.
I am all in favor of understanding, so where is your problem?
Michael
Vinaire wrote:
> No rules substitute proper understanding. A rule works 100% of the
> time only when one totally understands how it was derived in the first
> place.
>
> Today's trend appears to be toward robotic application of rules
> without understanding. Understanding is considered to be "too complex."
>
> Anything that is not understood would appear complex. Bad education
> will make most data appear very complex. With the advent of
> calculators, schools no longer teach mental math. So, we encourage a
> push-button mode of thinking, rather than a systematic thinking mode.
>
> Computers are a boon, but let us not depend upon them so much that we
> dumb down the development of thinking ability in our education.
>
> Vinaire
>
> ----- Original Message ----
> From: Michael S. <M.Suesserott@ gmx.net>
> To: Math4u@yahoogroups. com
> Sent: Tuesday, December 11, 2007 2:52:51 PM
> Subject: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on
> Evaluation Order
>
> There still seems to be some confusion as to the correct order of
> arithmetic operations. Having been involved in the implementation of
> computer languages in my professional career, perhaps I may be able to
> shed some light on this.
>
> It is true that there are several conventions that have been handed down
> to us from the olden days. Among them are
> the "My Dear Aunt Sally" convention (multiplication, division, addition,
> subtraction) and the "BODMAS" convention (bracketed operations,
> division, multiplication, addition, subtraction) . These two, though
> different, each had a large following among mathematicians. Then, more
> than 50 years ago, with the ascent of computers, another convention
> became prevalent, one that was deemed the most straightforward to
> implement in machine and assembly language. It was the
> precedence/associat ivity convention that is being used today in
> practically all computer languages on the planet, and, in consequence,
> in software products such as Excel.
>
> IMHO, mathematics teachers of this present generation have a
> responsibility to teach their students what is actually being used by
> the overwhelming majority of people. Some two billion computers and
> about one billion mobile phones have been sold worldwide. C, C++, Java,
> Python, etc. are the languages used to create the software for them,
> such as Excel which is used in countless technical and commercial
> applications. All of these abide by the precedence/associat ivity
> convention. Nobody cares about My Dear Aunt Sally any more.
>
> How is it that Google comes up with 17 when you input 16/2(8-3(4-2) )+1 ?
> The Google user interface is programmed, to a large degree, in Python,
> and of course Python obeys precedence/associat ivity.
>
> If PurpleMath promulgate a different standard, they are doing their
> users a grave disservice. These users will be rudely surprised when they
> find out that what they learned there is not what the real world follows.
>
> Our respected moderator, Brian, wrote he was waiting for "some authority
> such as the American Mathematical Society" to issue a convention. I
> don't think this is likely to happen. Mathematicians are, in a sense, a
> very liberal lot. As long as a definition is logically consistent, it is
> OK for them to use. Let me give you an example.
>
> If you read a mathematical paper on advanced algebra or topology topics,
> one of the first things you have to find out is the convention this
> author uses for function concatenation. Does (f o g)(x) equal f(g(x)) or
> g(f(x))? No divine decree has ever been issued from heaven, or, failing
> that, from the AMS, forcing an author to use one convention over the
> other. Both are acceptable, and both are being used to this day in
> scientific literature.
>
> In looking for a de facto standard, aside from considering what Excel
> and Google do, in things mathematical you might look at MATHEMATICA
> (www.wolfram. com <http://www.wolfram. com/>) for guidance. This is the
> most respected software in
> the field. We'll leave the question which convention MATHEMATICA follows
> as an exercise for the reader. Hint: It's not "My Dear Aunt Sally." :-)
>
> Hope this helps a little.
>
> Michael
>
>
>
>
>
> Yahoo! Groups Links
>
>
> mailto:Math4u-fullfeatured @yahoogroups. com
> <mailto:Math4u-fullfeatured @yahoogroups. com>
>
>
>
>
> ------------ --------- --------- --------- --------- --------- -
> Never miss a thing. Make Yahoo your homepage.
> <http://us.rd. yahoo.com/ evt=51438/ *http://www. yahoo.com/ r/hs>
the question I am dealing with here is why 16 / 2 * (8 – 3 * (4 – 2)) +
1 evaluates to 17, not 5, as some have suggested, and why this is a de
facto convention nowadays.
I am all in favor of understanding, so where is your problem?
Michael
Vinaire wrote:
> No rules substitute proper understanding. A rule works 100% of the
> time only when one totally understands how it was derived in the first
> place.
>
> Today's trend appears to be toward robotic application of rules
> without understanding. Understanding is considered to be "too complex."
>
> Anything that is not understood would appear complex. Bad education
> will make most data appear very complex. With the advent of
> calculators, schools no longer teach mental math. So, we encourage a
> push-button mode of thinking, rather than a systematic thinking mode.
>
> Computers are a boon, but let us not depend upon them so much that we
> dumb down the development of thinking ability in our education.
>
> Vinaire
>
> ----- Original Message ----
> From: Michael S. <M.Suesserott@ gmx.net>
> To: Math4u@yahoogroups. com
> Sent: Tuesday, December 11, 2007 2:52:51 PM
> Subject: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on
> Evaluation Order
>
> There still seems to be some confusion as to the correct order of
> arithmetic operations. Having been involved in the implementation of
> computer languages in my professional career, perhaps I may be able to
> shed some light on this.
>
> It is true that there are several conventions that have been handed down
> to us from the olden days. Among them are
> the "My Dear Aunt Sally" convention (multiplication, division, addition,
> subtraction) and the "BODMAS" convention (bracketed operations,
> division, multiplication, addition, subtraction) . These two, though
> different, each had a large following among mathematicians. Then, more
> than 50 years ago, with the ascent of computers, another convention
> became prevalent, one that was deemed the most straightforward to
> implement in machine and assembly language. It was the
> precedence/associat ivity convention that is being used today in
> practically all computer languages on the planet, and, in consequence,
> in software products such as Excel.
>
> IMHO, mathematics teachers of this present generation have a
> responsibility to teach their students what is actually being used by
> the overwhelming majority of people. Some two billion computers and
> about one billion mobile phones have been sold worldwide. C, C++, Java,
> Python, etc. are the languages used to create the software for them,
> such as Excel which is used in countless technical and commercial
> applications. All of these abide by the precedence/associat ivity
> convention. Nobody cares about My Dear Aunt Sally any more.
>
> How is it that Google comes up with 17 when you input 16/2(8-3(4-2) )+1 ?
> The Google user interface is programmed, to a large degree, in Python,
> and of course Python obeys precedence/associat ivity.
>
> If PurpleMath promulgate a different standard, they are doing their
> users a grave disservice. These users will be rudely surprised when they
> find out that what they learned there is not what the real world follows.
>
> Our respected moderator, Brian, wrote he was waiting for "some authority
> such as the American Mathematical Society" to issue a convention. I
> don't think this is likely to happen. Mathematicians are, in a sense, a
> very liberal lot. As long as a definition is logically consistent, it is
> OK for them to use. Let me give you an example.
>
> If you read a mathematical paper on advanced algebra or topology topics,
> one of the first things you have to find out is the convention this
> author uses for function concatenation. Does (f o g)(x) equal f(g(x)) or
> g(f(x))? No divine decree has ever been issued from heaven, or, failing
> that, from the AMS, forcing an author to use one convention over the
> other. Both are acceptable, and both are being used to this day in
> scientific literature.
>
> In looking for a de facto standard, aside from considering what Excel
> and Google do, in things mathematical you might look at MATHEMATICA
> (www.wolfram. com <http://www.wolfram. com/>) for guidance. This is the
> most respected software in
> the field. We'll leave the question which convention MATHEMATICA follows
> as an exercise for the reader. Hint: It's not "My Dear Aunt Sally." :-)
>
> Hope this helps a little.
>
> Michael
>
>
>
>
>
> Yahoo! Groups Links
>
>
> mailto:Math4u-fullfeatured @yahoogroups. com
> <mailto:Math4u-fullfeatured @yahoogroups. com>
>
>
>
>
> ------------ --------- --------- --------- --------- --------- -
> Never miss a thing. Make Yahoo your homepage.
> <http://us.rd. yahoo.com/ evt=51438/ *http://www. yahoo.com/ r/hs>
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