formula. This has nothing to do with how well people understand it. There
is a de facto standard (but no de jure standard seems to exist). If that
standard defines an order of evaluation that is consistent with something
that is easy to implement in computers, then that has nothing do to with
people not understanding. If you want to complain about bad education, go
ahead, but don't tell me something's wrong with the order of operations
everybody uses /because/ of bad education.
Grtz,
Rob
-------- Original-Nachricht --------
> Datum: Tue, 11 Dec 2007 13:13:51 -0800 (PST)
> Von: Vinaire <vinaire@yahoo.com>
> An: Math4u@yahoogroups.com
> Betreff: Re: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on Evaluation Order
> 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/associativity 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/associativity
> 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/associativity.
>
> 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) 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
>
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