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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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
Yahoo! Groups Links
<*> To visit your group on the web, go to:
http://groups.yahoo.com/group/Math4u/
<*> Your email settings:
Individual Email | Traditional
<*> To change settings online go to:
http://groups.yahoo.com/group/Math4u/join
(Yahoo! ID required)
<*> To change settings via email:
mailto:Math4u-digest@yahoogroups.com
mailto:Math4u-fullfeatured@yahoogroups.com
<*> To unsubscribe from this group, send an email to:
Math4u-unsubscribe@yahoogroups.com
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