Here is the reason why multiplication is senior to addition.
Multiplication is a shorthand for repeated additions.
7 + 2 x 3 = 7 + (2 + 2 + 2) = 7 + 6 = 13
It is 2 which is being repeatedly added to itself here and not (7+2)
Therefore, multiplication takes priority over addition in mixed operations when no parenthesis are present.
This convention is not arbitrary as you seem to think. This is what I meant by missing understanding.
Vinaire
From: Michael S. <M.Suesserott@gmx.net>
To: Math4u@yahoogroups.com
Sent: Tuesday, December 11, 2007 6:42:31 PM
Subject: Re: [Math4u] 16 / 2 * (8 – 3 * (4 – 2)) + 1 Again - A Word on Evaluation Order
Vinaire,
I see what you mean. Let me try to give a useful answer.
First of all, if we always used parentheses, we would not need any
complicated rules or conventions for evaluation order. For instance,
(2 * 3) + 5 is clearly 11, and
2 * (3 + 5) is clearly 16.
Only when we want to be lazy and get rid of a few parentheses will we
need rules. Then we will have to determine what 2 * 3 + 5 is supposed to
mean. Which of the above meanings shall we assign to this expression?
As you know, the generally accepted convention is that * has higher
precedence (or, as you say, seniority) than +. Mathematicians chose that
convention that, in the absence of parentheses, * binds more strongly
than +. It could very well be the other way around, but that's the
choice that has been made.
While there is no mathematically cogent reason for this choice, there
may be a linguistic one. In speech, we are used to saying, "3 apples and
5 oranges." Obviously, this means, "3 times an apple plus 5 times an
orange." It does not mean, "3 times the object [apple and 5 oranges]."
Nor does it mean "3 times an object [apple and 5], multiplied by
oranges," or any such nonsense. No, the number 3 binds to apples, and
the number 5 binds to oranges. So it is quite natural from the way we
speak to give multiplication precedence (a stronger binding) over addition.
But yes, it is still an arbitrary convention.
Michael
Vinaire wrote:
> 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
>
>
> ----- 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
>
> 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 <mailto:M.Suesserot t%40gmx.net> >
> > To: Math4u@yahoogroups. com <mailto:Math4u% 40yahoogroups. 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/
> <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% 40yahoogroups. com>
> > <mailto:Math4u- fullfeatured @yahoogroups. com
> <mailto:Math4u- fullfeatured% 40yahoogroups. com>>
> >
> >
> >
> >
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