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Topic: Please remind me why -3^2 = -9
Replies: 26   Last Post: Nov 18, 2012 7:51 PM

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 Phil Mahler Posts: 126 Registered: 12/6/04
Re: Please remind me why -3^2 = -9
Posted: Nov 18, 2012 3:57 PM

George, thank you for both statements, the mathematical, and the list courtesy one.

About the conventional order of operations, I agree that it most likely developed as algebraic notation developed, and the advantage for writing a polynomial as simply as possible became clear by what we now call "the order of operations".

In fact it took hundreds of years to come to where we are. Florian Cajori, in his wonderful book "A History of Mathematical Notations" wrote, in 1928 or 1929, "If an arithmetical or algebraical term contains ÷ and X, there is at present no agreement as to which sign shall be used first". And quotes an author from 1892: "It is best to avoid such expressions". Probably everyone on this list today would say that in say 24 ÷ 4 X 2 we do them from left to right, and so would I, but this consensus is recent, and who is to say it is truly universal? Better to use ().

You mention Excel, and in Excel the expression -3^2 produces the answer 9, not -9. So even today there is no consensus. I hope no dogmatic mathematician uses Excel to design an X-ray machine to which I will be subjected. :-)(http://http://en.wikipedia.org/wiki/Order_of_operations talks about Excel).
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As for "You're a teacher & you had to post this question?": I can see why a person ignorant of the issues around notation, and perhaps seeing the question in isolation, might get indignant. I had to chuckle a bit, myself. :-)

For this person and others I'll reiterate why I posted my original question (with the possible danger of pushing the tolerance of some good readers past their limit ):

1. Most students never learn which is the "correct" result for -3^2. if something is that broken, do you look at the users or the item itself?
2. In fact for many, perhaps most, the intuitive interpretation is "the square of -3", which is "incorrect".
3. The answer -9 is due to an arbitrary choice of which comes first, ^ or "-". Many students obviously agree with Excel that "-" comes first.
4. There is no reason why we should make either choice, based on the axioms of arithmetic, any more than saying we do ÷ and X from left to right. It is totally arbitrary and not intuitive at all.
5. Therefore my point is that we should not be spending so much wasted energy on this, but, if anything, should stress using symbols of grouping in the interest of clarity. OK to write 12 - 3^2, but never just write -3^2 except to discuss ambiguity and the correct use of Excel and some programming languages.

Phil

On Nov 18, 2012, at 10:57 AM, "Matthews, George" <matthewg@sunyocc.edu<mailto:matthewg@sunyocc.edu>> wrote:

Not mentioned in previous responses to the question is the rationale for the conventional order of operations:
Exponents (powers) before Products (or divisions) before Sums (or subtractions), in the absence of parentheses (grouping symbols),
is convenience in writing polynomials.
ax^p+bx^q+..., where x is a free variable, is much easier to read than
(a(x^p))+(b(x^q))+...

Those familiar with APL notation or Excel formulas are surely aware of these issues.

That said, this forum should be open to any mathematical, pedagogical question without judgment about motives or intelligence.

Just my two cents.
________________________________________
From: owner-mathedcc@mathforum.org<mailto:owner-mathedcc@mathforum.org> [owner-mathedcc@mathforum.org<mailto:owner-mathedcc@mathforum.org>] on behalf of EddieC [info@kopworld.net<mailto:info@kopworld.net>]
Sent: Sunday, November 18, 2012 6:28 AM
To: mathedcc@mathforum.org<mailto:mathedcc@mathforum.org>
Subject: Re: Please remind me why -3^2 = -9

You're a teacher & you had to post this question?

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Date Subject Author
10/18/12 Phil Mahler
10/18/12 John Peterson
10/18/12 Guy Brandenburg
10/18/12 Wayne Mackey
10/18/12 Phil Mahler
10/18/12 RotmanJ
10/18/12 Paul Hertzel
10/18/12 Phil Mahler
10/18/12 Clyde Greeno
10/19/12 Clyde Greeno
10/19/12 Alain Schremmer
10/19/12 Wayne Mackey
10/19/12 Alain Schremmer
10/19/12 Clyde Greeno
10/19/12 Alain Schremmer
10/21/12 Wayne Mackey
11/14/12 Beth Hentges
11/15/12 Clyde Greeno
11/16/12 Alain Schremmer
10/18/12 Collinge, Peter (Mathematics)
10/18/12 Guy Brandenburg
10/18/12 Phil Mahler
10/18/12 Alain Schremmer
11/18/12 EddieC
11/18/12 Matthews, George
11/18/12 Phil Mahler
11/18/12 Alain Schremmer