New Cur 6: Necessity is the Mother of Convention

ACARA’s draft mathematics curriculum contained the following elaboration from Year 8 Number:

using expressions such as \color{RawSienna}\boldsymbol{\frac{3^4}{3^4} =1}, and \color{RawSienna}\boldsymbol{{3^{4-4}}=3^0} to illustrate the convention that for any natural number đť‘›, \color{RawSienna}\boldsymbol{{n^0}=1}, for example, \color{RawSienna}\boldsymbol{{10^0}=1} (old AC9M8N02)

This has been changed for the approved curriculum:

using examples such as \color{OliveGreen}\boldsymbol{\frac{3^4}{3^4} =1}, and \color{OliveGreen}\boldsymbol{{3^{4-4}}=3^0} to illustrate the necessity that for any non-zero natural number 𝑛, \color{OliveGreen}\boldsymbol{{n^0}=1} (new AC9M8N02)

Give ’em another few years and ACARA just might land upon proper wording. And the proper use of commas. And brackets. And logic.

15 Replies to “New Cur 6: Necessity is the Mother of Convention”

  1. I know not everyone agrees on certain conventions, but I strongly believe that 0 is NOT a natural number.

    I did a quick survey of school Mathematics teachers at a recent gathering and found I was very much in the minority. Again, small sample size…

    Necessity though…? Yeah… ACARA continues to provide plenty of head-scratching moments!

      1. I couldn’t get past Efficient Strategies: “The choice of suitable approaches (mental, written, using digital tools), and methods within these. The means used for calculation reflect the purpose and context.”

        But thanks – I see even ACARA cannot decide whether 0 is a natural number or not!

        Although I never agreed, the IBO in the years I was teaching the Diploma Program made a point of defining 0 as a natural number. They may have changed their stance since (my login to their system no longer works, so I cannot check their glossary of command terms – anyone want to help…?). Such actions solve a lot of problems.

        I do remember searching the International Standards Organisation website to see what they had to say… I never did find an answer though.

        Maybe the topic of a future blog post and let the wisdom of the masses decide?

    1. I agree that 0 is not a natural number, but other authors regard 0 as a natural number; e.g. Bertrand Russell, “Introduction to mathematical philosophy”, p. 3.

  2. Back to the main point though… is a^{0}=1 either a “convention” or a “necessity” for non-zero values of a? (Not just non-zero natural numbers… in case ACARA wants to correct that oversight…)

    I certainly don’t think it is a convention.

      1. I do agree with you Terry that it is a definition and it makes sense in relation to the other index laws.

        Perhaps the latter drives the former? I don’t know.

  3. Am I missing something?

    What counts as a “natural number” is not even a convention: different people may take it differently (so one has to pay attention): and still the world keeps on turning.

    Much more worrying here is the move from the word “convention” in the old version (which is on the right lines) to “necessity” (which is a nonsense) – so given “a few years” in this direction and things will get worse.

    At the risk of boring the cognoscenti, once we give a naive (but reasonable) definition of positive integer powers, the index laws are a necessity.
    But only for positive integer powers.

    Any move beyond this involves a decision as to “What are the mathematical gods trying to suggest to us as the *best* choice/convention?”.

    The answer for *zeroth* powers is a (very sensible) decision, which students can fumble their way towards for themselves, and so experience the way mathematics ventures into new territory.

    More delicate is the move into fractional powers, where the old index laws break down – in some sense. (Things are more-or-less OK for fractional powers of positive reals, but the naive version comes unstuck for fractional powers of negative numbers. Think carefully about f(x) = x^(2/6) and g(x) = x^(1/3): the first seems to be “two-valued”).

    The details are for the few and later, and explain why we have to move to the definition based on first understanding the exponential function e^x, and then defining a^b as e^(b.ln(a)). This is almost never explained: if others know good, readable references, it would be good to know them.
    Some readers might enjoy the final Chapter IV.3 (“What is an exponential function?”) of my book “Infinite processes” (Springer 1982) – reissued as “Understanding infinity” by Dover (2006?). But it is probably out of print so may be hard to find.

    1. Thanks; I will read “Infinite processes” with interest; also useful is Mendelson, E. (2008). “Number systems and the foundations of analysis”. Dover.

    2. That x^{\frac{2}{6}} vs x^{\frac{1}{3}} chestnut reminds me of an old(ish) Cambridge Methods 3&4 textbook…

      I don’t have a copy any more, but I’m sure Marty and others remember it well.

      1. Thanks, RF. I posted about the problem here, and the review of the textbook with the blatant error is here. The error been removed for the current edition, but with no real explanation of what is going on. I have no idea about the edition for next year.

  4. In the glossary, ACRA defines the set of natural numbers this way.

    “The set N = {0,1,2,3 …} or N = {1,2,3 …} depending on whether counting is started at 0 or 1. “

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