It isn't so hard to convince students of grade 8. 9 that $-2$ is greater than $-4$, but it still make some confusion with an axiom says that ( The whole is greater than part ), which is true in the positive numbers. For example if we consider the ( Whole) to be $4$ and the half to be ( The part ), then 4 is greater than its half which is 2. The question is how to convince students that the converse should happen in the negative numbers, i.e, the half of $-4$ ( $-2$) is greater than the ( Whole ) which is $-4$ ?!
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asked Dec 18, 2014 at 21:45
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$\begingroup$ I think I have some idea of what you are asking; to use similar language to yours: When you remove all of a number, then you are left with $0$. When you remove some (but not all) of a number, then you are left with something closer to $0$. With this conception: Going from a positive number to a new number closer to zero means a decrease; going from a negative number to a new number closer to zero means an increase. (I would probably model this on the number line.) $\endgroup$– Benjamin DickmanDec 18, 2014 at 22:00
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$\begingroup$ Thank you very much. But I have no problem in explaining that $-2$ is greater than $-4$. The problem is how to convince students that the converse is happen in negative numbers ? $\endgroup$– Abdallah AbusharekhDec 18, 2014 at 22:07
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$\begingroup$ If your textbook has axiom "the whole is greater than the part", then it had better not be a textbook on real numbers. Perhaps it is a textbook on geometry or something. $\endgroup$– Gerald EdgarDec 18, 2014 at 22:44
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2$\begingroup$ I believe your statement there is the key: -2 is indeed smaller than -4 -- it's smaller in MAGNITUDE. The magnitude of the part is smaller than the magnitude of the whole. $\endgroup$– DavidButlerUofADec 18, 2014 at 23:34
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1$\begingroup$ I like it @mweiss, but we do need to remember that the official names for those symbols are "less than" and "greater than", so we do need to tell students what people mean when they say those things! $\endgroup$– DavidButlerUofADec 19, 2014 at 3:22
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2 Answers
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Based on the comment by @GeraldEdgar and others, I'd like to answer that "The whole is greater than the part" is just not a valid axiom about numbers (of any sort). This statement is relevant for collections (e.g. finite sets, or everyday physical objects), but it doesn't quite make sense regarding numbers.
To provide a concrete example, a statement that 2 is a half of 4 has in its essence a very different meaning than saying that the set {Eve, Sarah} constitutes a half of {Adam, Eve, Abraham, Sarah}. Speaking about parts of numbers appears to me to be an abuse of notation that inevitably leads to confusion.
Perhaps you could provide us with a context for the statement of this "axiom"?
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Reflecting on this, I think the difficulty comes because there are 2 ways of interpreting integers which have different ordering functions.
The first (and standard) interpretation is as a totally ordered set, from $^-\infty$ to $\infty$. From this perspective $^-4 < {^-2}$.
The second way (as used in high school physics) is to interpret integers as a 1 dimensional vector. The direction is given by $^+/_-$, and the magnitude is given by a totally ordered set from $0$ to $ \infty$. From this perspective, it would make sense to think of $^-4 > {^-2}$, as a velocity of $^-4$ is clearly bigger than a velocity of $^-2$.
However, to be mathematically consistent, interpretation 2 integers are only partially ordered, with no overall supremum, as $^+4$ and $^-4$ are not able to be compared. It may be best not to call directed natural numbers such as these integers, and refer to them as 1D vectors. However, they do conform to your whole/part intuition, and are routinely used according to this intuition, while still (incorrectly?) being called integers.
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$\begingroup$ This is an interesting direction (pun intended) that I haven't considered. How then would you respond to the OP's question? Does the statement "The whole is greater than the part" make sense as a learning aid given this viewpoint? $\endgroup$– yoniLaviDec 22, 2014 at 20:17
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$\begingroup$ @yoniLavi Integers are learned by middle school, while 1D vectors are used in high school, so I can't imagine the vector interpretation actually being useful for real life teaching of integers. They would just confuse the standard definition. $\endgroup$– RichardDec 22, 2014 at 23:01