"Subtraction" and "Negative"--Same Sign, Different Concepts?
Date: 12/14/2005 at 06:51:33 From: Chris Subject: Difference between 'subtraction' and 'negative'? Is there a difference between the operation "-" when used in the expression "y = a - b" as compared with its meaning in the expression y = -b? I don't doubt the truth of the statement 0 = -1 -(-1) but the reasoning I have seen has never quite convinced me, because it seems that the - sign is being used to mean two different things. On a number line these two things are something like 1) -b => Rotate the following number by 180 degrees. 2) a - b => Continue (or count) in the direction of the following number, rotated by 180 degrees. Now while these are fairly similar they are not the same thing. Taking another approach, we are allowed say "y = a x b" but we are not allowed to say "y = x b". So "-" is being used in a "syntactically" different way to "x". This indicates to me that we are allowing "-" to be used to mean two different things. I think that, as it happens, the rules for combining the two different meanings of the sign allow us to get away with saying "-(-) = +" without distinguishing between the two, because it just happens to work out like that. The reason I'm so hung up on this is that I think if we temporarily used new signs - say, p and n, for positive (no rotation) and negative (180 degree rotation)--and continued to use + and - for the operations of counting on or counting with rotaion, it all becomes a lot easier to explain what is actually happening. That is, "pa - pb" can be shown to be the same as "pa + nb" and also it is much easier to grasp that "pa - (nb) = pa + pb". Is there anything to all this?
Date: 12/14/2005 at 11:37:24 From: Doctor Peterson Subject: Re: Difference between 'subtraction' and 'negative'? Hi, Chris. You're exactly right that "minus" and "negative" are two different operations; that's why they have different names (and we _try_ to get kids to use the right names ;-). "Minus" is a binary operation (performed on two numbers), while "negative" is a unary operation (performed on a single number). They have the same symbol because they are _very_ closely related, and it doesn't cause any trouble to use the same symbol. However, some texts have done just what you suggest, and used different symbols for them; one common choice is a raised "-" to mean negative, so an equation might look like _ a - b = a + b This does help some students to get a better sense of the distinction before they move on to the normal notation. I might also add that scientific calculators have separate keys for the two operations; on those that most closely mimic written math notation, the "negative" key is commonly labeled "(-)" to distinguish it from the "minus" key, "-". Computer programming languages don't need to make that distinction, since they can determine the meaning from context; I'm not sure why calculators don't do the same, but it's probably due to little differences in the way users think of keys on a calculator vs. characters on a page. Regardless of whether we use distinct notations for the two operations, it is important to distinguish the operations themselves. The negative is called the "additive inverse", and is defined by a + -a = 0 That is, the additive inverse of a is the number you can add to a to get 0. Having defined that, we actually _define_ subtraction in terms of this: a - b = a + -b That is, we define subtraction as _meaning_ addition of the additive inverse; so, as you said, subtraction means "turn around on the number line and walk the given distance in the opposite direction". This is the connection between the two operations, and the reason it is helpful to use the same symbol. When I read "a - b", I see it as "a + -b", because addition has important properties, such as commutativity, that subtraction lacks, making it very useful to forget about subtraction and use only addition. (Note that if we used a very different symbol, like your "n", it would be harder to see the connection and to learn to see it this way: it is not obvious that a - b = a + nb.) There is a similar issue with regard to multiplication and division. If we wanted, we could use the division symbol "/" to indicate the multiplicative inverse (also called the reciprocal): a * /a = 1 We define division as multiplication by this inverse: a / b = a * /b in much the same way as for subtraction. For some reason this has never caught on, as far as symbols are concerned. Similarly, although we can use "+" as a unary operator (which has no effect on a number, as +a = a, meaning 0+a), we don't happen to use multiplication, "*", in the same way, so that *a = a, meaning 1*a. It wouldn't hurt to do so, but has not been found useful! Getting back to negatives, the fact that -(-a) = a (effectively, turning around twice and ending up in the same direction) allows us to easily rewrite -1 - (-1) as -1 + -(-1) = -1 + 1 = 0 This is much harder to explain in terms of subtraction only; by seeing subtraction as adding the negative, it becomes relatively simple. In summary, your thoughts are valid, and not really new; seeing things this way is actually essential to learning algebra well. So you're in good company! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 12/14/2005 at 12:14:54 From: Chris Subject: Difference between 'subtraction' and 'negative'? Fantastic! To some small degree I have been hung up on this for years and despite moderate searching I have never come across any of these texts, so it is quite a relief for me to find that it is a valid way of thinking. Thanks again for your time.
Date: 12/14/2005 at 22:58:25 From: Doctor Peterson Subject: Re: Difference between 'subtraction' and 'negative'? Hi, Chris. I wanted to confirm my experience that some (many?) texts use the raised negative sign, so I tried finding references on the web. Here is the only explicit reference I found: http://homepages.ius.edu/MEHRINGE/T101/Notes/Section4-1.htm I didn't find enough evidence to determine how widespread this is, or whether it is a current phenomenon. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 01/03/2006 at 09:11:06 From: Chris Subject: Thank you (Difference between 'subtraction' and 'negative'?) Thanks for this. Visually, the look of the expressions which mix the raised "-" and normal "-" are exactly right for me and express the meanings quite intuitively. The distinction between 5 - 6 and 5 + ^-6 is clear, as is the fact that they give the same result. Regards, Chris.
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