Talking About Zero, Absolute Zero and Negative Numbers
Date: 11/12/2003 at 22:30:58 From: T. Ball Subject: Absolute Zero, Zero, & Negative numbers Zero is a tricky number to explain, especially when children are interested in why we have negative numbers. It is difficult to conceptualize taking something away from nothing. It becomes even trickier when thinking about absolute zero. Are there any useful strategies other than temperature to introduce negative numbers, and also to teach about zero, its properties, and how they differ from absolute zero?
Date: 11/13/2003 at 08:49:13 From: Doctor Peterson Subject: Re: Absolute Zero, Zero, & Negative numbers Thanks for writing to Dr. Math! I'm not sure what you are asking about absolute zero; I'm only familiar with that term as a temperature, and as far as numbers themselves are concerned, absolute zero is just zero. The only special thing I see about it is how it fits into the scale: in the case of Kelvin temperature, we have a scale with a definite starting point, and negative temperatures do not exist (though I understand that this is not really quite true if you dig deep enough into the physics of temperature!). On the other hand, in scales where no (known) lowest value exists, negative numbers have to be allowed, and zero becomes not an absolute end point, but a mere reference point along the scale from which quantities are measured in both directions. If the concept of absolute zero temperature had not been discovered, then we could not have a Kelvin scale, and all temperature scales would have to, at least theoretically, allow for negative temperatures. There are some situations where only positive numbers make sense, and in those cases we have the equivalent of a Kelvin scale, with an absolute zero. For example, a person's height can't be negative; no one can be less than zero meters tall! But altitude, which on the surface sounds the same, does allow negatives; I can be 100 meters above sea level, or 100 meters below sea level (a negative number). Again, we can find an alternative scale for altitude that has an "absolute zero", namely the distance from the center of the earth; so altitude referenced to sea level is just a convenience for people who live near sea level (as Celsius is convenient for people who "live" in the zone where water is liquid). But not all scales can be made absolute; coordinate systems for space offer no (known) "farthest left" location, so negative numbers are required no matter what you do. Which leads into my next comment ... Apart from temperature, I think the only good way to introduce negative numbers is with a number line (of which temperature is just a familiar example we can point to in a child's environment, especially when they live in a cold climate). If you look at what we say about negative numbers, such as Positive and Negative Integer Rules http://mathforum.org/library/drmath/sets/select/dm_pos_neg.html you will find a lot of references to the number line. The basic idea is that, if we want to locate every point on a line by associating it with a number, positive numbers just aren't enough! In order to put any numbers on the line in numerical order we have to have a zero point from which we start counting; and in order to label points in both directions, we need negative numbers. Once you have that idea of negative numbers as labels for points to the left of zero on a number line, everything else falls into place. After building that view of negative numbers, you can return to counting situations and ask whether negative numbers ever make sense there. That's when ideas of "owing" or "debt" arise as uses for negative numbers. But probably those ideas won't be clear without the number line model to make the concept concrete. In a sense, we are then modeling counting in terms of the number line, extending the idea of number to allow for negatives that result from subtracting more than you have. Such modeling is really the essence of mathematics. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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