
RE: negative * negative
Posted:
Apr 2, 1996 12:26 PM


A critical question has been lurking behind a couple questions that people have raised. Before we talk about multiplying integers, we must have some understanding of what an integer is. We act as if this is a trivial, obvious idea, but it is most certainly not! Witness the historical controversy in an earlier post by my good friend John Owens. To get really technical, an integer is an ordered pair of whole numbers, where (a,b) is equivalent to (c,d) iff a + c = b + d. Now I am not advocating teaching middle school students using set theory, but the point is that an integer is a RELATIONSHIP between two whole numbers. That's why they are so difficult. [Side note: Ditto for rational numbers.] From a mathematical point of view: We generally name an integer using the representative of its class so that one of the whole numbers is 0, that is in relationship to zero. So for example, if a>b, then (a,b) = (ab,0); which we can consider a positive number. If a<b, then (a,b) = (0,ba); a negative. [Remember: the ordered pairs must be whole numbers.] Thus, when we teach integers on the number line, we are really saying that an integer is the displacement above or below zero, which is the same as the displacement between other pairs of whole numbers. So, 3 = (0,3) = (1,4) = (2,5) ... When we use money, 3 is the change that occurs when we go from $5 to $2 or from $4 to $1 or from $3 to broke. When we try to apply operations to integers, things get strange because we think of operations in terms of whole numbers. Now we are applying them to RELATIONSHIPS between whole numbers. Adding is not bad  join together displacements (as on the number line). It feels right, much like joining whole numbers (but not the same). For 3 + 4, we move 3 units to the left, then 4 more units to the left, for a total of 7 units to the left. [Note: Hitting on the point named 7 is just a shortcut for saying that you ended up 7 units to the left of where you started from, assuming that you started at 0. If you started at 1, you would end up at 6, which still is the right answer; i.e., a displacement of 7 units from where you started. An integer is NOT a location!] Subtracting is manageable  it is the inverse of addition, so displace in the opposite direction. This is analogous to take away with whole numbers. The real problem comes with multiplication. There is a trap, because when we multiply by a whole number, we can think of this as repeated addition. BUT this is not integer multiplication! This is whole number multiplication. While the whole number a is related to the integer +a or (a,0) but IS NOT THE SAME, because an integer is a relationship between two whole numbers. Thus, if multiplying by a negative does not seem to flow nicely from previous ideas of multiplication, there is a reason. We need to add in the idea of displacement. If we take multiplying by a positive to be like whole number multiplication, then multiplying by a negative must have the opposite effect. SUMMARY: There is a reason that integer multiplication is hard to explain. Take plenty of time to develop integers so that the students have a chance to figure out how they work. IT IS NOT OBVIOUS! PS: This does not mean I know how to do teach this! Just that I believe it is much harder than we often act. :)
Gary Martin Univ. of Hawaii

