# The Product of Two Negative Numbers 1

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3 Stewart s external interpretation of plus and minus as saving and debt doesn t quite make sense from a mathematical perspective. Possessing no money and having no debt adds up to zero. If I now borrow five dollars from the bank, I have five dollars in my pocket, but I also have a debt. This means that my borrowing resulted in both (+5) and (-5). There is no actual mathematical operation of this kind. Forgiving is forgiving of the debt created by borrowing. Given the (+5) + (-5) resulting from borrowing, forgiving erases the (-5), keeping the (+5). This too has no mathematical analog. Stewart s internal account, on the other hand, makes eminent sense, even to a layman. The point he makes is that the convention of the product of two negative numbers being a positive number is consistent with the conventions of algebra that we are familiar with. Following some other convention say, two negatives making a negative, or for that matter, two positives making a negative would result in an inconsistency that would force us to throw away a great deal of what is valuable in algebra (pp.163-4). This internal account would be appropriate for an older child, but not quite so for a seven-year old. When he saw an earlier draft of this write up, Chong Chi Tat, a colleague of mine, ed me the following internal argument, probably easier for a child to understand: The product of ( 3) x ( 5) is either ( 15) or (+15). If it is ( 15), then ( 5) x ( 3) = ( 5) x (+3). Canceling ( 5) from both sides, we get ( 3) = (+3). This is an absurd result so it couldn t be that the product is ( 15). Hence the product of ( 3) x ( 5) is (+15) As Stewart says, The important thing is not to say to the student, That s how it is. Don t question it. Just learn it. But to my mind it would be even worse to leave them with the impression that there was never any choice to be made, that it is somehow ordained that minus times minus makes plus. All of those concepts plus, minus, times are human inventions. (p. 164) 1.5 The meaning of multiplication with negative numbers A few days ago, Chandra, Uttam, Walter, Cindy and I were talking about science and math education, when the issue of the product of two negative numbers came up yet again. I recounted its history beginning with what happened twenty years ago, went on to Adam s question, and ended with Ian Stewart s unsatisfactory solution. And Uttam said, But doesn t it follow from the textbook idea of multiplication as repeated addition? No, I said, We can view five times three as adding three five times, but what does it mean to say minus five times in minus five times minus three? After a minute s pause, Uttam said, We can view multiplication with negative numbers as repeated subtraction. What follows is the working out of this idea. 2. The Explanation 2.1 Ambiguity in the meanings of plus and minus To understand why the product of two negative numbers is a positive number, we have to begin with the meanings of the symbols + and in mathematics. Depending on the context, they may refer either to the categories of positive and negative numbers, or to the operations of addition and subtraction. 3

6 The analogy of a leaping frog can cover this situation as well. Given below, for instance, is (+3) x (4) When multiplying by a negative number, the frog leaps in the opposite direction. 2.4 The meaning of division by a negative number Figure 5 This line of thinking naturally leads to the question of what division by a negative number (e.g., six divided by minus four) means, and what happens when we divide a negative number by a negative number. I leave it to you to look for an answer. But here are some clues: Just as subtraction is the reversal of addition, division is the reversal of multiplication: (+4) x (+6) is the result of adding four to zero six times (+24) (+6) is the number that, when subtracted six times from 24, results in zero. Mental animation: (+4) x (+6): A frog begins at zero, leaps to the right six times, each time taking four steps. It arrives at 24. (+24) (+6): The frog s return journey: it begins at 24, leaps to the left six times, and arrives at zero. The number of steps for each leap is four. 3. Concepts, metaphors, and reasoning in mathematics I would like to spell out some of the lessons to draw from the above discussion: A) Young learners need help to understand why we accept certain beliefs and practices, and reject others. For instance, it is important that children, when they are exposed to the belief that the earth spins on a tilted axis and revolves around the sun, understand why we believe this. Likewise, it is important for young learners to understand why the product of two negative numbers is positive, though the sum of two negative numbers is negative. Conventional education fails to provide such understanding. B) Mathematical inquiry includes formal reasoning as well as conceptual reasoning. Solving algebraic equations like x 2 + 4x + 9 = 21 requires formal reasoning, while seeing that in spherical geometry, two straight lines can intersect at more than one point requires conceptual reasoning. Conceptual reasoning is itself enhanced by visual reasoning (e.g., the diagrams in sections 1 and 2), and analogical reasoning (the ideas of ladder, debts, and leaping frogs). C) A great deal of the math that we teach our children focuses on the skills of making mathematical calculations, with no attention to the modes of mathematical thinking, mathematical intuitions, and mathematical concepts. It is important that children understand the concepts of numbers, addition and subtraction, multiplication and division, and use that understanding in their reasoning. In exploring the basic arithmetic concepts in the preceding section, we used the number line as the crucial image to systematically pursue the concepts of positive and negative numbers, addition, subtraction, multiplication, and division. A more common metaphor used in classrooms to understand arithmetic operations is that of objects in boxes and baskets. While the number line metaphor conceptualizes numbers as locations ordered along a scale (the line) as in figure 6a, the box/basket metaphor conceptualizes them as unordered dots in two dimensional space, as in figure 6b: 6

7 number line Figure 6a Figure 6b See how the idea of the leaping-frog-on-the-number-line changes in figure 6b. We could think of the dots in the boxes in figure 6b as frogs. Addition and subtraction would then not be leaping from one location to another, but leaping into and out of boxes. (+3) + (+2) (+5) (+2) Figure 7 The idea of multiplication as repeated addition can now be expressed as: a number of baskets (circles), each with the same number of frogs (dots); an empty box (rectangle); and all the frogs leaping into the box. 4 x 3 (three times four) Figure 8 And division can be expressed as: a number of frogs in a box; a number of empty baskets; and the frogs leaping into the baskets such that every basket has the same number of frogs (twelve divided by three) Figure 9 The conceptualization and visualization of numbers as counting the frogs in boxes and baskets, represented visually as dots in rectangles and circles, rather than in terms of a number line, would perhaps make it easier for younger children to understand arithmetic operations on natural numbers (positive integers). However, moving on to operations on negative integers, we would need the more abstract number line. Dots, rectangles, and circles do not lend themselves to a representation of negative numbers such that the representation tells us why the product of two negative numbers is a positive number. 7

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AN OVERVIEW & NUMBER SYSTEMS Lesson No. 01 Analogue versus Digital Most of the quantities in nature that can be measured are continuous. Examples include Intensity of light during the da y: The intensity