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AnswerA few years ago I spent some time looking in beginning or "popular" math books whenever I could, to find an explanation of why 1 x 1=+1. All the answers I found were unsatisfactory to my understanding. An important piece of history that is left out of math class is that some very brilliant people in centuries past wondered if negative number could be used or even exist. My point is that from grade school on it was expected that we accepted 1 x 1 = 1 and that it made sense. I just don't think it does. Then I looked at What is Mathematics? An Elementrary Approach to Ideas and Methods by Richard Courant and Herbert Robbins Although it didn't look too "elementary" I did find this one part that really struck at the heart of the matter. Here is the excerpt. It took a long time for mathematicians to realize that the “rule of signs” together with all the other definitions governing negative integers and fractions cannot be “proved.” They are created by us in order to attain freedom of operation while preserving the fundamental laws of arithmetic.
… Even the great Euler resorted to a thoroughly unconvincing argument to show that (1)(1) “must” equal +1. For, as he reasoned, it must either be +1 or –1, and cannot be –1, since –1 = (+1)(1).
But from other sources with myriad explanations I felt they were no explantion at all. Some resorted to using grammer to justify math, such as saying if I don't have a negative five boxes of a negative five books it must mean I have a positive twenty five books, since I used a double negative in my sentance. Others touched on the communative or distributive laws to prove it, but that strikes me as a bit of a tautological proof. In the sense that it this case the problem comes from a more fundametal level than the premise of the communicative or distributatie laws in math. Here are what some Math giants thought about negative numbers coming soon) I think all the explanations below are totally useless From http://www.mathsisfun.com/multiplyingnegatives.html Multiplying Negatives Why does multiplying two negative numbers make a positive? Well, first there is the "common sense" explanation: If I say "Eat!" I am encouraging you to eat (positive), but if I say "do not eat!" I am saying the opposite (negative). Now if I say "Do NOT not eat", I am saying I don't want you to starve, so I am back to saying "Eat!". So, two negatives make a positive, and If that satisifes you, then you don't need to read any more. Then it goes on with this totally lame explaneation Direction It is all about direction. Remember the Number Line? Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 3 x 2 = 6 steps forward: Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times: Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times: Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times: That's All ! From http://www.math.toronto.edu/mathnet/questionCorner/minustimesaminus.html Why is the Product of Negative Numbers Positive? Asked by an anonymous poster on March 18, 1997: The answer has to do with the fundamental properties of operations on numbers (the notions of "addition", "subtraction", "multiplication", and "division"). Your 7th grader's question is an important and fundamental one (which I am both surprised and sorry that he has not been able to find an answer for yet). Each number has an "additive inverse" associated to it (a sort of "opposite" number), which when added to the original number gives zero. This is in fact the reason why the negative numbers were introduced: so that each positive number would have an additive inverse. For example, the inverse of 3 is 3, and the inverse of 3 is 3. Note that when you take the inverse of an inverse you get the same number back again: "(3)" means "the inverse of 3", which is 3 (because 3 is the number which, when added to 3, gives zero). To put it another way, if you change sign twice, you get back to the original sign. Now, any time you change the sign of one of the factors in a product, you change the sign of the product: (something) × (something else) is the inverse of (something) × (something else), because when you add them (and use the fact that multiplication needs to distribute over addition), you get zero. For example, is the inverse of , because when you add them and use the distributive law, you get . So is the inverse of , which is itself (by similar reasoning) the inverse of . Therefore, is the inverse of the inverse of 12; in other words, the inverse of ; in other words, 12. The fact that the product of two negatives is a positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again. The answer to this question is accessible to a 7th grader (and should, in my opinion, be explained as part of every student's arithmetic classes). However, as an aside, he may be interested to know that more advanced versions of this question are studied at a university level: there is a subject called Abstract Algebra (usually only covered in a junior or senior level undergraduate university course) which studies the properties of operations on numbers in complete generality, even in contexts that have nothing to do with numbers at all. Even in such general, nonnumerical contexts, the property that the product of two negative things is positive still holds. Followup Comment by Buzz Breedlove on May 9, 1997: This is a comment on your answer to the question: "Why is a negative number times a negative number a positive number?" As a volunteer teacher for a prealgebra class of sixth graders, I addressed the same question with the following practical demonstration. I randomly handed students each a bunch of red and black checkers. I announced that the blacks (hypothetically) represented each correct answer the student had given during the class. The red checkers represented (hypothetically) their wrong answers. I told them that for each black checker, I owed them a dollar, and for each red checker they owed me a dollar. They excitedly calculated their respective balances. I then advised the students that my accounting had been wrong and I had incorrectly given each student more red checkers than I should have given them. I went from student to student, taking back (subtracting) n red checkers. After each example, I had the student recalculate their balance. For each red checker ($1) subtracted, the students realized their balance increased by $1. After just one example, all the students cheered in unison with the joy of understanding subtracting negative numbers. I then subtracted 2 red checkers three times from the next student. Again the students cheered realizing that subtracting 2 red checkers three times was like adding six to the balance sheet. They then understood that . After this demonstration, students used negative numbers in their algebra with understanding. From my experience at Cosumnes River Elementary School Rancho Murieta, CA Ms. Lung's sixthgrade class of 1996 Do you have a place to make comments like these? Buzz Breedlove breedlov@calweb.com Thank you for your comments; I have placed them as followup comments to the question. Followup question by Ms. White, Community College on October 3, 1997: The key point in your explanation is that should be the same thing as (4)  (4)  (4). The question left to answer is, why? Everybody can accept that taking 3 times 4 is the same thing as adding 4 together three times. The question is, why does this imply that taking 3 times 4 should be the same thing as subtracting 4 three times? The answer is precisely because of distributivity. should be the negative of : that number which, when added to , gives zero. Since is 4 added together 3 times, what you'd need to do to it to get zero is to add 3 copies of (4) to cancel them out. That is why, from the fact that it follows that (3)x 4=(4)(4)(4). . 