Repeated addition allows us to multiply a positive number and a negative number. Using piles and holes this looks like:. Interpreting negative times a positive and negative times negative through repeated addition, however, is problematic. The truth is that multiplication has no meaning here in context of repeated addition. We have entered new territory and if we want to open up our world to new types of numbers it is not surprising that previously concrete, literal definitions begin to flail.
So we have to engage in a sophisticated shift of thinking, letting go of the question What is multiplication? How would we like multiplication to behave? Comment: Let me stress this point. To approach this we first have to be clear on what features of arithmetic we feel should still be true.
That's also consistent with the intuition of adding negative three repeatedly five times, now look above above us slightly higher so you can see ideas of multiplying two negatives, but we can do the exact same product experiment. We want whatever this answer to be consistent with the rest of mathematics that we know so we can do the same product experiment.
What would negative two times six plus negative six to be equal to. Well, six plus negative six is going to be zero. I'll leave you there and I'll see if I can make a few other videos that can also give you a conceptual understanding of why these are true. Dividing negative numbers. For example, under addition, 0 is the additive identity. Under multiplication, 1 is the multiplicative identity. The full set of axioms required is below. From these axioms, we can prove that a negative times a negative is a positive.
Prerequisite knowledge : While I went through and added the justification for each step of the proof that was missing, I needed a fair bit of fluency with the original set of axioms.
I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms. This algebraic proof from Benjamin Dickman is much simpler than going back to a proof based on the axioms of arithmetic. From this, we can show that ab and — ab have opposite signs and therefore that a positive times a negative is a negative. Using the fact multiplication is commutative, a negative times a positive is also negative.
Prerequisite knowledge : The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures. Using the number line again, and considering just -1 as a multiplier and p as some positive number:. Next fun question: we tell students to reverse an inequality when multiplying both sides by a negative. March 15, — pm. I also like to show students that 9 x 9 which they know is 81 is the same as Expanding to give the first 3 terms using the distributive law or the grid or box method yields — 10 — 10 which gives Not a proof though!
March 25, — pm. If you add weight pos. If you remove weight neg. June 1, — pm. July 14, — pm. It seems to me that the entire debt is a negative, so why would I suddenly classify any part of it as a positive? Make a large circle on a piece of paper. When the circle is completely empty, it represents 0. Keep in mind that there are multiple ways to represent positive and negative integers with this model.
But the net effect of these is identical. This suggests that subtracting a negative removing red chips is equivalent to adding a positive adding black chips. AND, because multiplication is basically shorthand for addition, i. Late yesterday, I found a Reddit discussion on this topic. There was a link to a module on negatives at purple math. They had an interesting suggestion for understanding this, which got me a little further down the road:.
Instead, you control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes. If you add a hot cube add a positive number to the pot, the temperature of the stew goes up. If you add a cold cube add a negative number , the temperature goes down. If you remove a hot cube subtract a positive number , the temperature goes down. And if you remove a cold cube subtract a negative number , the temperature goes UP!
That is, subtracting a negative is the same as adding a positive. Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes add three-times-positive-two , the temperature goes up by six. And if you remove two triple-cold cubes subtract two-times-negative-three , you get the same result. Thanks again to you, and to Ben, for your generous patience in working with me. Never too late to learn, good golly!
Congrats on making sense of a difficult, abstract idea. There are many. Whatever works. I never stop learning new math, a subject I slept through in high school, avoided in college, then embraced personally and professionally in my thirties.
Perseverance, patience, searching for good models online, and belief in your ability to learn and grow at any age are powerful, helpful attitudes and practices. I am 67 and always pushing my mathematical envelope. Think about this, Michael Paul. It was nice of you to do it. A pleasure and privilege. Feel free to ask for ideas anytime: mikegold umich. It equals This becomes even more useful when dealing with variables.
I did, for years. I believe that it is. Michael Paul, the idea of opposites is interesting. Maybe the number-line is not our friend. This is really working for multiplication for me, without having to turn my mind into a pretzel thinking about hot and cold cubes.
But it falls apart with division for me. I must be missing something. This was the lightbulb for her, the opposite of the opposite is the the thing itself. Thank you for the super helpful article!!! Thank you for clarifying.
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