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Inverse property of multiplication

Simple idea that multiplying by a number's multiplicative inverse gets you back to one. 5 × 1/5 = 1. Created by Sal Khan.

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Video transcript

Let's say that I have five lemons so that's [counting to five] ... five lemons and I were to ask you: what do I have to multiply times five to get one? or in this case: what do I have to multiply times five lemons to get one lemon? and so, another question you might ask because really multiplication and division are two sides of the same coin is what do I have to divide five by to get to one lemon or yellow circle, or whatever I have drawn right over here Well, if you have five things and you divide by five, you're gonna have five groups of one so if you divide by five, you're gonna have [counting to five] ... five groups So you could say five divided by five is equal to one take five things and divide it into five groups then each group is going to have one in them or you could say five times one fifth is equal to one (and I use the dot for multiplication) I could also say five times one fifth is equal to one these are all really saying the same thing Maybe what's kind of interesting here (although it's not some huge learning) it's really just another way of writing what you already probably know, is this idea that if I have a number and I multiply times it's multiplicative inverse (and most of the time when people talk about inverses in mathematics they are talking about the multiplicative inverse) then I'm going to get one so five time one fifth is equal to one but that's just because five times one fifth is the same thing as five divided by five if you were to actually multiply this out you actually take five times one fifth this is equal to five-over-one times one-over-five you multiply the numerators: five times one is five multiply the denominators: one times five is five so you have five fifths, and five fifths is the exact same thing as one So if someone where to ask you a question, they say "Hey, I have the number 217 and I want to multiply it by something, and I want to get one after multiplying it by that something" Well then you say - Well look, If I took 217 and divided it by 217 that would get me to one and dividing by 217 is the exact same thing as multiplying by one-over-217 multiplying by its multiplicative inverse which is, once again, a word that is fancier than the actual concept you are just multiplying by the inverse of this number Another way to think about it is if I have five things and I take one fifth of those things, how many things do I have? Well, if I take one fifth of five things I have exactly one thing right over here But the general idea is super-duper-duper simple if I have some crazy number ... 8,345 that's actually not so crazy, let's turn it to something in the millions ... and 271 ... so 8,345,271 And I say, what do I have to multiply by (and now I use this multiplication symbol right now) what do I have to multiply that by in order to get one? I just have to multiply it by the inverse of this the multiplicative inverse of this so one-over-8,345,271