 The last of the elementary arithmetic operations is division and in many ways this is the most complicated not for any particular reason other than we actually combine two very different ideas when we talk about division. So again mathematics begins when we take some sort of concrete process and make abstraction of it and in this particular case the process is well sometimes you have some sort of quantity and you want to allocate that to a group of people equally and there's actually two different ways we have of doing this. So for example let's say I have a whole bunch of things and I want to distribute it among several recipients so there's two distinct problems that are going to emerge. First I might know the number of recipients I might know the number of people that I'm giving this to and so for example say I have this collection of objects and I want to divide this among three recipients and the question that I want to answer is how many does each person receive. Now because we're dividing the amount into a given number of parts there's three parts which will be given to each of the recipients then this particular question is known as a partative division. The other possibility is I might know how much each recipient is going to receive so I'm going to take this amount here and I'm going to divide it up so that each person receives this much and then the question that I want to answer is well how many people can I give this much to and because I'm dividing the amount into parts of a given size this much this is called the quota because I know the quota then what I have is what's called a quotitive division. So for example let's take that partative division so I have this collection here and I want to split it up among three recipients and I do want to make sure that I give the same amount to each person so one way we can solve this problem might be the following. So I'll set up some bags some some places that I can put these things that we're going to distribute and so here I have one two three recipients so I'm going to have one two three bags one two three places that I can drop these things into and I have a whole bunch of things here this is my collection that I'm going to try and distribute. So I want to distribute these so that all the bags have the same amount. Now one way to guarantee that all the bags have the same amount is to put the same amount in each bag which means I'm going to take out sets of three of these so that I can put in one two three. So I'll take a set of three and distribute them. I'll put another one in each bag so again that's going to require one two three more so this set of three is gone and what I have left at this point I can't distribute this last one because if I give this one out then somebody has more than somebody else than everybody else. So we're left with some number remaining and I can't distribute this last one evenly. So we can summarize our steps again mathematics occurs when we make an abstraction of some sort of concrete process and so what I did is I took this amount I divided it among three recipients and each one got this amount and there was this left over and so the abstract feature of this set well that's a seven the abstract feature of this set that's two the abstract feature of this one and that says seven divided among three gives two with one remaining and our notation for this process is going to remind us that when we did this we've removed sets so again to go back to that here's my original and I tried to distribute it equally among those three places so I threw one in each I threw another one in each and that removed them from the original set well that's a subtraction and so the notation that I'm going to use is going to be a variation of my subtraction symbol now I'll either write it this way and write seven divided by three is two with the remainder of one or the other option is I can use something that looks like the subtraction symbol but it's a slash seven divided by three is two with a remainder of one well I can do the same thing quotatively and again the quotative problem is if I know how many each person is going to get then the question that I want to answer is how many persons receive this amount so here this time I'll proceed in the same way and I'll set down a number of empty bags and fill as many as I can with the understanding that each bag has to have three of these things so I'll fill the bags one by one here's a set of three which I'll drop into the first bag here's another set of three which I'll drop into the next bag and well now I'm left with one and that's not enough to put into this last bag so I can fill one two bags and I'll have one left over and again we'll make abstraction of the process again this amount seven divided into collections of this amount three I got two collections one two and there's one thing left over and as before we did this by removing sets we have our original amount and we removed a set of three we've removed a set of three and again we'll use our symbol for that division and so again we have seven divided by three is two with remainder one using either that symbol or the slash now let's take a look at how we can define division so now when we divided seven things among three recipients each received two and there was one left over the other way we can look at this is we had three sets of two and one more gave us our original and so that means that there's a relationship here when I found seven divided by three gave me two with remainder one it's the same as saying seven is three twos plus one more now that was our part of division my quotitive division will give me something slightly different when I divided seven into sets of three I got two sets and there was one left over so that says that two sets of three with that one thing left over is going to give me seven and this time we have the relationship seven divided by three remainder one seven is two sets of three with one more left over now because multiplication is commutative these two are really the same relationship the only difference is this is three times two this is two times three and we can use that to give our actual definition of division and so suppose I have a is bq plus r so there we have it bq plus r where r is something between 0 and less than b and that emerges because if you think about what we did here if I still had three left over I would have distributed them so the only reason I stop giving things out is I've run out of things to give out so I have the remainder has to be less than b well then it tells me that a divided by b is q with remainder r and some terms here this amount that I'm giving out that's our dividend this amount which either represents the number of sets or the size of each set is going to be our divisor the value here the number of sets that we have or the number in each set that's our quotient and then what's left over is going to be our remainder