Thursday, June 7, 2012
Set Theory, Units and Why We Can Multiply Apples and Oranges but We Cannot Add Them
You have, probably, been told that you can not add apples and oranges. Why is that? But, you may realized or may have been taught that you can multiply them. How is that possible? And this state of affairs may be following you through your school, education, and even (non-mathematical!) career. Here is the explanation.
Let’s say there are some apples on the table and let’s say we want to count them. We decide we want to count apples. And there we go. Suppose there are eight apples on the table, and we correctly count them, thus obtaining count of eight. Eight apples. The most important thing here is what we decide to count. We decided to count apples. And nothing else. It is the apples count we are interested in and not any other objects. That’s our definition what belongs to our set, specific set that (qwe decided!) will contain apples only. Now, if we see pencils on the tables, pears, oranges, books, they don’t match our definition, they are not apples, and hence they will not be added to our “set”. That’s the reason we “cannot” add oranges and apples. It is our decision that we want to count apples only, and our decisions if more apples are put on the table we will add them.
If we decide that our set, the things we want to count, will have other objects and that we want to have a total number of objects we are interested in, then we have to specify that in our definition. We have to say, now, that we have decided to count, as members of our set, say apples, books, and oranges. We may not be interested at all how many of each are there, we just want their total number. IN this case, we clearly can add apples, books, and oranges together, because it is our definition of what belongs to a set that determines elements and number of elements in that set. And, with this set definition, we clearly can add apples and oranges, and for that matter any object we decide will belong to our set of interest.
Let’s see again in which scenario apples and oranges can be added again. Suppose that, on our table, we have 8 apples, 5 books, 7 oranges, and 3 pencils. And suppose that we define the set as “count all fruits on the table”. IN that case we will not count books and pencils, but we will correctly add together apples and oranges, because they are fruits and that’s the definition of a set membership. Hence, our set will have 8 fruits (apples) and 7 fruits (oranges), giving the sum of 15 fruits.
The conclusion is that the set membership definition determines what will belong to a set, what kind of objects, and that this definition will determine which objects we can add together. Definition of the set membership is essential to determine which objects we can count together.
Ok, so, we clarified that, when the set definition says “count only apples” we can not add apples and oranges together. But, when you say “multiply apples and oranges” we can do that. Why?
The answer lies in the two step process we always do, but we may not be aware of that. And, in some language imprecision as well. You do not multiply apples and oranges, You multiply the numbers obtained by counting apples and oranges. Let’s suppose you want to multiply 5 apples with 3 oranges. But then, let’s, for a moment, focus only on 5 apples. Or, even there is a basket of apples, say around 30, beside the table. You can say “I want 5 apples on the table”. You take five apples from the basket and put them on the table. Now, you can say, I want 3 times 5 apples on the table. Then you take, from the basket, groups of 5 apples, 3 times. You essentially took 3 x 5 = 15 apples from the basket. But, where that number 3 came from. Ok, you can say, and you will be right, it came from your head, you just imagined number 3 and decided to count 3 x 5 = 15 apples from the basket. So, you have this, 3 x 5apples = 15apples. But, notice! While you arbitrarily imagined that number 3, it can also come from counting another objects! You can say, you have counted people in the room, there were 3 of them and each of them will have to have 5 apples. Hence, you obtained number 3, this time not from your head, but from real counting of the people in the room. And, again you will have 15 apples on the table, from the basket. We can write that as 3 people x 5 apples = 15 [ people x apples ] . The “unit” here is [people x apples ] and essentially it tells us HOW we have obtained numbers used in the multiplication! By these “units” we keep track what we have counted. So, it is not at all that we have “multiplied people and apples”, but that we have multiplied numbers obtained by counting people and apples. If we use oranges and apples, and say, I want to put 5 apples beside each of 3 oranges, how many apples I will need to take out of the basket, the answer will be 3 x 5 = 15 [oranges x apples ].
Only numbers can be multiplied, added, divided, subtracted. Objects, concepts, like apples, oranges, people, cars, pencils, books, can not be ‘multiplied”, they can be counted only. By counting them we obtain the numbers to work with. It is with these numbers only that we do mathematical operations. You know that 4 + 3 = 7 no matter if we count apples, or oranges, or cars. It is an universal result. When you write down 4 + 3, a friend beside you doesn’t know are you counting in your head CDs, dollars, or apples, but he knows that the result will be 7 no matter what.