Friday 17 October 2014

Post 4: The Shape of Numbers 1: Addition is commutative

An important idea in math is commutativity. Addition of numbers is commutative. In the simplest of terms, a + b = b + a. And though this may sound obvious, and a child may "know" it, it may not be so obvious as to why it is true. So, my son and I explored this idea a little. 

You pick a number A and you add it to number B. You get a number C. 

Now you do the reverse. 

You take the number B, and you add it to number A. You again get the number C. Why?

There could be several ways to understand it. (Also prove it formally). But intuitively, here is some of the stuff we did, as "geometric" proofs. 

First, we look at the "shape". Here is 4+3 on the first two rows, and 3+4 on the next 2 rows on an abacus. 

And we see that the "shape" of 4+3 is the same as the "shape" of 3+4, except that one is the flipped or mirrored or reflected version of the other. So, we see that the "shape" has not changed, it is the same shape, we have only somehow moved it, and therefore, 3 + 4 = 4 + 3.

And what if now we increase the number of numbers being added? Here is 1+2+3 and 3+2+1. Again, the "shape" is flipped, but since it is the same shape, we can say 1+2+3 = 3+2+1. 


Of course, this does not immediately show that 2+1+3 will also be the same "shape", so we need to explore it in another way too. 


What if we look at another shape way? Here are 7 rectangles forming a large rectangle. 



And now, you can cut up 3 rectangles leaving 4, or cut up 4 rectangles leaving 3, like this. 


And now it becomes clear: because to make the larger rectangle, you have to put together smaller numbers of rectangles, but the order of how you put them together does not matter, you can either put 3 first and then 4, or 4 first and then 3, both make the same "shaped" larger rectangle of 7 smaller ones. And this idea scales now to more numbers than two: using this idea of cutting up a large rectangle into several pieces, you can show that 1+3+3 = 3+3+1 = 3+1+3. And so on. 

Another similar "proof" is that you have a bucket that needs to be filled up by say a certain number of balls. Say the bucket can contain 10 balls altogether. Then, it does not matter whether you throw in 4 balls followed by 6 balls, or throw in 6 balls followed by 4 balls. The end result is that the bucket gets filled up. 

This property says that addition of numbers is commutative. 

And before we as adults start saying, "this is obvious", we should think up some counter-examples, where changing the order of operations does not lead to the same final result, and we find that most of our everyday processes are actually non-commutative.  

Eating your food and then digesting it is not the same as digesting the food and then eating it. In fact, it is not possible. In the same way, putting a ball in front of you and then kicking it, is not the same as kicking a ball and then putting it in front of you. However, when you are in Manhattan, going down 1 block east followed by 1 block north gets you to the same point as going 1 block north followed by 1 block east. This is commutative.

I asked my son to think up of one commutative example and one non-commutative example. He said, putting the engine on and then flying a plane is not the same as flying the plane and then putting the engine on, because you cannot fly a plane without first putting the engine on. Whereas, if you have a plane and a helicopter, putting on the engine of the plane followed by putting on the engine of the helicopter, is the same as putting on the engine of the helicopter and then the engine of the plane, because now you have both engines on.  

So, the important idea to grasp is that when things commute, the order does not matter. In whichever order you choose to do things, you reach the same end result. Whereas, in things that do not commute, the order matters. If you change the order of doing things, you get different end results. 

We had good fun discussing examples and counter examples of commutative behaviour: including this one that made us both laugh. He said, you know, at school yesterday, the computer had my name as Last name, First name, while I know my name is First name, Last name. And I asked him, so does your name commute? He said yes, because it is the same name! :) 

4 comments:

  1. Som, I don't know if you are following this rather interesting debate in India about mathematical education and pedagogical techniques and precepts. Here's the link; http://www.thehindu.com/opinion/op-ed/decolonising-maths-education/article6528274.ece?homepage=true and you can read the earlier two articles and other related links down below on he page.

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    1. Wow, I certainly had no clue about this religion-math interface being so debated in India right now! I did read the article, however, the whole debate on western versus indian versus any other regional approach seems to focus on the wrong kind of pedagogical issue, at least to my mind. To me, if I am not spending serious time on the history of evolution of mathematics in different parts of the world, is rather peripheral - and there is sufficient history and evidence of cross cultural collaborations to also make it even more peripheral to me (Ramanujan and number theory and graphs for example). The focus that is important to me, at least when we think of young kids, is whether we are teaching them maths in a way that they can enjoy it, and develop the discipline of working hard with the enjoyment. I mean seriously, would Tendulkar be thinking of how "English" or how "Indian" cricket is, when he practices for hours to master the sport?

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    3. Totally agree with you, I guess the whole issue of epistemology should be viewed within the perspective of context (the relevant/dominant/counter-dominant social, political and cultural views or the zeitgeist of that era) So we have so many ancient philosophers like Plato, Descartes, Leibniz who were also mathematicians. So they would view mathematics as metaphysics in a certain sense. Also if you read closely, what CK Raju is talking about is the exchange of ideas back and forth between essentially dominant superpowers (and hence with similar intellectual heft) in essence... That is my takeaway, that knowledge is not greek, roman, chinese, arabic or hindu. The society which produces it views it from that focal lens and the optics of discussion is framed within its ambit. So when I try to study, learn and analyse it I also view it from my own references and that might not be always viewed as very appropriate by someone somewhere else 150-200 years from now.

      Apropa teaching at schools, I am no expert on pedagogical systems and perspectives. But do believe many a time a certain dumbing down is done and we know very well that a very specific ideological framework was dominant in the Indian school education you and me received. The discussion is about bringing back "objectivity" and "balance", but not overshooting it in the direction of nationalist patriarchal chest-thumping. I personally believe these are very important, because hidden in those innocuous lines in textbook are the seeds for weltanschauung ( a german term, whose weak english translation is an entire world view) for a young mind and how they will then interact with the wider world. For eg., the Peano's axioms are not rocket science, but I encountered them only in senior high school. Outside the school's textbooks. also, searching through book fairs in middle school, I first came across the term "vedic" maths and interesting ways to calculate squares for natural numbers for eg., amongst other things. That could perhaps have been done earlier in the textbooks of school only. In India we have a habit of treating young ones as too young to understand what adults deem as complex or unnecessary, wherein we should be a little more mindful of them as young adults. And I agree with you how to make the entire curriculum more fun-like and not-boring is very important, to make sure that students with varying degrees of skills, ability and intent feel comfortable in the school environment is the crux of the matter.

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