Meowthematics: Zero or Infinite treats?
Hellow hoomans!
It's been a while. Where have I been all this time you ask? Nothing special really, just performing my regular cat duties: eat, sleep and the like. It's hard to break the natural cycle you see. Everytime I think of exploring some meowthematics, my hooman just HAS to bring me some tasty goodies (What a capable servant!😼) and this happens...
Can you really blame me(ow) though? I am a cat, I'm supposed to be 'lazy', as some hoomans say - which I should clarify is a grave misconception- for I am simply following nature as my buddy Newton theorized: "An object at rest remains at rest..." (Quick side story: It was ME(ow) who dropped the apple from the tree on his head just so that he could go ahead and formulate the theory on gravity)
But enough of the chatter! I came here to discuss something very interesting, almost as interesting as a rolling ball or a moving laser pointer! Okay, maybe not THAT much but still interesting nonetheless. While I was scrolling through Instagram using my very own account (which all you hoomans should follow: @meowthematics), I stumbled upon this meme, or as I like to conveniently call it me(ow)me(ow) [Reminder: I'm a cat] :
What blasphemy! If my hooman pulled off this sinful act, I would definitely give him a taste of these paws 🐾 and put him up for adoption😾. But funnily enough, I don't find it surprising that this dog is at its wits' end in this scenario. Stoopid doggo. Of course, it's still one treat! They still add up to one! I would confidently wager all nine of my lives to claim that all these doggos would happily jump at the sight of a second, albeit broken treat. Their intellect only goes so far after all! They can't be compared to cats, let alone one as intellectual as I am.
But this does provide Meow to explore an interesting proposition. I suppose you all can rationally understand why the two broken-off treats would still add up to one in total or the case where the broken treats are broken into half yet again as long as you're not a stoopid doggo. But what if this process continues?
Let's not stop at some finite, measurable length. Go far beyond what even these exceptionally sharp cat eyes of mine can pick up. Let's break the treats into infinitesimally small pieces : something so small in length that it is virtually zero but still has a non-zero size. Essentially, it should be the smallest quantity nearest to zero but still not zero itself. At face value, it's not too grandiose of an idea but at a deeper level, it's very weird to think about this infinitesimal quantity. Say this quantity has a value of ε. But I could simply define a new quantity of value ε/2 which would undoubtedly be smaller than ε. So, was ε even the smallest quantity in the first place? This 'infinitesimal' quantity, despite having a clear-cut definition of the smallest non-zero quantity has a somewhat dubious and confusing value. Furthermore, I can technically keep on dividing this quantity with 4, 8, 16, 32, .... (and essentially breaking them off into two halves every time) to create smaller and smaller quantities. Presumably, I should be able to do this countless times to produce more infinitesimally smaller objects. Does that give me infinitely many number of such pieces? Or does it mean that after a certain point, we move past the idea of non-zero smallest quantity and simply concede that it reaches zero?
Now, the more important question that follows after this: when this procedure is carried out on my treat, can I generate infinitely many treats, or do all my treats eventually disappear into nothingness?
Interestingly, the answer seems to be a bit of both. Yes, an infinite number of iterations on the aforementioned procedure eventually leads to zero. At the same time, I will generate infinitely many pieces of the original object, all of which seem to have zero size (magnitude/length) themselves. The most confusing part, however? Remember how I started off this procedure by breaking off one single treat into 2, then into 4, and so on? So logically speaking, the sum of those infinitely many pieces with a size that approaches zero should still add up to the very same single treat. Of course, this should be true and it will continue to be so for a finitely large number of pieces. But weirdly, it seems to suggest that adding infinitely many zeros results in one. The thing to note, however, is that infinity, in itself, isn't exactly a number but more of a limit: a value that a sequence of numbers tend to (and may not reach). Dealing with infinity is more of a formal procedure that involves mathematical analysis and isn't necessarily a case of 'using common sense'. (But of course, the analysis still involves a rational procedure that follows mathematical logic). Looking at the problem at hand, it's more of a case of no matter how many pieces we break it into; to the point it might eventually reach zero but still isn't zero itself yet, it will still add up to one.
For your information, the concept of infinitely many pieces being generated is closely related to the density of real numbers (or more specifically, how an interval in the number line contains infinitely many numbers inside it). Besides this, suppose the original length was 1 and I'm strictly working in 1D, then the first break produces a piece of length 1/2, the next of 1/2^2, and so on. At the n-th step, the length should be 1/2^n. When I say the length goes to zero at the infinity step, I can represent it mathematically as the following:
Officially, this is referred to as 'taking limit'. This is a fundamental concept for Calculus and Infinite Sequences (Be on the lookout for future posts!).
Meowthematics aside, on a more personal note, I would NOT entertain the idea of 'infinitely many treats'. I'm not like that stupid doggo from earlier. For me, that would be akin to a 'Zero' treat since I wouldn't be able to lick it up and taste anything anyway. Instead, how about I show you guys a trick to actually get double the treats? Every hooman falls prey to this charm:
It's about time I get my (double ;) treat and get back to my nap schedule. Catch you guys later!
Meows and Scratches,
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