Sunday, August 14, 2011

LOL

The beautiful lady is my grandma! Ever since I was a little kid, I used to call her "Amma".

Just a while back, I was lying in her bed while she uploaded pictures of her recent trip to Lhasa on facebook. She was trying to tag Chris, Jillian, Siobhan and Jessica in some picture of hers. These people were travelers Amma had met there. She wasn't friends with them on facebook, so she just wanted to tag them to place their names there.

So after a few minutes of tagging, she asked me to make sure she did it correctly. When I read the screen, I saw the tags Jessica Alba, Chris Brown, Siobhan Magnus and Jillian Michaels! Madam tagged all the stars without knowing it! 

We had a solid laugh when I explained what happened!

A great memory of our time together :)

Friday, August 12, 2011

Is 0.999... really equal to 1.000...?

What does (1/3) mean? It means you break the number 1 into 3 equal parts.

When you sum up these 3 parts, or perform the multiplication operation (1/3) x 3, you get back unity or 1. That makes perfect sense. But under decimal theory, 1/3=0.333...,  where the “3..” represents an endless recurrence of 3s. And then 0.333....x3 = 0.99999...

This equality implies that 0.999....=1.000...

Yes, I realize that Mathematics speaks for itself, but how can we understand this strange equality and how do we know that this equality is true?

In order to answer these questions, we firstly need to understand the concept of real numbers. So the question I ask you is this: how can you quantitatively define 2 distinct real numbers? Think about this for a second before moving on.

Here’s the crunch!

Any two distinct real numbers A and B will always have an infinite number of numbers lying in between them. Under mathematical formalism, one would say that the set of real numbers between A and B has an infinite cardinality. Cardinality refers to the number of elements in a set. But anyways, we don’t have to delve into strict mathematical terminology to understand the ideas here.   
Now I’m going to prove this idea to you. Let’s just take any two arbitrary real numbers for analysis. Let’s take the numbers 1.000... and 1.0100... So how many numbers are there in between these two numbers? Well, there is 1.0090.., 1.00990.., 1.009990..., 1.0099990... and so on. This already shows you that there can be an infinite number of numbers in between the two numbers we are studying. Of course, there could be even more combinations and permutations of numbers, leading to even more possible numbers (but still infinite). There could be 1.0080..., 1.00912330.., 1.0091241240.... and many many more. Don’t bother listing them, you’ll go on forever... literally.

Try this out for other distinct real numbers. You’ll realize the same thing. There are always an infinite number of numbers in between 2 distinct real numbers. The reason for this is the ability of real numbers to expand, or to be more precise, contract in decimal places forever and ever. 

So now I ask you, how many numbers separate A and B, when A and B are exactly the same? Quite intuitively, the answer is none. If they are the same, no numbers should separate them.

Now, let’s look at our strange equality 0.999...= 1.0000...

So I ask you now, how many numbers separate 0.999... and 1.000...?

To answer this question, let’s just try to generate a possible number that could separate them. Let’s try the number 0.999990... Does it separate them? Nope. 0.9999999... is greater than 0.999990... and hence it can’t separate the two numbers.

Let’s try something smaller. Perhaps 0.990...?  Nope, 0.999... exceeds 0.990... and hence 0.990... cannot be in between the numbers.

This same pattern of analysis can be repeated and one will realize that for any real number you try to define in between 0.999... and 1.00..., 0.999.... will always be greater than it or 1.000... will always be smaller than it.

We can only conclude that there must be no real number separating 0.999... and 1.000... And according to what we deduced from the earlier part, since no real numbers separate 0.999... and 1.000..., they must be equal.

QED

One final question you may ask is this: even after this proof, why does it still intuitively seem that 0.999... cannot be equal to 1.000...? Well this is because when we were taught basic arithmetic in school, we were not exactly taught about real numbers and their analysis, but rather simpler forms of numbers like whole numbers. 

Thus, when we see "0.99..", we automatically tend to reject the idea that it could be equal to "1.00.." , just like we would reject the idea of a whole number like 99 being equal to 100. The key idea here is to unlearn our previous  ideas of numbers as whole numbers or simple numbers and to replace our understanding with a more complex one, such as that of the real numbers. Only after that will this strange equality start to make more sense intuitively. 

If you liked this mini-proof that I gave and want to learn more, do check out the following links:

Thursday, August 11, 2011

My demons and me

Just like any other human being, I have my faults and weaknesses. I’d like to call them my personal demons. Sometimes, they are very powerful and impede my growth. And sometimes, I outgrow them and they seem miniscule in the distance.

I always believed these demons of mine were chaotic and meaningless- that they came up sporadically to haunt my life. But recently, I’ve come to realize something very interesting about them. They follow some sort of a pattern!

When I’m bored, uninspired and sad, I get tempted by these demons into wasting my life and wasting my time. I start watching loads of TV and surfing the internet. My appetite multiplies tenfold and I begin napping like a maniac. I get angry very easily and I become restless, sometimes walking back and forth my room, and sometimes even taking 2 hour long walks to kill time. I start skipping school and stop all forms of useful work. 

This hit me really hard at the start of 2011. In fact, I recall I wasted an entire week or two just vacillating over what to do. It was absolute purgatory, so much so that even my mother felt it.

When I finally manage to pull myself out of the hellish state that I described, my demons somehow return to haunt me! This part is really interesting and to explain it fully, I’ll have to go off tangent and tell you a bit more about myself.

I really really love Mathematics and Physics. I think the physical and mathematical worlds are both structurally elegant and I enjoy studying their beauty. In fact, I do a lot of independent research on these subjects and because of that, learn many new and groundbreaking ideas. I rarely post them on this blog because I find it is very tedious to write about them and I don’t really have the leisure of time at the moment.

But anyways, these demons use my love and passion for Mathematics and Physics against me! They use my knowledge and natural scepticism to make me feel that my understanding of the world is useless. They question certain axioms in Mathematics and Physics and in the process, make me feel that the knowledge I so painstakingly built up is obsolete. Many a time, I get pushed back to the hellish state of life that I so painfully made my way out of, and then the whole process repeats and sustains itself.

I’m sure there are other people who have had it worse than me, but yeah, it’s still been a long and hard road. Fortunately, things changed for me just a few weeks back. For one, I came to realize how my demons functioned. That’s the source of inspiration for this post. Secondly, I came to understand that I have to ignore these demons in order to overcome them. That’s precisely what I’ve been doing and I’ve never felt better in my life.

As to what these demons are... I’d like to think of them as bits of negative Karma that I’ve accumulated over time through my misdeeds. In the end, they are still a part of me; I cannot escape from them. In Buddhist thought, this can be understood as the intertwining of good and evil.

I guess the best I can do now is to work around these demons- to ignore them and to become a better human being, both for myself and for all life around me. I mean it’s either that or I accept defeat, and I’m not going to settle with the latter.

Picture from: http://shipinthenight.wordpress.com/2011/07/22/gates-of-hell-torn-asunder/

Wednesday, August 10, 2011

Love, sex and lust

Hi everybody! It’s been an awfully long time since I wrote, but I’m really happy to be back on this blog. I’ll do my best to keep updating from now on.

As the title suggests, I’ll be sharing my views on love, sex and lust. If these topics make you uncomfortable, well then, grow up I guess? Nah, I’m just kidding. Read at your own leisure. I’m mainly going to be talking about my ideals of love and sex and the problems behind lust. I came up with these ideals through studying myself and my follies over the years. I’ll start off with love.

Love

So what kind of love am I talking about? Well, I’d like to focus on the love between a man and a woman- romantic love, that is. I’m doing this because this kind of love seems to be the “in-thing” in the crazy teenage world of mine.

A very common misconception is that romantic love is all about dependence and obsession. Many teenagers actually believe that true romantic love is found through depending on the guy or girl and literally “sticking” around them persistently, such that almost all free time is spent with the girlfriend or boyfriend.

Here’s an analogy to illustrate the problem here- two people, who are rather unhappy and unfulfilled in their lives, come together and further drain each other’s happiness by wasting time together. 

I hope you can see the inherent flaw in this typical teenage view of love. True love is not found through dependence. Quite to the contrary, it is found through absolute independence. It is attained when two independent people, who are both very happy and fulfilled in their lives, come together and share their happiness, only to make the net amount greater. They must also love and appreciate each other for the “crude” humanity they both possess, and not for any other reason. 

Sex

When I mention sex here, I don’t just mean the act of sexual intercourse. I actually refer to all acts of physical intimacy between a man and a woman.

There is another common misconception in the teenage world, especially among boys, that sex is purely about fun. Most teenage boys actually believe that sex is all about making use of a woman to get private pleasure. Their views have been heavily influenced by pornography and its portrayal of women as “sluts”, “bitches” and “whores”.

And again, this view is false. Sex is after all a physical manifestation of the love that I described earlier. Therefore, sex must also inherit the ideals of love. Sex cannot be governed by carnal desires. It must be governed by absolute appreciation and affection for the person whom you love.

Lust

Lust is known as one of the seven deadly sins. I feel that lust is any man’s greatest flaw. Lust corrupts a man’s mind and changes the way he looks at himself and at women, and sometimes even children and other men. Lust puts innocent women at risk. Lust induces recklessness in men, who just so happen to be the rulers, governors and policy makers of civilizations, and consequently destroys civilizations as well.  

Sometimes, men live their entire lives without realizing that lust is a big problem. They entertain the carnal desire without realizing that it “eats” into their humanity. In this sense, I feel that women, in general, are very fortunate to not be riddled by lust the way men are.

Mahatma Gandhi actually realized this huge problem of lust and fought his own for a large portion of his life. Through his own brahmacharya or sexual control, he managed to keep his lust in check and attained chastity. I’m sure he became a much greater and purer person through this endeavor. His successful control of lust serves as a very important example for all men.

That being said, I don't mean that all men should eradicate their lust by attaining chastity like Mahatma Gandhi did. Mankind will wither away if we did that. I mean to say that all men should eradicate their lust by governing their love and sexuality with absolute affection for the other person, and not with a carnal desire for the other person. 

You know, in my country Singapore, all boys from 16 years of age and above are made to undergo conscription. There is a saying here that once the boys finish their military service, they come out as men. I don’t think this should be the only benchmark for manhood. I believe that a true man has the ability to control his lust, just like Mahatma Gandhi did. I believe it is a true man’s lifelong duty to keep his lust in check and to create a clean and pure world for himself and his loved ones.