Skip navigation

A pathetic way of living

Being around pro people all the time can be really annoying, especially when one is not very pro. I spent this entire term being envious of other people’s proness - it started of as admiration, but later gave way to envy and self doubt. But then I realized that this is an absolutely pathetic state of being. Self doubt and envy wasn’t getting me anywhere, and in fact doubting myself was increasing the gap between my friends and me, since I stopped attempting to become more pro; my line of reasoning was “What’s the point? I’ll never be as pro as the others”. Looking back, that was a pathetic way of thinking – giving up is clearly not the way.

I realize now that the only solution to my unproness is proness. I need to be more pro, and it is my belief that proness in a field comes from passion – if I am passionate about something, then eventually I will become pro at it. Self doubt is nothing more that doubting one’s passion – I promise to myself not to doubt my passion ever again, especially my passion for math.

One might think that what’s the point of making such a promise when you know you are going to break it, but not making the promise is even more pathetic than making a promise knowing you will break it.

Geometric and Arithmetic Mean

I don’t know why I always have trouble proving this.

x,y \geq 0, \sqrt{xy} \leq \frac{x+y}{2}, with equality if and only if x = y

There are a lot of proofs for this that I consider to be ugly because of the type of algebraic gimmickry that they use. The best proof that I could think of is as follows.

0 \leq (x - y)^2 (with equality if and only if x = y)

0 \leq x^2 - 2xy + y^2

0 \leq x^2 + 2xy - 4xy + y^2 (1.1)

4xy \leq x^2 + 2xy + y^2

4xy \leq \left(x+y\right)^{2}

Taking positive square roots, and by the strict monotonicity of the positive square root function,

2\sqrt{xy} \leq (x + y)

\sqrt{xy} \leq \frac{(x + y)}{2}

Note that, the strict monotonicity of the positive square root function is required for the equality condition.

On the other hand, if in (1.1) if we said adding 4xy to both sides, this would have become an ugly proof. I “have a low tolerance” for algebraic gimmickry like that.

Here’s another ugly proof:

We have x^2 - 2xy + y^2 = \left( x - y \right)^{2} \geq 0 (1.2) with equality if and only if x = y. This yields

xy \leq xy + \frac{1}{4} \left( x^2 - 2 xy + y^2 \right) (1.3)

= xy+\frac{1}{4}x^{2}-\frac{1}{2}xy+\frac{1}{4}y^{2}

= \frac{1}{4}x^{2}+\frac{1}{2}xy+\frac{1}{4}y^{2}

= \frac{1}{4}\left(x^{2}+2xy+y^{2}\right)

= \frac{1}{4}\left(x+y\right)^{2}

Taking roots yields the desired inequality. It is clar that we have equality if and only if the second summand in (1.3) vanishes; by (1.2) this is only possible only if x = y.

(This proof was taken form Fall 2004 and Winter 2005 notes by Volker Runde)

Pets and slavery

Molly the pet Dog

Meet Molly. Isn’t she the cutest thing ever? Well, I went camping this weekend with some friends, and Molly came with us.

Observing Molly stimulated some questions. What gave us human beings the right to domesticate dogs and put them on leashes? Does Molly long to roam free with in the wilderness like her ancestors used to? Is Molly a slave whose purpose is to entertain us human beings, her masters?

I am aware that since Molly has been bred, etc, she probably wouldn’t survive in the wild/the streets. But it still distubed me that when we walked around with Molly, she did not choose her own path, she was forced to follow our path. Also she was trained to sit when asked to.

I guess I am supposed to take comfort in the fact that dogs may not think like humans do. and they may not have the same thoughts of freedom that humans would.

Proness

My last post had a list of things that I wish I was pro at, but how exactly do you become pro at things, given that you are not naturally endowed with a given talent?

The model that I think that works is that in order to became pro at something, you must at first become obsessed with the thing. Then you keep being obsessed till you are pro (or dead).

For example, I believe that in order to become pro at a musical instrument, you must obsessively play it till you are pro.

Or to become pro at math you must constantly think about problems and the structure of things.

I’ll end the post with a story about Newton that is probably not true:
A lady asked the famous scientist (and infamous mathematician) how he came to discover the law of gravitation and he is said to have replied, “By constantly thinking about it.”

Things I wish I had learnt before university

Now that I’ve completed one year of University, I think I finally realize how short life really is. I say ‘I think’ because it is possible that I might realize that life is even shorter in the future. So I’m going to create a list of things that I wish I had learnt before I started college, because life is too short to be wasted on learning things I should have learnt already. I will update this list as I go along

Things I wish I had learnt before university(going to try to make this a ranked list)

  1. \LaTeX
  2. Epsilons and Deltas :)
  3. Dvorak Keyboard layout
  4. Vim

For things I did learn, but not very well… Things I wish I had become pro at before starting university:

  • The Violin
  • C
  • Unix/Linux
  • Japanese
Follow

Get every new post delivered to your Inbox.