Beauty of Proof by Contradiction

What is Proof By Contradiction -

To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction.

Axioms

An Axiom is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

1. Additive Identity
[ a + 0 = a ]

2. Multiplicative Identity
[ a . 1 = 1 ]

3. [ 1 != 0 ]

4. Additive Inverse
[ a + (-a) = 0 ]
or, it could be written as a – a =0

5. Multiplicative Inverse
[ a. 1/a = 1 ]

6. [ a .0 = 0 ] (this axiom has a proof)

We all know that we cannot divide by 0. Can this truth(which we simply accept) be proved?

We will start by assuming we can divide 1 by 0.

From Axiom 5, the multiplicative inverse, for any n

n. 1/n = 1.
So, when n = 0 , then
0 . 1/0 = 1

Lets choose 2 numbers, x and y, such that x != y.
From axiom 6, we can say

x . 0 = 0
y . 0 = 0
x . 0 = y . 0

Lets, multiply each side by inverse of 0 i.e. 1/0.

(x.0) 1/0 = (y.0) 1/0
We can rewrite it,
x(0 . 1/0) = y (0 . 1/0)

Now, 0. 1/0 is 1(multiplicative inverse), which means
x = y , but this contradicts our assumption that x!=y.
So, this means it was wrong to assume that 1/0 exists.

There are some famous proofs using this method -
1. There are infinite prime numbers ( Euclid proved this in Elements)
2. √2 is irrational

Sources:
1. Proof by Contradiction
2. The Art Of The Infinite ( our lost language of numbers)

Life Purpose – A quote

The main purpose of life is: To have a job in whose purpose you can believe; To have friends whose immediate purposes you can trust; To have some spot on earth to which you can return as home; To be at the same time a citizen of a larger world.”

- James A. Michener

Silence – a powerful tool

Well Said

In response to , Programmers: Before you turn 40, get a plan B, I found this great comment -

Disclaimer: I didn’t read the full article. I can’t stomach these types of “career strategy” articles anymore. This approach to life in general is very misguided (maybe less so if you just want a career to fulfill other goals), but for those who want to pursue a more idealistic life, just pursue your passions. Plan B is being an improv comedian. Plan B is writing the great American novel. Plan B is traveling the world as a war correspondent.

I left corporate world precisely because I felt “planning my career”, as I was encouraged to do, was an absolute false way to live (for me). I couldn’t reconcile the absurdity of it with the severity of the consequences if I didn’t do it. But the consequences are all socially constructed, and they go away as soon as you change your mentality.

When I think of my heroes, I don’t think of people who’ve hedged their bets here or there – they just did what they felt they had to.

Well Said !!

Just do it

The author says,

Embrace new and uncomfortable experiences, and look forward to change. Just do what feels right.

One of the comments on the page ,
Just do it -

change is challenging because there are inherent risks involved and a fear of the unknown. It’s a lot like standing on the edge of a cottage dock before you jump in the water. You know it’s going to be cold upon arrival but pretty soon it’s comfortable. The same goes for making many decisions.

T.S.Elliot

We shall not cease from exploration, and the end of all our exploring will be to arrive where we started and know the place for the first time

Beautiful Poem

By Wordsworth,

My Heart Leaps Up
When I behold a rainbow
In the sky

So was it when my life began
So is it now I am a man
So be it when I shall grow old
or let me die

The child is the father of the man
and I wish all my days could be
Bound each to each in natural piety

Effortlessness

From “When The Shoe Fits” -
” Only that which is attained through effortlessness will never be a burden to you, and only that which is not a burden can be eternal. Only that which is not in any way unnatural can remain with you forever and forever. “
” And whatsoever you achieve through will, will always remain a burden to you. “

New year resolutions

As I entered my room at office, my colleagues were waiting to give new year bumps.Someone asked me – “Any new year resolutions?” . I had made 2 – improve my health – not just losing weight but work on mind, body and soul.My slogan for the year is “I am a peaceful soul”.
The other resolution is to work on my chess.I want to read atleast 2 chessbooks this year, and teach chess to kids more often.I have been teaching chess to 4-5 kids of my building.We get together, when both they and I, are free.I want to increase the frequency of our meeting for chess lessons.
I enjoyed receiving bumps from colleagues – getting launched into the air and on way down getting caught. A very apt way of starting new year, knowing that the year ahead will have up and down moments, but enjoy each moment.

Find your place in the Life Pattern

From Inner Peace

“You have a part in the scheme of things. What that part is you can know only from within yourself. You can seek it in receptive silence. You can begin to live in accordance with it by doing all the good things you are motivated toward and giving these things priority in your life over all the superficial things that customarily occupy human lives.”