Wednesday, February 26, 2014

I no longer pray per say but..

Waking up in the morning as I greet my day,
I no longer pray asking HIM if HE can turn it my way.

I get onto my knees not with the hope that,
my day will turn any different or better to say, If I pray..
For I know that HE has already planned my day.

I fold my hands more in gratitude and,
to acknowledge the unfathomable power of nature and the divine.
With the theory of karma being mysterious to me,
but hearing all the time that it is working in the background,
I sometimes give up in fighting over my thoughts on,
how things "should have" turned out,
as opposed to how HE has made them happen. 

This is not to say that I am ungrateful for waking up to another sunrise,
Or that I am by any means disappointed seeing not much control over destiny,
But a mere understanding and acceptance that there are few things beyond my control.
With every event that is happening,
Is it a blessing in disguise for I might not know what is best for me,
given my limited human vision.
Or just a rough patch that will make me emerge stronger and turn better?

Though there's one thing that I can vouch for -,
Everything has its own time,
A time that it is destined to happen,
As the day unfolds, I should just let it flow.
As long as I feel happy and contend with the way my day went,
If I see that I did whatever I could do to the best of my ability,
If I I did not let myself run down on my self-esteem,
It was a successful day.

Provided that I was honest with myself,
For anything that did not go as I wished for,
I just need to keep my mind and heart open,
I will get the answers at the right time.

There is usually something better in store for us than we think,
Or if that is not how it seems like or turns out even in the longer run,
then it has to be one of the following two cases -
either it made us tread in a direction that we would not have explored otherwise,
and emerge stronger, or
It's still not the end..?

Tuesday, February 18, 2014

Getting mathematical - Understanding negative numbers intuitively

Hello there!

This is a second part of the series - "Getting Mathematical".Here I will be sharing some fun facts about negative numbers.

Question that we would try to answer is the following:

With positive numbers used to measure tangible things - what we can touch or hold (six cars etc.). What do we possibly mean when we flip the sign of a natural number? Intuitively.


What could -3 possibly mean? Or what could 10-12 mean? We can not say -2 apples. Then?

The answer lies in recognizing that apart from being used as a means of measurement, numbers can also represents concepts/relationships in the real world. While natural numbers are used for counting, negative numbers are more for capturing concepts like debt or comparisons [1]. So, in the question above, 10-12 can be used to model a situation where you have 10 apples but your friends needs a dozen. So, you are short of two apples. 

Another example: Negative numbers are helpful in modeling alternating patterns[1]. Consider a number x. If you keep multiplying it by -1, you will get the pattern x,-x,x,-x..The reason that we are considering multiplication by -1 is because it does not change the size of the number. So, allows us to focus on the concept of negativity.

Considering that the weather forecast of the day during a month is oscillating between rainy/sunny. Then knowing today's weather, you can ascertain the weather 20 days ahead without counting - rainy, sunny, rainy, sunny.....

Leveraging negative numbers back and forth behavior: we can say (rainy) * (-1)(power 20)= rainy.

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That's pretty much about it for this post. I would appreciate if you write back! - questions/suggestions/comments?

Dippy

References

[1] "A Visual, Intuitive Gide to Imaginary Numbers", betterexplained.com, K. Azad.

Getting mathematical - Understanding complex numbers intuitively

Hello everybody!

I will be posting a series sharing some intuitive ideas about the concept of number systems in mathematics - natural numbers, rational numbers, irrational numbers and complex numbers.Essentially, what led to their discovery and their relevance in the real-world.  How to explain the notion behind them to a kid.This post is focused on complex numbers. The question that I am raising in this context is:

What does bi denote in the form a+bi? We call it imaginary part? But it has to have some real world significance? What is it? 

As simple as it sounds but understanding number system in mathematics does require quite some thought. Things which we (at least I did) took for granted and never questioned when they were taught. I remember the multiple sessions that we had in high school, solving problems involving  complex numbers - a+bi. Remember? But leaving aside the details about how to multiple them, find its conjugate etc, I wonder if I understood their real-world relevance until few days ago. I recognized a gaping hole in my education seeing that I was not aware of the things that I am going to talk about in this post earlier - the time when I actually studied these concepts in school.

Although I would love to have said that the concepts in this post are from my own understanding and research but that is not true! Please know that everything that I write in this post comes from what I learned from the book [1] and the blog [2] that I followed to solidify my understanding.

Getting started...

Here is one word answer to the question above: Rotation. 

Details: So, we have learned in high school that this imaginary number i (pronounced as iota) is a number whose square is -1. This raises the question that what number can it be whose square is a negative number. Lets try to understand it by using the concepts that we do understand.

In the equation square(x)=9. What is x? It would be 3 and -3. 
Another way to read this is: scale the number 1 by 3, then scale the result further by 3 => giving us 9. Do you see that? 

In other words, I am saying that you can observe this square operation as two steps:
1) 1 * 3 = 3 (This pertains to scaling the number by 3)
2) 3 * 3 = 9 (Scaling the result 3 by 3 times again)

Now, consider solving the following equation:

                                   1.x.x = -1

The question arises : What transformation (just like scaling in the example above) does x represent?

Answer: Start with point (1,0) on x-axis. Now, imagine rotating the point by 90 degrees. What point will you get?

(0,1).. Right?

Rotate it again by 90 degrees. Where will you end up?

It will be (-1,0). Right? Which is what we wanted. We wanted to see how we can transform 1 to -1! 

See the figure below for a pictorial representation of what I said above:

[2] Understanding complex numbers

So, the number i represents rotation. Complex numbers are a way to measure a number in terms of rotation.

They are mostly employed in solving differential equations or physics equations.

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This was all that I wanted to share with you all. I really enjoyed learning about them. I hope you also found it interesting?

Dippy

References

[1] "College Trigonometry", Richman, F., Walker, C., and E. Walker, 1970.
[2]  "A Visual, Intuitive Gide to Imaginary Numbers", betterexplained.com, K. Azad.