Nishka's first steps

One evening Nishka and I were taking an evening stroll in the park when we spotted the swimming pool at a distance. Laughing in delight, Nishka ran towards the pool. But as she approached the pool, she had to stop. There were a few small steps near the pool and she needed help to get down. She looked back to see where I was. Seeing that I was at a distance, she took a quick decision. Holding her balance on one of her legs, she put her other leg forward and gently tugged herself down the mini step. The jump caused a jerk and she took a few seconds to steady herself on the lower step. But she had done it - she had climbed down the step on her own! Once she found her balance, she smiled widely, and looked back to see if I had noticed. Seeing my proud smile, she said gleefully, "Mummy..! Clap.. !!!"

She then set out on her next task. She turned around, put one of her foot forward and pulled herself up on the top of the stair. Both of us clapped in delight again! We were both lost in the climbing up and climbing down and celebrating and cheering the good climbs for a few minutes, when I looked up. An older couple, maybe in the early forties were looking at us indulgently and smiling. I smiled back - in embarrassment and pride. After all it must have been a sight to an onlooker, to see a mother cheer her daughter climb up and down such a small step.

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Another day, Nishka, and I came down to the garden. We had walked only a few steps when we saw a group of young boys sit on one corner with a football. The leader of the gang was a 8 year old and the rest were around 5. Nishka as usual broke herself free from me and ran towards the children and started to talk to the boys in her baby language.

The 5 year olds tried to shoo her off. But the eight year old kicked the foot ball at her gently. The ball came and bounced at her feet. She giggled. She picked up the ball and bounced it back to the eight year old , saying, "Anna ! Catch..!" He smiled back and kicked again.

Nishka giggled as she would go in pursuit of the ball whenever he kicked the ball to her. She would then pick up the ball and throw it back to the boy. "Nishka. This is called a kick. Kick the ball back to Anna. Don't throw it," I said, showing her a kick.

Nishka looked at me with a question mark on her face for a few seconds. Then she slowly came to the ball, and very gently nudged the ball with her little leg. All of us there, couldn't contain our excitement. Nishka had kicked! After that the boy and Nishka kicked ball to each other for another 10 minutes.

It was a beautiful sight to watch a 8 year old boy play football with a 1.5 year old girl.

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Prerna

"My husband is a investment banker in the US," said my friend on the chat. I smiled. I had suspected that she was wealthy by looking at her pictures in front of their two-storied well designed house in upscale NewYork city. The clothes that she wore were also expensive.



I remembered the time we were girls. Those days most students went for tutions during their high school. The tution teachers were generally professors in the local university. They were acknowledged experts in their subjects and had several published papers in renowned forums to their credit. Needless to say, the quality of teaching in these tution classes was much higher than the quality delivered in the schools. Mysore, being a university town, was especially known for its teachers and the tution classes were much sought after by students like us.



Prerna and me went together for most of the tutions, including the Trigonometry tution. The Trigonometry tution teacher, besides being an subject matter expert, was also an excellent communicator. She was able to connect to the students very well and also had a good sense of humour. However sometimes she would be inappropriate.



Prerna's father was a middle level officer in the near by textiles factory. The cost of the multiple tutions was always a strain on the family every month. But the family managed - managed with pride. Prerna was always taken into confidence whenever there was a financial strain in the family and I never noticed her feeling small or guilty when she was short of cash.



That month, the family had got a new car. A car was a necessity for the family of five- two adults, two grown up children and Prerna's grandmother. Hence the cash flow that month was a little tight. During the month end, when it was the time to pay for the tution fees, Prerna, very innocently told the teacher, in front of the class, that her father wouldnt be able to pay that month and she would pay the complete amount the next month. Maybe she should have been discreet, but childhood is not discreet. A child trusts unconditionally. Which was why we, her fellow classmates, understood what she was saying.



The teacher did not. None of us were prepared for our teacher's reaction.



"If your family is short of cash are you not buying any groceries this month? If you are, how come you cannot pay the fees? Next month if you cant pay, maybe you should discontinue the tutions," she said. The class suddenly became silent. If someone dropped a pin at that moment, we all would have heard it.



The teacher and her family stayed in a huge bungalow and were quite well off. Her husband was a director in a company.Her response was totally uncalled for.



I dont know what Prerna felt that day. But I was shaken. That day as we walked back home, I told her, "If I ever become a teacher when I grow up, I will give tutions for free." I consulted my parents in the evening, and offered to pay her fees, but Prerna declined. She paid up the whole amount the next week.



Prerna was a very beautiful, graceful and well mannered girl. Other than the financial crisis the family faced, I did not find her lacking in anything. We were the best of friends for 3-4 years more. But eventually we lost touch with each other.



Therefore one fine day last year, when I got a friend request from her on facebook, I was more than glad to accept. I saw the pictures of her various Europe vacations and I was very happy for her. I could see that she was doing well in life. I knew she was not working, so I was curious about had caused the wheels of fortune to turn for her. Which is why I asked her where her husband worked.



And when I saw her reply, I couldn't help smiling.



"God ultimately levels everybody's life" I thought.`

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Note: Some parts of this post has been fictionalized to protect privacy of the people involved. 

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The history of pi, sine, cosine

Call it the effect of reading 'The man who knew infinity – A life of the genius Ramanujan'. Of late I have been reading a lot of mathematics and trying to find out the origin of certain constants and functions ( like pi, sine, cosine etc) that we have been brought up to accept as a standard.

And the discovery was amazing. Most of the constants are not a result of some complex scientific assumptions or have not been born as a need to compute any complex mathematical equation. In fact these constants have always existed in nature and have been discovered by mathematicians over the years.

The History of Pi

Everybody knows that the value of pi is 3.14…er, something, but how many people know where the ratio came from? Actually, the ratio came from nature--it's the ratio between the circumference of a circle and its diameter, and it was always there, just waiting to be discovered. But who discovered it?

It's hard to pinpoint who, exactly, first became conscious of the constant ratio between the circumference of a circle and its diameter, as human civilizations seem to have been aware of it as early as 2550 BC. The Great Pyramid at Giza, which was built between 2550 and 2500 BC, has a perimeter of 1760 cubits (one cubit is about 18 inches, though it was measured by a person's forearm length and thus varied) and a height of 280 cubits, which gives it a ratio of 1760/280 or approximately 2 times pi. Egyptologists believe these proportions were chosen for symbolic reasons, but, of course, we can never be too sure.

The earliest textual evidence of pi dates back to 1900 BC; both the Babylonians and the Egyptians had a rough idea of the value. The Babylonians estimated pi to be about 25/8, while the Egyptians estimated it to be about 256/81.

The Ancient Greek mathematician Archimedes of Syracuse (287-212 BC) is largely considered to be the first to calculate an accurate estimation of the value of pi. Archimedes didn't calculate the exact value of pi, but rather came up with a very close approximation of 3.1485.

Pi, or, at least, the approximate ratio, also appears in the Bible:

"And he made a molten sea, ten cubit from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." --1 Kings 7:23

Etymology of 'pi' 

The Greek letter pi (p) was first used to denote the constant ratio in 1707, by Welsh mathematician William Jones, though the use of the symbol was not popularized until 1737 by Swiss mathematician and physicist, Leonhard Euler. Pi was taken from the Greek word for perimeter.

The History of Sine , Cosine

For centuries, mathematicians have tried to find the length of a chord given the length of an arc of a circle. This study resulted in the forming of the "chord function" which gave the result mentioned above. The chord function was discovered by Hipparchus of Nicaea (180–125 BC) and Ptolemy of Roman Egypt (90–165 AD).

The evolution of sine and cosine function was the result of the study on the same line.

Aryabhata(476–550 CE) formed a table giving the values of "half-chords"( length of chord/2 ) on arcs formed on a circle of a radius of 3438 units. Values of half-chords for a given angle of arc is what is today called a sine function of the angle. The Sine was called Jiya, Cosine was called CotiJya.

Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics.The now lost tables of Hipparchus (c.190 BC – c.120 BC) and Menelaus (c.70–140 CE) and those of Ptolemy (c.AD 90 – c.168) were all tables of chords and not of half-chords.  Āryabhaṭa's table remained as the standard sine table of ancient India.

Etymology of Sine, Cosine

Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جــيــب, abbreviated jb جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جــيــب, which means "bosom", when the Arabic text was translated in the 12th century into Latin by Gerard of Cremona. The translator used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold") [7][8] The English form sine was introduced in the 1590s.

Sources :
http://www.techhive.com/article/191389/a_brief_history_of_pi.html
http://en.wikipedia.org/wiki/Aryabhata%27s_sine_table
http://nrich.maths.org/6908

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