Pythagorean Theorem and Bhaskara's Proof

The Pythagorean Theorem is one of the most celebrated mathematical theorem in the world. According Guinness Book World of Records, it boasts more than 500 proofs. Most famous of the proof is the proof provided by Euclid in Elements

Many mathematicians from antiquity to modern times had marveled at this rule and proved it in different ways. Here we will see how Bhaskara, one of the most important mathematicians of old India proved it geometrically.
The theorem says a² + b² = c² : where a and b are sides of a right triangle and c is its hypotenuse
Bhaskara gives the following diagram to prove the theorem:

Bhaskara’s description of Pythagoras Theorem

From the picture, you can see that
Area Large squire is = Area of all 4 triangles + area of smaller squire
c * c = 4 * ( 1/2 ) a * b + (b-a) * (b-a) = 2ab – ( a² + b² + 2ab)
c² = a² + b²
which is the Pythagorean theorem

Is Book a Failed product?

A good product creates and maintains two way communications between the user and the product. A successful product, software or otherwise, creates a more natural form of interaction, an interaction that can take place subconsciously, without effort, whereby the communication in both directions is done so naturally, so effortlessly, that the result is a smooth merger of person and the product, jointly performing a task.

Given that definition, see what Socrates had to say about Books, one of the most enduring and successful product ever in human history:

SOCRATES: You know, Phaedrus, that’s the strange thing about writing, which makes it truly analogous to painting. The painter’s product stands before as if they are alive, but if you question them, they maintain a most majestic silence. They seem to talk to you as if they were intelligent, but if you ask them anything about what they say, from a desire to be instructed, they go on telling you just the same thing forever. And once a thing is put into writing, whatever it may be, drifts all over the place, getting into the hands not only of those who understand it, but equally of those who have no business with it: it does not know how to address the right people, and not address the wrong.

—Plato: Collected Dialogues, 1961.

Is the product Book, a failure? We know its not. Then how does it follow the definition of a good product, described in the beginning of this note?

Credits:

The Design of Future Things by Donald A. Norman

Books.Google.com for complete quote by Socrates

Panchathantra: The best of Indian stories

Dear Malu,

Panchathandram is full of stories. Very good stories. This is a book that has a lot stories about birds, animals, and many people. There are stories of a crow that will stay with owls and then defeat them in war. There are stories of people who are so greedy and bad things happening to them. There are stories of people who are so good, but will go through difficult times, but in the end many good things will happen to them.

Once upon a time, there was a kingdom called Mahilaropya. It was ruled by a good king. He had three sons. All of them were not very good at studies. They just liked playing around and teachers could not teach them anything. King was so sad. “How can they become kings themselves when they grow up, if they are so stupid and not good at studies?” King worried.
King thought and thought and decided that his sons must be taught all the good lessons as soon as possible. “But who can teach my sons all the lessons needed to make them a king?” King asked his prime minister. Prime minister Said “Oh King, only a very smart teacher can teach your sons all the lessons needed. We need a great teacher to teach them well”

So King ordered Prime Minister to find such a teacher. He decided that if a teacher can teach his sons well, he will give a lot of presents to the teacher. Prime Minister, after searching a lot, found such a teacher. His name was Vishnu Sharma. Prime Minister brought him before the king.

“Can you teach my children all the great things they need to know as Kings?” King asked
“Yes, your illustrious King, I can”, Vishnu Sharma Said.
“How can you teach them all the lessons to those kids who only like playing and not learning?”
“I will teach everything to them by telling STORIES…” Vishnu Sharma Said.

And thus, Vishnu Sharma taught five big lessons to Kings Sons by telling them many stories. All these stories together came to be known as Panchathandram. Each story has very good “moral” or a lesson attached to it.

Here is a story from Panchathandram. Ask Mummy to tell you the moral of the story.

Little Rabbit who defeated the Big Lion

Once upon a time there lived a ferocious lion in the forest. It was a greedy lion and started killing animals in the forest. All the animals together went to the Lion and begged him to stop killing. They told him that everyday one of them will come to his house and he can kill that animal and eat. So every day it was the turn of one of the animals and in the end came the rabbits' turn. The rabbits chose an old rabbit among them. The rabbit was wise and old. It took its own sweet time to go to the Lion. The Lion was getting impatient on not seeing any animal come by and swore to kill all animals the next day.

The rabbit then strode along to the Lion by sunset. The Lion was angry at him. But the wise rabbit was calm and slowly told the Lion that it was not his fault. He told the Lion that a group of rabbits were coming to him for the day when on the way, an another angry Lion attacked them all and ate all rabbits but himself. Somehow he escaped to reach safely, the rabbit said. He said that the other Lion was talking very ill of this Lion and was telling that this Lion is a very bad lion. The Lion was naturally very enraged and asked to be taken to the location of the other Lion.

The wise rabbit agreed and led the Lion towards a deep well filled with water. Then he showed the Lion his reflection in the water of the well. The foolish Lion seeing the reflection thought that there is a real Lion in the well. The Lion was furious and started growling and naturally its image in the water, the other Lion, was also equally angry. Then the Lion jumped into the water at the other Lion to attack it, and so lost its life in the well. Thus the wise rabbit saved the forest and its inhabitants from the proud Lion.

With lots of Love,
Papa

About our Flags

Dearest Malu,

You remember the apartment complex where we stay in US. Pappa was coming back from office and slowly driving on Capitol Avenue. I could see the apartment from far and first thing you see from far, is a big flag fluttering over the apartment. If you remember, it is the national flag of America. It is a beautiful flag and when the wind blows, it starts fluttering and dancing. If you keep looking at the flag, you will see RED and WHITE stripes. They are almost like the stripes on your frock. Start counting all the RED stripes. How many are there? If you get 7, then you are right. If you haven’t, try counting again. Then count all the white stripes, you will get 6. Now if you count all the RED and WHITE stripes, how many stripes will you get? You will get 13. America had 13 different places in the beginning. That was very very long back. Look at the stars. They are all white and what is the color of the “box” where they have placed the stars? It is Navy blue. Remember the uniform Ammama had stitched for you long back? I think the color of the uniform frock was navy blue. Now start counting all those stars. It is very difficult to count those stars. You need to know how to count up to 50. But Akuchechi will know. Let her count. There are 50 stars in that flag. It is because US has 50 states. California is the state where we live. We used to live in Virginia too. That is another state. You can see picture of all states in US in your LeapPad Book.

Dubai where Akkuchechi lives also has a beautiful flag. It has four colors. Count the number of colors in this flag. There are four colors. Can you name each of the colors? Yes Red is there. Then Green, White and Black. If you really look at this flag, you will see letter E there. Dubai is Unites Arab Emirates. That is the real name of that place. And Emirates starts with a letter E, like America starts with A and our India starts with a letter I.



What about our India? Don’t we have a flag like all other countries? Yes we do have. It is a really beautiful flag. It has three colors, Saffron (it is like red), White (every flags seems to have white color, isn’t it?) and Green. Look at that circle in the middle of white stripe. It is called Chakra. Chakra is another word for Wheel. Your cycle has two big wheels and many small wheels right? But this wheel on our flag is a special wheel. It is called Asoka Chakra. You can see that it has got spikes. Those little blue lines inside the chakra. Count all the spikes and you will be surprised that it has 24 spikes. For big people chakra with 24 spikes means many things. For you it means movement and speed, like when you get on to cycle (you can move fast and fast right?). When we are at it, can you count the colors in our flag again? There three colors, no there are four colors. Saffron, White, Green and Blue used for chakra. But we call our flag “Tri-color” flag. It just means flag with three main colors. But since it is our flag, it is also a pet name for Indian flag. Like malu is for Jhanvi.



Our flag was not always like this all time. When Gandhiji wanted a new flag, he asked an uncle named Pingali Venkayya to make a flag for him. Pingali made one flag, but just with two colors. Green and Red. Gandhiji didn’t like it. He wanted to add white also into the flag. Then he looked at and thought he will add a “Charka” also in it. You can see that flag here. It is almost like our flag. But what is there in the middle? It is called Charka. It is a machine used to make clothes. There still are some people using this machine to make clothes. They make cloth from cotton. The dress you are wearing now is made of cotton. You can make threads from cotton using charka. Gandhiji was really fond of making strings using charka. Using that strings he used to make clothes. He never sold it but made dress for himself. After some time people did not like charka on our flag. They then put chakra (wheel) and that is how it became our flag.

Making the Indian flag is very difficult. Only two places in India can make the flag cloth. And flag cloth must be always Khadi. Remember the strings made of cotton? By sticking all the strings together, you get a cloth. That cloth is called Khadhi. In Kannur, there are places called Neythu Shala. You can see how people make cloth by visiting there. The cloth they make is white. Then how will they add colors to it? You can do it yourself. Take a white cloth and pour some blue ink into it. Mummy has a pen with blue ink. After sometime, your white cloth will become blue color. That is how people add different colors into the Khadi cloth. It is called dying. After adding saffron and green color carefully into the cloth, our flag cloth is ready to use. Don’t they have to add white color into it? No No, because cloth is already white. After that they will make Chakra on the white part of the cloth. They do it by embossing. Take a coin and keep a paper on the top of it. Using your pencil keeps rubbing on the coin through the paper. After sometime, you will see a picture of your 50p coin on the paper! They do something like that using a chakra on the flag.

But one big problem with our flag is that, it has to have a definite size. If you look at the flag, it has length and it has height. You know how to measure your height, right? You can stand up against the wall and mummy will make a mark on the wall where your head touches it. And then she will take a tape and measure how much is your height. Ok what is your height now? Let me know. You can also use a scale to measure length and height. Akkuchechi must have a scale with her. Using that scale, you can measure the length of this paper. Its length will be somewhat 8 inches and height will be 11 inches. For our flag, if the length is 3 inches then its height must be 2 inches. Such flag will be small. You can also make big flags. When you grow up you will learn proportions. Then you will know when your flag is 30 inches long, what its height should be.

But now we can do some coloring of the flags. I have attached a big picture our flag along with this mail (www.girisree.com/BloggerData/Flags/indiaflagbig.gif. You must have three color pencils. Find out the proper colors. Look at the color picture of our flag. See where white comes. It comes in the middle. Where is the Green? It is on the top. No No. It is at the bottom. Now you know where Saffron is. Careful when you color Chakra. All 24 spikes should be colored properly. When you are done on one side, you have to do the other side also. I have not added picture on the other side. That you have to make yourself.

After you have colored, get a scissor from mummy. She will give it you only if you be very careful with it. Cut along the flag carefully. Now you have a India flag, colored on both sides. Isn’t pretty? Now all you need is a flag post. Flag post is long stick on which a flag can be hoisted. Get a long stick from Ammama. Mummy also might have one. One with which she threaten you to beat with! Also get some gum. Apply gum on one side of your flag and paste it on one end of the stick. Hold on it on other end. Hey now you have the flag on the flag post. You can run, holding one end of the stick. Sometimes, your flag will start fluttering like the one we have here in the apartment. You can also shout “Jai Hindh” while running with flag. That is going to be real fun!



With lots of love,

Pappa

March of penguins

{Reflection on March of penguins based on the film of the same name}

Dearest Malu,
We all live on this earth and as you have seen in pictures it is almost like a ball. It has oceans and land. This is not a small ball, but a very big one. It is almost like the beach ball you have, with both ends little flattened. At two ends of this ball are poles, North Pole and South Pole. And both places are full of ice. The South Pole is called Antarctica.

Antarctica is full of ice. In Antarctica you can see vast mountains and sheets of ice there. Most of it is actually an ocean (with lots of water). But almost all the time this ocean is frozen to ice.
Antarctica - That is where Emperor Penguin lives. There are many kinds of penguins. The penguins we saw in Monterrey Bay, when we all went to see big fishes were from South Africa. But Antarctica has one special penguin and it is Emperor penguins. They live in the ocean. They are birds, but can not fly. The small little wings that you see them having can not make them fly. But they can swim very well. They catch fishes in the ocean and eat them as and when they like. They, with friends, swim in the ocean, play in the ocean and live in the ocean.

An emperor penguin baby (chick) is born out of a big egg that mummy Penguin lays. From the egg, a small chick will come out. Mummy and Pappa feeds the baby and it grows and become real emperor penguin. The story of that egg, chick and the baby becoming a grown up penguin is a beautiful story and that is the story the Pappa is going to tell you now.

Antarctica is very cold. Very very cold. Even the big ocean, most part will become ice on the surface. It is ice every where. White ice, where ever you look. It is the coldest (you take a cup of water and pour outside, it will become ice then and there), driest (all water is ice, so you will not even get water to drink), windiest (wind blows so fast, it can beat pappa’s car even in the freeway) and darkest (sun does not come for full day here at Antarctica, it will be there for only very small time and it is night almost always) place on our earth. That is where Emperor penguin lives. Can you image how difficult it will be living there? Of course they have a kind of fur coat all over them, but even that is not enough to ward off the cold and wind.

But every year, in the month of February, when real winter season is about to start, emperor penguins will come out of sea and will start marching to a place where they can lay egg. Not one or two penguins, but many, hundreds and hundreds of them will come out of sea. The place they have to go is far away. They will not take a bus or car since there are no roads to that place. They will not take train, since there are no rail roads to that place. They can not take planes, since they do not have one. All these seven thousand penguins will have to walk to reach a place where they can lay egg. They will walk day and night, sometimes seven days or sometimes even more, to a place where they can safely lay egg and bring up their babies. The place they are going to is a vast icy place where there is no ocean. They will walk. When their little feet get tied, they walk on their belly.

The place they go is not an ocean, so they will not get any food there. So before they started walking, they will catch fishes for three months and keep all those fishes in their tummy so that they can eat them one by one when they reach the place where they can lay eggs. The place they are going is the place they themselves were born. That was the same place their papa and mummy were born and their grandpa and grandma were born. And after long long march, day and night, in the very very cold, they all, thousands of them, will reach their birth place almost same day every year.

After reaching there, after some weeks of “pennukannal”, a Pappa penguin will marry a mummy penguin and they will dance for sometime. During that time Pappa will teach mummy penguin a special song which she will memorize. And then they will wait for egg to form in mummy’s tummy and for it to come out. While they wait winter will grow harsh and huge wind will blow to destroy them. All penguins, thousands of them will huddle together to heat up their body and looking at them far, you will see a large mass of penguins huddled together and icy winds blowing over them.

Now it is May and it is dark almost all time. During all these tough times, an egg is forming in mummy penguin’s tummy and in June she will lay a big egg. She can not keep it on the ice, because it is so cold that it will freeze and break the egg. So she will keep the egg on her feet, under her fur coat. Because egg is on her feet and she must never show it, she will not even able to walk properly. But now three months are over, all the fishes mummy caught and kept in the tummy are also over. Mummy is now very hungry and she has to go to ocean to catch fish again. So she will ask Pappa penguin to take care of egg while she is away to ocean to catch fishes. The handing over the egg from mummy to Pappa is very difficult one. You can try it by passing a ball balanced on your feet to Akkuchechi to her feet. Those mummy penguins who has handed over the eggs to Pappa penguins will go back to ocean. She will walk again weeks to reach ocean. This time more ocean has become ice so they have to walk more and more to get near to the ocean. And she will be back after 3 months to Pappa penguin.

Meanwhile Pappa penguin will tend the egg balancing it on his feet, protecting against harsh winds and standing there on the icy ground. He will do this for 3 months. Without blinking eyes, without eating anything for 3 months, while winter grow violent and wind blows over them every day and night.

Days and weeks will pass. And one day sun will start coming up in the sky, ending long nights. Papa penguin has not eaten for 4 months now and he has been keeping the egg between his feet. And then one day, a crack will appear on the egg and you will see a small beak coming out of it. Slowly the egg will crack open, and there will come a little chick. Pappa penguin is so happy. But he is hungry too. Pappa is waiting for mummy to come back from sea with food for the new baby. He has kept a small food in his throat all these months to feed the baby until mummy comes home. And they together will wait.

And one day soon mummy penguin will arrive with her tummy full of fishes, krill and squids, after walking again many days from sea to this home. When mummy penguins reaches where Pappa penguins and babies live, mummy has to find the Pappa penguin. All penguins looks same: same color, same beaks, and same coats. So how can mummy find Pappa? Remember the special song Pappa taught mummy penguin before mummy went to sea for food. All Pappa penguins will be singing aloud when mummy penguins come marching to them. And each mummy penguins will find its Pappa penguins based on this song. And mummy will see her baby’s face first time he was born. And first time, family is together.

Now mummy is back, Pappa has to go back to sea for food. It is difficult for him to go away, but he has not eaten for four months and he also has to bring back food for baby. Before going away, Pappa will teach a special song to his baby. You now know why, because by the time he is back, baby will have grown up. This song will be the only way, by which he can find his baby.
Now baby penguin is having real fun. They are growing up too. He is getting stomach full food from mummy. Mummy will keep him between her feet and show him the world around. She will clean him up and when the wind blows hard protect him with her fur coat. But she will not allow him to touch the ice below her feet. After teaching him walking, keeping his tiny feet on her feet, one day baby will take his first steps on ice with mummy’s help. Then he will make friends. A lots of baby penguins will start running around and toppling about and learn many tricks under the watchful eyes of their mothers.

Mummy penguin will now have to go back to sea to gather food and eat. Now babies are old enough, mummy will leave the baby alone and will start back to sea. Now babies are alone. But before long Pappa penguin will be back with his tummy full of food. All babies will gather together and welcome their Pappas back home, all singing that song each of their Pappas taught before they went to sea. And now it will be Pappa who will feed grown-up child. After some days, Pappa will leave to sea, but child will not be alone for long because his mummy is back. This cycle will happen many times.

Sun is hot and it is summer too. Ice starts melting making sea to come closer and closer to the home. Now children are big and now they have coats like Pappa and mummy. And in September, Pappa and mummy will decide to go away. They will leave their grown-up children in the home and will go away for ever. The child penguin will grow more and grow stronger without his Pappa and mummy around. In December, the sea will reach almost near to their home.

All boys and girls will stand at the edge of sea. They have never seen ocean in their lives. They will stand at the edge of sea and they will look into the clean blue water of sea. And one day, one by one, they will take a plunge into sea. And first time they will be in their true home. Ocean.
They will live in the water for four years and after that they too will repeat the story that I have been telling you so far.
Take care,

Lots of Love
Pappa

Monsoon Rain

{ A personal memory about Monsoon Rains in Kerala}

Dearest Malu,

Malu, I hope you are fine and having fun. Mummy told me that you have cold and is going to doctor. This is rainy season and sometimes kids and even big people get cold and fever. That is ok so far as you have medicine and keeps yourself fine. Is it raining outside now? I can imagine you standing near the grills and watching the rain pouring outside. If you listen, you can hear tut-tut sound of the rain drops falling on “muutam”. They make lots circles in the puddle and it is fun to watch a naughty rain drop breaking the circles now and then. Sometimes it is sunny. And then… it becomes sloooowly dark and sun hides behind clouds. Is he playing hide and seek with his friends in the sky? Maybe. You could feel no sound for sometime and can see dark clouds filling the sky. And suddenly it starts raining again. Take a deep breath and you can feel nice smell when the rain falls on the grass.

It is fun playing in the rain. You can slap your feel in the puddle and can hear plup-plup sound. Extend you palm into the falling rain from the veranda and drops will do chuk-chuk on your little palm. It is cold also. But be careful, the muttam is wet and you might slip over. Also make sure that mummy dries you out well, otherwise you can get cold. You can see water flowing slowly in the muttam and if you go outside the gate, you can see water running fast in the drainage. You can sit inside and can even make a paper boat. I have attached a note on how to make a paper boat along with this mail. It is easy and fun. Mummy will help you. Then you can keep it in the water outside and can see your little boat move up and down in the water. And if you keep the boat in the drainage outside, you will see it going faster. Do not try to get it back, we can make another one.

Sometimes you think rain is really boring. Always raining and raining. Have you wondered why we need rain? Look outside, you can see the Guava tree full of green leaves and muttam is getting covered with grass. They are so because of water rain gives. Without rain, we will not have rice, and vegetables. You might think then that we can eat chicken biriyani all day. But even chicken needs water and you drink water too. That water comes from rain. But raining too much also causes problems. Water will come everywhere, sometimes even come inside houses. It is called flood, but we are ok since Ammama’s house is good and water will not come inside.

If you listen again, you can almost hear rain singing to you. It sings almost like your dance teacher saying tha-thei-thithathei. Some people think music can make rain. There was singer who lived long back called Thansen. He was living with a big king and one time there was no rain in his place. So he sang and sang a particular song and that made clouds so dark. He kept on signing and then sky opened up and rained.

But really, from where does this rain come? Ya! You know that it comes from sky. But from where sky gets that water? Everybody gets water from the sea. There is a big ocean near Ammama’s house. You ask mummy to take you there. You can go to baby beach and can see Sea. It is called Arabian Sea. It is big pot of water. It so big, you can’t see the other side of it. It is so big and you can see waves hitting against the land. It is fun to watch. Watch Ammama making “chooru”. She takes water in a “chembu” and keeps it on the stove. You will see that pot is almost full when she keeps it on stove. Sea is like that pot with water. And then sun heats up sea and water from sea goes up in the sky. If you watch Ammama’s rice pot, after stove is turned on, you can see this yourself. See that whitish smoke coming out of the pot. Where are they going? They are going UP. It is not smoke, it is vapor. Sun does the same thing with water in the sea and water becomes vapor and goes up and up in the sky. And they become clouds. Clouds with a lot water vapor will be dark in color. Pappa does not why. When the rice is on the stove, you take a clean lid and show it over the pot. See the inside the lid, what do you see? Water drops!!. See how you moved water from the pot to the lid without touching water in the pot!. Big big hills and big winds are like the lid in your hand. When big wind from sea moves these clouds in the sky, they become cool and they become water again. And that is what rain is. Big people call this, Water cycle and some other big people call this as Hydrologic cycle.

After the rain is stopped, you can go outside and see. Most of the time you will not see much water in the muttam. It rained all day and where does the water go? Some of it goes under ground and trees and plants drink it from there. Sometimes we dig a big hole in the ground to fetch water. This is called Wells. You can see a well in Achamma’s house.

The rain that you get now is because of Monsoon winds blowing from sea nearby. It will rain until September. So in Kannur now is Monsoon season, where as for Pappa in US it is summer. Funny isn’t it? The Monsoon is an Arabic word. As you know Arabic is a language they speak in Dubai, where Akkuchechi is. We speak Malayalam in Kannur and this letter is written in English. The Monsoon rains are real fun, you see sun outside and thinking that it won't rain, you dress up. You are about to get out of the house, suddenly it will start raining. Sometimes there will be a big wind - storm - that comes with rain, you will see big coconut trees almost dancing. Sometimes there is lightning- that big flash like somebody is taking a picture of you- and thunder too. That is why climate scientists - people studying monsoon- call it whimsical, unpredictable.

There are so many songs about rain. If you go to internet, you will get many many songs there. You already know “Rain Rain go away” song. Pappa really don’t like that song. Remember the old man who was snoring when raining and bumping his head on the top of the head song. Mummy knows it and she will sing it for you. Here is a funny song for you from US

Doctor Foster went to Gloucester
In a shower of rain,
He stepped in a puddle,
Right up to his middle,
And never went there again.

You can read Gloucester as “gloster”. You see it rhymes with Foster.

Pappa got another song about a mummy and a girl going to school in rain. This is a Japanese song. Japan is a country like India and US, but it is in Pacific Ocean. Read this aloud and it is real fun

Rainy day, rainy day, I like it;
My mother will come here with my umbrella,
Pitch pitch, chap chap, run run run!

Bag on my shoulder, I follow my mother;
A bell is ringing somewhere,
Pitch pitch, chap chap, run run run!

Oh oh, that girl is dripping wet;
She is crying under the willow,
Pitch pitch, chap chap, run run run!

Mother, mother, I’ll lend her my umbrella;
“Hi girl, use this umbrella,”
Pitch pitch, chap chap, run run run!

I am all right, don’t worry,
Mother will take me in under her big umbrella,
Pitch pitch, chap chap, run run run!


Hey! “Pitch pitch, chap chap, run run run” is the sound of rain!

With lots of love
Pappa