Mathemagic

by Rohini Chowdhury

  The children were so excited that they did not notice that Mehul had suddenly and quietly become his usual self again. They danced around the room, listing out all they could measure.

  ‘Very good!’ said the Wizard. ‘Now tell me…’

  ‘Tell you what?’ asked the twins, pausing in their dance around the room.

  ‘Tell me…’ said the Wizard, taking a deep breath. ‘WHAT is a number?’

  5

  Numbers Everywhere!

  Megha sighed. ‘We’ve told you—we use numbers to count, to put objects in sequence, and now we know, to measure things Isn’t that enough?’

  ‘That is certainly a beginning,’ replied the Wizard, ‘but it does not answer my question completely. Think harder, children! Come on—there are two of you, and two clever heads are better than my one old head!’

  ‘Well, numbers are not something we can eat, or touch, or smell, or hear, or see,’ said Mehul slowly.

  ‘So does that mean that numbers do not exist?’ asked Megha, frowning.

  ‘That makes no sense!’ said Mehul. ‘How can we use numbers to do all those things you told us about if they didn’t exist?’

  ‘Oh, numbers exist, they most certainly exist!’ replied the Wizard. ‘They exist as much as you and I! They tell us about the world we live in! Remember—how tall, how short, how far, how near, how heavy, how light, how many, how much! Numbers are everywhere!’

  The children flopped down on the floor. This was so confusing. How could something exist if one could not touch it, or feel it, or sense it in any way? And yet, Sir Tzyphyr was right, numbers were everywhere.

  The Wizard sighed, and sat down cross-legged on the floor next to the children. ‘This sounds like magic, does it not?’ he smiled. ‘This is the wonder of numbers. We may not be able to see them with our eyes, or hold them in our hands, or even understand what they really are. But believe me children, numbers do exist. They exist on their own, they do not need us.’

  ‘They do not need us? How can that be?’ asked Mehul. ‘Our teacher told us that human beings invented numbers!’

  The Wizard looked shocked. ‘No! No one invented numbers! Not you human beings, not us Mathemagicians! Not even your teacher!’ he declared.

  ‘Then who?’ asked Megha. ‘Numbers can’t just exist! Someone must have made them up!’

  ‘No, child,’ replied the Wizard, suddenly looking very old. ‘No one made them up. Numbers existed before any of us were born! Numbers are at least as old as this universe, or maybe even older. They have always existed, they have always been there. And they will still be there when you and I have grown old and died.’

  The children thought over this in silence. It was hard to imagine a world before they were born, and harder to imagine one when they were grown-up, and old, and dead!

  ‘Oh, I can’t think any more!’ groaned Mehul, holding his head. ‘I’m so tired, it’s late, and I feel as if I have only half a brain right now,’ he said. ‘Can we please go to bed now? I am anyway half-asleep!’

  ‘Half a brain! Half-asleep! How carelessly you use words!’ grumbled the Wizard. ‘In mathematics, we must be exact! Megha! Can you tell me—what is the meaning of the word half?’

  Megha sighed. ‘Half is when you share something between two people,’ she answered sleepily. ‘Please, Sir Tzyphyr, can we stop now?’ she added, yawning. ‘We’re so tired!’

  ‘Tired? Tired of numbers? That is impossible!’ exclaimed Sir Tzyphyr astonished. ‘All you need is some energy,’ he declared. Ignoring the twins’ sad faces, he dug into his pocket and pulled out a large bar of chocolate, wrapped in shiny golden paper. ‘Here we are,’ he said. ‘Chocolate for the two of you!’

  6

  Share and Share Alike

  The children sat up, interested.

  ‘Not magic chocolate, I hope!’ said Mehul.

  ‘No, don’t worry,’ smiled the Wizard. ‘Ordinary chocolate! No side effects! It’s good for tiredness!’ He broke the bar into two and handed the larger piece to Mehul, the smaller to Megha. ‘Here you are—half each!’

  ‘That’s not half!’ protested Megha.

  ‘Of course it is,’ said the Wizard. ‘You told me so yourself, half is when you share something between two people. And that is exactly what I have done!’

  ‘No, no,’ answered Megha patiently. ‘You have to share it the same! Only then would it be half!’

  ‘Half is when you share something EQUALLY between two people,’ explained Mehul, who was feeling a little guilty at having been given the larger piece.

  ‘Aha! Now we are talking!’ exclaimed the Wizard, pleased. He waved his hands in the air, and to the children’s surprise, the pieces of chocolate flew out of their hands and joined together into one large bar again.

  This time the Wizard broke the bar into two equal pieces. ‘Here we go, half a chocolate bar for each of you!’ he said beaming. ‘NOW do you see why it is important to be exact in everything you say and do, especially in mathematics?’

  The children nodded, happily munching their chocolate, which was delicious.

  But the Wizard was not done with them, not yet. ‘So, let’s see, what else can be divided into half?’ he asked. ‘Half a kilometre? Half a bag of sugar?’

  ‘Half a cake, half a loaf of bread!’ cried Mehul.

  ‘Half an hour! Half my pocket money!’ added Megha.

  ‘Now then, everything that we count or measure is not the same—we can divide some things into halves, but not others,’ said Sir Tzyphyr, holding up a hand. ‘For instance, we can’t have half a cow, or half a cat!’

  ‘Just as we can’t have half a girl,’ continued Megha.

  ‘Or half a boy!’ added Mehul.

  ‘And definitely not half a Wizard!’ declared Sir Tzyphyr, with a sudden twinkle in his eye.

  The children giggled and agreed. No, definitely not half a Wizard, they didn’t want that!

  ‘But we CAN have half an apple!’ said the Wizard. He dug into his pockets again, and pulled out a shiny red apple from one, and a small, sharp knife from the other. ‘Here we go, half each for the two of you,’ he said, deftly cutting the apple into two equal pieces.

  The twins looked at each other and knew they had the same thought.

  Mehul spoke for both of them. ‘No, Sir Tzyphyr,’ he said. ‘We can’t eat this apple, not unless you also have some!’

  ‘Oh, don’t worry about me, I have plenty of food in my pockets,’ said the Wizard. ‘Go ahead and share the apple between the two of you.’

  ‘Only if you share it with us,’ declared Megha.

  ‘Oh very well,’ sighed the Wizard, pretending to be tired of the argument, but secretly very pleased. He put the two halves of the apple together, and just as the pieces of chocolate had fused into one large bar again, the apple too became whole. This time the Wizard cut it into three equal pieces. ‘Here we are, one-third for each of us,’ he said. ‘Of course,’ he added apologetically as he handed them their pieces, ‘a third is smaller than a half, as I am sure you understand.’

  ‘Yes, of course,’ said Megha. ‘A third has to be smaller than a half—the apple is now being shared between three of us, not two of us. So because there are more people eating the same apple, each one will get less. That’s logical.’ And she bit into her piece with a satisfied crunch.

  ‘Not that we mind at all, Sir Tzyphyr!’ said Mehul quickly, just in case the Wizard thought they were unhappy about the smaller pieces.

  The Wizard nodded again, this time not bothering to hide that he was pleased. The three munched in silence till the apple had vanished into their tummies.

  7

  Parts and the Whole

  ‘Oooh, I’m stuffed!’ said Megha, rubbing her tummy happily.

  ‘Yes, a fraction can be quite satisfying,’ said Sir Tzyphyr, finishing off his third of the apple, and sighing happily.

  ‘A fraction? What is that?’ asked Megha, puzzled.

  ‘Something to do with numbers, I a
m sure!’ guessed Mehul.

  The Wizard nodded. ‘Yes, a fraction is a special type of number. It is a part of a whole. Like half, or a third. So just as we can say we have one of something, we can also say we have a fraction of something—like half a chocolate, or a third of an apple.’

  ‘And what if we divide something into four equal parts? What do we have then?’ asked Megha.

  ‘A fourth!’ declared Mehul. ‘Isn’t that right?’

  ‘Yes,’ nodded the Wizard. ‘And the greater the number of parts we divide something into, the smaller each part becomes. Get it?’

  ‘Yes,’ nodded the twins. That was easy to understand. After all, if one apple was divided equally between five children, each would get a smaller share than if it had been divided between four children. The children didn’t need the Wizard to tell them that! It was just common sense!

  ‘Do you notice another fact about fractions?’ asked the Wizard, putting on his serious voice.

  ‘Ummm … they are not very big?’ suggested Mehul, who was still a tiny bit hungry despite the chocolate and the apple.

  ‘That’s right,’ agreed the Wizard. ‘Fractions, proper fractions that is, are always less than one, because they are not the whole, only parts of a whole.’

  Megha frowned, trying to understand. ‘So, if we put the two halves of the apple together, we get a whole apple,’ she said.

  ‘And if we had put the three thirds together, instead of eating them, we would have got a whole apple again!’ said Mehul.

  ‘Correct,’ said the Wizard. ‘But what if I had put a half and a third together? What do you think we would have got then? Do you think we would have got the whole apple back?’

  The children were silent, thinking hard. ‘No,’ said Mehul at last. ‘A third of an apple and half an apple would not make a full apple.’

  ‘And why is that?’ asked the Wizard.

  ‘Because,’ explained Megha slowly, ‘a third is less than a half, and so, if we put a third together with a half, there would still be a part of the apple missing!’

  ‘That is absolutely right!’ exclaimed the Wizard, delighted. He gave a sudden little hop of happiness, and to the delight and surprise of the children, broke into a dance. He tapped his feet and twirled on his toes, and as he tapped and twirled he sang a little song:

  Numbers, numbers everywhere!

  How many stars in the dark night sky?

  How many apples on an apple tree?

  How many coins in a rich man’s purse?

  One, two, three!

  Who comes first, the father or the child?

  What came last, the egg or the bird?

  What comes next, the night or the morn?

  Tell me, tell me—first, second, third!

  How far to the sun? How far to the moon?

  How tall? How short? How big? How small?

  How heavy? How light? How cold? How warm?

  Tell me now, don’t take more time!

  The children laughed and joined in the dance, making up the words as they went along.

  Numbers, numbers everywhere!

  Numbers for counting,

  Putting things in order

  Numbers used to measure

  Ali Baba’s treasure!

  We can’t eat numbers, we can’t touch numbers

  We can’t see numbers, we can’t smell numbers

  We can’t hear numbers—so don’t even try!

  So what are numbers? THINK, THINK, THINK!

  A whole, a half, a third, a fourth

  Less than one, but numbers still!

  Oh, numbers are magical, numbers are fun

  Numbers are logical, numbers are … AWESOME!

  The children collapsed laughing on to the floor.

  8

  Nothing and Goodbye!

  ‘Humph,’ said the Wizard, suddenly serious again. ‘Numbers are no laughing matter,’ he declared. ‘There are more numbers in the universe than we can imagine, there are big numbers, and very big numbers, and very, very big numbers! There are small numbers and very small numbers, and teeny tiny numbers.’ The Wizard paused, and took a deep breath.

  The children were listening intently, their eyes big and round. Were there really so many different kinds of numbers in the world?

  ‘Yes, there are, and many other kinds too!’ replied the Wizard. ‘Of course, ruling over them all is zero, the king of all numbers!’

  ‘Zero? That means NOTHING!’ declared Mehul. ‘How can zero be a number?’

  ‘You just told us that we use numbers to count, and we use numbers to measure,’ said Megha. ‘How can we count or measure nothing?’

  ‘Ah, but we can, and sometimes we must!’ answered the Wizard.

  ‘How? Why?’ asked the twins.

  ‘How many sweets?’ asked the Wizard, pulling out a toffee wrapped in pink paper from the air.

  ‘One, of course,’ said the twins together.

  The Wizard carefully unwrapped the toffee and popped it into his mouth. ‘And now? How many sweets?’

  ‘Zero!’ said the twins.

  ‘So there you go! That’s one reason why we need zero—to understand what happens when we have one of something, and we take that away!’

  ‘We can say we have “nothing” left,’ protested Megha.

  ‘And “nothing” is one meaning of “zero”, is it not?’ replied the Wizard.

  The children nodded, unconvinced.

  ‘It’s almost morning,’ said the Wizard, glancing at the brightening sky through the window. ‘I must go soon, for I cannot stay beyond the dawn. But first, let me tell you a story.’ And he hopped on to the bookshelf again.

  The children yawned, suddenly sleepy again. They climbed into bed, and snuggled into their pillows. But they had no plans of sleeping, not yet! They both loved stories and wouldn’t miss the Wizard’s tale for anything.

  ‘Long ago,’ began the Wizard, ‘even though we knew about numbers, and knew that they existed, we did not know about zero. We knew that “nothing” existed, but we didn’t really understand it fully.’

  ‘Did you say “nothing” existed?’ interrupted Mehul. ‘How can “nothing” exist? It is nothing. It is not there!’

  ‘Ah! But by not being there, nothing becomes something, and we need ways to talk about it,’ explained the Wizard. ‘When I ate up the toffee, from being there, it was not there—and it was necessary that we be able to describe that, was it not?’

  The children nodded. Yes, the Wizard was making sense. Almost.

  The Wizard smiled, and continued with his story, a faraway look on his face. ‘The ancients realized that even though they did not understand what “nothing” was, it was important. They tried ways and means to learn more about it, and used poetry and stories to describe it.’

  ‘How did they do that?’ asked Megha sleepily.

  The Wizard was silent for a moment, lost in memories of a time so long ago that he had lost count of the number of years that had passed. ‘They looked up at the sky, and wondered about the vast darkness of space, and they called it “shunya”—which means “empty”,’ he continued softly. ‘Slowly they began to understand what “nothing” meant, that even though it was nothing, it was empty, it was shunya, it existed. Slowly they began to understand that it was not just emptiness. They thought of it as similar to the sky—which is empty, but which is still there.’

  The children were still listening, though half asleep. The Wizard continued his story. ‘Slowly, people all over the world began to understand the importance of “nothing”. Each nation gave it its own name. The Greeks called it “ouden”, the Romans called it “vacuus”, the Arabs called it “sifr”—from where my own name “Tzyphyr” has come…’

  Sir Tzyphyr paused. The children had fallen fast asleep.

  The Wizard smiled to himself. He had made a good beginning tonight—there were two more minds in the universe now, thinking about numbers. And numbers, as he knew, were the best and the most important things t
o think about.

  Sir Tzyphr pulled out his magic pencil, and waved it over the sleeping children, filling their dreams with numbers. As the first rays of the rising sun came in through the window, the old Mathemagician whispered quietly, ‘Sleep well, children! We shall continue our story another night!’ Then he disappeared in a puff of silver smoke.

  Acknowledgements

  I would like to thank Professor A.N. Maheshwari for his encouragement and invaluable guidance during the writing of this book. I am deeply grateful to him for his time, his patience and for the mathematical insights and understanding that he provided.

  My gratitude also to Dr Asha Maheshwari for her help in conceptualizing this book, and to my daughters Vipasha and Vidisha for their comments and suggestions.

  Finally, a big thank you to Sudeshna Shome Ghosh who helped me develop the idea for this series, and to Mimi Basu who made this book her own and saw it to completion.

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  This collection published 2012

  Copyright © Rohini Chowdhury, 2012

  The moral right of the author has been asserted

  ISBN: 978-0-143-33206-0

  This digital edition published in 2012.

  e-ISBN: 978-8-184-75685-2

  This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.