Posts from the ‘Philosophy of Math’ Category

Discussion on π

Discussion on π

Following is a continuation from a discussion between two high school students. Gopal is the one who is explaining about Pi and Tejas is the opponent who’s trying to cross-question and knock down Gopal.

TEJAS: Why did they (mathematicians) consider Pi as a ratio between circumference and the diameter? Why not circumference and the radius? Even circumference/radius would be a constant which gives the approximate value as 6.2832…

GOPAL: For any given circle or sphere where the center is not given, how would you find the radius? Without knowing the center, we cannot directly get the radius. But even if we do not know the center, we can calculate the diameter through many ways. Hence they took the ratio as c/d.

TEJAS: But area of a circle is πr^2 and not in terms of diameter?

GOPAL: We can have a formula with diameter which will come to π (d^2)/4. But that’ll make the formula more complicated. πr^2 is easier to remember and faster to calculate.

TEJAS: You said that the diameter of a circle can be calculated even when the center is not known. Can you explain, how?

GOPAL: Trace the circle into a paper and fold it into half…you the diameter.

TEJAS: What if tracing is not possible or the paper cannot be folded?

GOPAL: Draw a tangent to the circle. At the point of intersection, draw a perpendicular to the tangent. The chord hence formed will be the diameter of the circle.

TEJAS: What if it’s a sphere? How would you trace that?

GOPAL: Good question. If the sphere is small, use a screw gauge to measure it’s diameter. If the sphere is big, keep four blocks on its four sides such that it becomes a circle inscribed in a square. Now measure the length of the side of the square.

TEJAS: What if the sphere is so big that you can’t do any of these?

GOPAL: In that case, we’ll have to invent any new measuring equipment for measuring that huge sphere. Since that is an impractical solution, we need to think of some other way. When man had to deal with such calculations, he felt the need to devise a formula. He observed that all circles are similar and all spheres are similar. Hence, if the ratio between circumference & diameter of a circle is constant, the ratio between all circumference & diameter of all circles & spheres has to be a constant. That constant is π.

TEJAS: When the value of π is non-recurring and unending, how can say that its value is constant? It decimal places goes on and on and on.

GOPAL: What are numbers? Are 1, 2, 3, 4.7, ¼, etc numbers? They are ‘numbers’ expressed in digits. ‘4’ is a symbol that represents the number which has the value 4. Think! The number 4 can be expressed as IV or FOUR or 8/2, etc. These are different ways of expressing four. Four can also be geometrically represented by showing four shapes or the fourth stroke from the Origin in a number line. We can also write an equation x + y = 4 which will represent 4 algebraically. So, numbers, arithmetic, algebra, geometry, etc are different languages in mathematics. And languages have got their own challenges. There are some words or sounds in a particular language which can only be expressed correctly in that very language. No other language can exactly express that exact word meaning or sound, and mathematical languages are no exceptions. That is why you can exactly draw √2 or √3 on a number line but you cannot write the value with a finite number of decimal places. The reason is that the language of geometry is conveys the idea of √2 or √3 rather than numbers. In the case of irrational numbers, we might not be able to write its correct value up to the last decimal point but we can certainly plot them correctly on a number line.

TEJAS: Talking about irrational numbers, how can you say that π is not rational? Are there any proofs for that?

GOPAL: There are many proofs that prove that π is not rational but most of them are not very simple. One simple proof is that – a rational number can be expressed in the form p/q where p & q are integers. But if you look at circumference and diameter, if the circumference of a circle is an integer, the diameter isn’t. If the diameter is an integer, then the circumference is not. Then how can you say that π is rational?

TEJAS: Fair enough. But just because π is not rational, it does not mean it is irrational? There are numbers like Complex numbers which comprises of an imaginary number (√-1). Can’t π be a complex number and not an irrational number? And why does π become irrational when it is the ratio between two rational numbers?

GOPAL: Suppose the length of the circumference is √2, then the ratio of circumference and diameter would be between two irrational numbers. So, your second argument is not correct. π is a constant number which is expressed as a ratio. A ratio need not be ‘only’ of integers. √2:1 is a ratio.

The dialogue continues…

The above is a real discussion that had happened in one of our classes. As you can clearly see, the result of this discussion brought out many concepts about numbers and made the students inquire and investigate more.

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STORY OF NUMBERS

Once upon a time, when man had not discovered mathematics, there lived an emperor by the name Guna. He was a very wise ruler. One night, a space ship landed in one of the jungles in his land and out came a huge army of numbers. These numbers were of different types. Very soon, they spread all over the earth. Since nobody knew the nature of these numbers, everyone was scared of them except Guna. He decided to study them and bring them under his control. After many years of struggling with the numbers, he captured them and made them his slaves. But a few years later, there was a prison break. Somebody set free all the numbers. The task of bringing back all the numbers was given to the Intelligence Bureau (IB) of the kingdom. Following is the story of what happened after that.

IB: Your Excellency, please tell us how do we identify all the numbers? They seem to disguise themselves in many forms.

Guna: Not to worry. If you understand numbers, then you can bring them under your control very easily. First go for the numbers that we use for counting. These numbers are not very crooked or shrewd. Those poor fellows are very natural and can be naturally found everywhere. Go and look for their leader – Number 1. When you see two 1s together, they are called as Number 2. Capture both of them. In the same way, you will see that these natural numbers might be written using different numerals. E.g. The number 23. It is nothing but a combination of 1s twenty three times. So, ultimately all natural numbers are nothing but an addition of its parent number – Number 1.

IB: Ok Sire.

(IB goes and captures the group of Natural numbers)

IB: Sir, we have successfully captured all the natural numbers. Who should we capture next?

Guna: These natural numbers had formed a new group with another leader and called themselves as Whole numbers. Now that you have captured the natural numbers, you just need to find their leader. He is a very tricky fellow. He is neither a negative person nor a positive one. To tell you honestly, he doesn’t even exist. But when he comes after any number, the number’s value and strength becomes 10 times. His name is zero.

IB (comes back after a few days): Sir, we have managed to take zero into our custody. As you said, it was difficult to find somebody who doesn’t exist when he’s alone, but still gives power to others when combined with them.

Guna: Where is he? Bring him to me?

IB: Here Sir. He’s standing right in front of you in these chains.

Guna: Oh! I almost forgot. He cannot be seen, right? Since I can’t see him, how do I believe that he has been captured?

IB: We have a solution Sir. We have brought a number 8 along with us. Now we shall put 0 after 8 and you can see for yourself how powerful 8 becomes.

(IB puts 0 after 8 and 8 becomes ten times powerful)

Guna: Oh yes, yes. I can feel the power of zero. Take him away. He scares me sometimes. Good that you did not divide 8 by zero. Or else, you would have seen the power going to infinity.

IB: Now who’s next my Lord?

Guna: Next you can target Integers. You see, these positive numbers and zero got into the company of many negative elements. They looked like natural numbers but had a negative effect all around them. Together, the positive natural numbers and the negative numbers formed a new group or a new set called Integers. They gave themselves a secret symbol ‘Z’. There’s an easy way to capture them. You can go in search of negative 1. He looks like this ‘-1’. Once you capture him, you multiply -1 one by one with all the natural numbers.

IB: Sir, what do you mean by ‘multiply’?

Guna: Multiply means to add repeatedly. When you repeatedly add -1 to any number, let us say 2; then what happens is 2 x (-1). So you end up repeatedly adding a negative number to a positive number. When you repeatedly add negative qualities, it means, your positive qualities gets reduced. Suppose, you are in the company of negative people for a long time, you end up becoming a negative person because you keep adding negativities into you.

IB: Oh! That’s awful.

Guna: Yes. So, once you capture -1, you multiply it by all the natural numbers. This will lead you to the negative set of numbers who helped the whole numbers to form the group of integers.

(IB comes back after a few days with the set of negative integers)

IB: Sir, somehow we captured all the negative numbers. We had to use a negative approach to capture them. But we managed to get all of them. Are there more numbers?

Guna: Yes, there are. You have now captured the easy ones. These integers had formed a bigger group called Rational numbers by combining an integer with another integer in the form of a fraction. Now that you have taken care of all the integers, you are left with the numbers which are not integers, but rational numbers.

IB: Sir, any specific reason why they are called Rational?

Guna: Yes. As their name suggests, they have a very rational and logical thinking. They combine amongst themselves in the form of a fraction like p/q where both p & q are integers but q will not be zero. When they discovered that with whatever numbers they have they can form portion of those numbers, they started dividing themselves into fractions. All rational numbers can be expressed in the form of fractions. So, once you see any fraction, i.e., any integer divided by another non-zero integer, please bear in mind that it will be a rational number. But be careful, these numbers can also disguise themselves in the form of decimals.

IB: How do we identify decimal numbers?

Guna: Very simple. They will have a ‘dot’ between any two numbers. But they are very clever. They change their original form completely when they change form from fractions to decimals. E.g. the fractional number ¾ when converted to decimals become 0.75. The 3 & 4 are gone and they are replaced by a decimal (dot), 7 and 5.

Some numbers have recurring numbers after the decimal point, like 4/3. It’s value in decimal is 1.111111…..It goes unending.

IB: Unending? How can that be? How can something go unending?

Guna: It is a small error that happens. You must have seen audio CDs which lose track sometimes and keep repeating the same song again and again. The same thing happens with some rational numbers. They keep repeating the same digits after the decimal point and they can be seen recurring. Such recurring decimals are also examples of rational numbers.
So, you have to capture the integers that appear in the form of fractions, the decimal numbers which do not recur (i.e., terminating decimals) and the numbers where digits keep recurring after the decimal point. Once you have captured them, then total of all the numbers you have captured till now, including natural, whole and integer numbers, will form the group of rational numbers. Wish you good luck!

IB: Thank you Sir.

(IB comes back after a month)

IB: My Lord, this was the most challenging task. All the numbers kept changing their forms when we went near them. But the way you defined these numbers made us understand their nature and helped us take them into custody.

Guna: Good. See, there is a big difference in just following orders and following orders after understanding the nature of the enemy very well.

However, your task isn’t over. There exists another group of numbers which are called Irrational numbers. They do not have rationale. They seem to be very illogical. Some of them disguise themselves as positive natural numbers with a hat which they call as ‘root’. When you ask them their value, you will see that they will express themselves in decimal form. But unlike, rational numbers which had recurring or terminating decimals, irrational numbers will have non-terminating numbers after the decimal place.

IB: Non-terminating? Non-recurring? Again unending? No wonder they are called ‘Irrational’!

Guna: True. Waste no time. Proceed immediately.

IB: Sir, but how far should we go checking if their decimals are terminating or recurring? What if they become terminating after many numbers?

Guna: Take this instrument with you. It is called a number line. Even though it might look small now, you can extend it to both sides as much as you want. The middle number is 0. To its left side are negative numbers, and to its right are positive numbers. As you go on extending, you will see that every rational or irrational number will have a position, a place, in this number line. Can you see a 3 here? Extend a bit, you will see 3.1. Extend it further, you will see 3.11. Now, divide the portion between 3 and 4 into three equal parts.
The first cut among the three divisions that appears between 3 and 4 is nothing but the rational number 10/3.

Whenever you come across any number, try to see if you can find a place for it in the number line. If you can present these numbers geometrically in the number line, it is like you are finding the exact location, exact value of the numbers. Once you have found out the exact location, go in with your search team and arrest the number.

(IB returns after 3 months)

IB: We have managed to capture almost all the irrational numbers. Like you said, when we came across any number with a hat called ‘root’ we tried to figure out if the number can be presented on the number line. This save a lot of our time or else, we would have been still calculating to see if the numbers after decimal point is recurring or non-recurring, terminating or non-terminating.

Guna: Hmm. But why did you say, ‘Almost’? Couldn’t you capture all the irrational numbers?

IB: Sir, we heard that there is a number called Pi. He lives in geometrical shapes which have got circular sides. We searched every ‘nook and corner’, I mean, we went round and round in circular shapes but still couldn’t find Mr.Pi. We searched every single circle, sphere, cylinder, cone and similar shapes but couldn’t find him.

Some gave his address as 22/7. When our agents figured out the value of 22/7, they found out that it is a rational number which gives the value 3.142857142857142857….and thus 142857 goes recurring. To cross-check, we went to the prison and checked the records and found an imposter of Pi whom most of the people in the world considered as Pi, a very similar look-alike, was already in our custody. He laughed at us when we asked him if he was Pi. He said, “Since I almost look like Pi, many people mistake me for Pi. They think Pi is 22/7. But Pi is only approximately equal to my value. You can never find him if you go searching like this. Nobody has ever found him even though they speak of him quite often, not even on the number line. Ha ha ha….”

Mr.22/7 was correct. We tried locating Pi on the number line but couldn’t find him. Sir, we are really sorry to have failed in capturing this one number.

Guna: Ha ha ha…don’t be depressed. Tell me, what is the definition of Pi that you came across?

IB: Pi is the ratio of the circumference of a circle to its diameter.

Guna: Is it mentioned anywhere that it is a number?

IB (surprised look): Er, no.

Guna: Say the definition again?

IB: Pi is the ratio

Oh yes, Pi is a ‘ratio’. It is not a number!! But the whole world seems to think of it as a number? Why?

Guna (Smiling): Pi is a ratio. A ratio can be between any two things, not necessarily numbers. And ratio is a concept. A concept can be without a form. But inorder for us to understand, we give it a form, a value. Thus we gave Pi an approximate value of 22/7. However, with frequent use, many people tend to forget that the value is only approximate and not exactly equal to 22/7. This happens because we are trying to find the nearest value of a relationship between two things – the circumference and the diameter.

Since Pi is very frequently used, it almost got the credibility of being a number. And they called him a Transcendental number. Transcend – something which is ‘beyond’. With the tools of numbers, expression of Pi is beyond the capacity. That is why you couldn’t find him.

Anyways, you have been working very hard to recapture all these numbers. You have successfully captured all Real numbers. There exists some numbers which are Unreal or Imaginary. But we will take care of them after some time. Now you may take some rest, spend some time with your families and tell the story of numbers to your children.

FUNCTIONS

What is a ‘Function’ in Mathematics? What is it that we mean when we write ‘function of x’ as f(x)?


Suppose a teacher is teaching a subject to five students. Even though the subject is the same, each person will value it differently according to the way they think, in other words, according to the way their brain functions. In the same way, if we take the subject as ‘x’ and it goes into five students – a, b, c, d and e, the way each of them function will be different because their brain functions differently. The way ‘a’ functions can be called as a(x), for ‘b’ we can call b(x) and so on.
Imagine a stock broker, Mr. f, who tells his customer, “Whatever money you give me to invest, I’ll return you Rs.100 less than thrice the amount”. For this statement, mathematically we write: f(x) = 3x – 100, where x is ‘whatever money is given’. Another stock broker, Mr. g, tells his customer, “I’ll return you double the money what you give me plus Rs.100 more.” This function can be written as g(x) = 2x + 100. Now, assume that you are the customer. You start thinking, “Who can give me more money?” If you give Rs.100 to Mr. f, he will give you according to his function, i.e., 3x – 100 = 3 x 100 – 100 = Rs.200. If you give the same amount of Rs. 100 to Mr. g, he will give you 2 x 100 + 100 = Rs. 300. So, depending on the different functions, the results differ for the same input ‘x’, in this case Rs.100.

Function of x means, how does x act in a function with the name f. This is what f(x) means. g(x) means, how x acts in another function with the name g. Basically, the mathematical word function has the meaning of the ordinary word function – ‘the way something works’.

To represent different functions (as we saw in this case of representation of different people functioning differently), we use different letters and write as f(x), g(x), h(x), etc.

The Reality in Irrational Numbers

Teacher: What is an irrational number?

Arya: A number which is not Rational.

Teacher: Good. In that case, how would you define Irrational numbers?

Arya: (Thinking) Okay. A rational number is a number that can be written in the form p/q where p & q are integers and q is not equal to 0. Hence, an irrational number should be a number which cannot be written in the form p/q where p & q are integers and q is not equal to 0.

Teacher: Smart! Can you give me examples of irrational numbers?

Students: √2, √3, √7, Pi, etc.

Teacher: Why are they irrational?

Students: Because when we convert them into decimal form, the decimals go unending and non-recurring.

Teacher: So, can we say, irrational numbers are those which have unending and non-recurring decimals?

Students: Yes, we can.

Teacher: But why do we get unending decimals when we take the square root or some numbers?

Students: (Wondering)

Teacher: Let’s analyze. What is Multiplication?

Student: (Surprised) What a question is that? Multiplication means multiplication.

Teacher: Why do we say 3 times 2 is 6?

Students: Because when we add three 2’s we get six.

Teacher: In other words, when 2 is added thrice, we get 6. So, how would you define ‘Multiplication’?

Akhila: Multiplication is nothing but addition.

Teacher: Good. Multiplication is repetitive addition. See, this is how we arrive at definitions.

Students: (Smiling)

Teacher: Now, let us see how the square of a number is obtained. 3 times 3 is 9. That means, when we add 3, three times, we get 9. Since 3 is an integer, we get √9 as an integer. Suppose we square 1.5, i.e. multiply 1.5 by 1.5. We get 2.25. Let us see if we get the same answer by adding the number 1.5, 1.5 times. We have to add 1.5 times, which means we have to add half (0.5) of the number 1.5 to 1.5. Half of 1.5 is 0.75. When we add 0.75 to 1.5, we get 2.25. Now, we can confidently say that repetitive addition of a number to that many number of times (as the value of the number) will result in the square of the number. E.g. Square of 3.6 can be got by adding 3.6 repeatedly upto 3.6 number of times.
So, if we take any number, it will be the result of some number (say ‘x’) multiplied by the same number (‘x’). Hence, the number 2 is the result of a number multiplied by the same number. So, there exists a number which is the square root of 2, i.e. √2 is a Real number, because it exists.
Now the challenge we face is, expressing it correctly in decimals. Why are we unable to do that? Suppose we have a thread of 1 cm length and we cut it into 3 equal parts. What is the length of each part?

Srinivas: 1/3 cms

Leela: 0.3333333… cms

Teacher: Okay. Leela, now add all the lengths of all the three pieces. What do you get?

Leela: 0.9999999…. cms

Teacher: Why aren’t you getting the length of the thread – 1 cm?

Leela: But 0.999999…can be rounded off to 1 cms.

Teacher: True. But it is still lesser than 1 cm. (Smiling)

Leela: (Sad look) Ya, that is true.

Teacher: Why is it so? Think!

(All the students start discussing but the confusion still remains in the air)

Teacher: We have different languages – English, Sanskrit, Hindi, Chinese, etc. There are certain words in every language that cannot be expressed in the exact sense through other languages. E.g. There is a word ‘Shraddha’ in Sanskrit. It has got many meanings like Belief, Faithful, Concentration, Devotion, etc. No other language can express the complete meaning of Shraddha in one word. One needs to study Sanskrit language and Sanskrit scriptures to understand it.
In the same way, Fractions, Decimals, Percentages, Algebra, Geometry, etc can be considered as different languages in Mathematics. Some things can be expressed better using the language of Fractions more than the language of Decimals. What we need to understand is that these Mathematical languages and numerical aspects are nothing but ways to express a particular object/number/aspect in Math. Sometimes languages fall short of words. How can you express your joy through words when your country wins the world cup? You might try to express it through sounds but you cannot express it accurately through words.

Srinivas: Sir, this is the first time someone is explaining Math in such a different way. It’s more like the philosophical side of Math (Smiling).
Suppose someone asks me ‘What is the Reality in Irrational Numbers?’ Would I be correct if I say that irrational numbers are Real numbers because even though they might look strange and we cannot express it exactly through numbers, we can surely express it through a geometrical construction? So, maybe we cannot write the exact value of √2 in decimals but we can show where exactly it lies on the number line. Hence, we can say that we can express it correctly through the language of geometry?

Teacher: (with a broad smile) You are perfectly correct.

Srinivas: In that case, we find Pi hard to understand because we are trying to find it through numbers. In fact, Pi exists in circular shapes in terms of a geometrical language. It is just that we need to have the eyes to see it.

Teacher: Your analogy is wonderful Srinivas. Irrational numbers are real numbers because they exist. They look confusing because we are trying to understand them only through the language of numbers, fractions and decimals. We might understand them better through geometry.

Philosophy of Math | THE IRRATIONAL PI

The 8th standard students of a school get a new Math teacher. The story below is of the first class they had with their new Math teacher.

Teacher: What is Pi?
Students: 22/7.

Teacher: Are you sure?
Students: No, it’s 3.14.

T: Which is the correct answer?
S: Both answers are correct.

T: Listen to my question again. What is Pi?
S: (murmuring) That’s what we told you.

T: Okay. What is the value of Pi?
S: 22/7 or 3.14

T: Correct. Why did you give this answer when I asked you a different question?
S: (blank faces)

T: I had asked you, what is Pi? And you gave me the value of Pi. It’s like I’m asking you (pointing to the duster in his hand) what is this and you are replying, it weighs 100 gms. I want you to answer my first question – What is Pi?
(one of the students shout)
S: It is an irrational number.

T: That was a better answer. Can you try refining the answer?
S: (blank)

T: Let me make it clearer. √2, √3, √5, etc are also irrational numbers and so is Pi. So, we can say, Pi is one of the irrational numbers. But the question still remains – What is Pi?
(Students discuss amongst themselves but couldn’t get a concrete answer)

T: Last year you saw this Pi in your Math text and you were told that its value is approximately equal to 22/7 or 3.14. You believed it immediately. Did anyone dare to question from where did this Pi come into existence?
S: But why should we question? It was given in the text book.

T: Many centuries ago, in the schools it was taught that the earth is flat. And everyone just believed it until one man challenged it and proved it wrong. We need to think, ask questions if required, when we learn something. Let’s come back to our topic. How did Pi come into this planet, or atleast how did it come into our text books? To understand anything, we need to enquire how it all began.
When man invented the wheel, his life gained speed. He started using circle and properties of circle in his daily life. Probably (it is just an assumption) it was then that he saw that a circle is more complicated than a square or triangle or rectangle. He might have found it difficult to measure the circumference of a circle, to calculate the area of a circle and sphere; and volume of a sphere.
When curiosity comes in, then the mind is unrest. People start working on deriving formulae for calculating the circumference, area and volume of circular shapes. Soon they saw that a circle with radius 0.5 or diameter 1 cm, has its circumference as a little above 3 cms. They tried increasing the radius and found that the circumference increased proportionately. They concluded that there exists a ratio between the circumference and the diameter (or circumference and the radius too) which remains constant. And with every change in the diameter, the circumference changes proportionately.
Then they started using 3 as the ratio between circumference and diameter but very soon they faced difficulties. They saw that it gave only approximate values. The correct values couldn’t be calculated for larger figures. This gave rise the need to get a more accurate ratio between the circumference and the radius.
In almost all civilizations (Greek, Babylonian, Indian, Roman, etc), they were using an approximate value for this ratio which they later mathematicians kept refining more and more. But none of them were able to come to a fixed number to give the value of Pi because the decimals went on unending.
S: Sir, but when we divide 22 by 7, we get 3.142857142857142857… which means that it is recurring (142857)?

T: 22/7 is a recurring in decimal but did we say that value of Pi is 22/7? We said value of Pi is approximately equal to 22/7 or 3.14. We take approximations for practical calculations.
S: Oh yes! Since we keep seeing the value of Pi = 22/7 in the text book, we forgot that it is only an approximation and not exactly equal to.
But why is it so? Why can’t we arrive at a fixed decimal place for the value of Pi?

T: That is why such numbers are called Irrational Numbers because they act very strange. We try to find out the logic behind them, but we cannot arrive at any reason for their strange behavior.
(There is pin drop silence in the class and the students sit in astonishment on listening to the explanations)

T: Okay! Now who can tell me – What is Pi?
S: It is the ratio between the circumference and the diameter. If the diameter is 1, the circumference would be little more than 3.

T: What is the value of Pi?
S: 22/7 or 3.14. No no…it’s approximately equal to 22/7 or 3.14.

T: Good. Why irrational numbers are called so?
S: Because of their irrational behavior.

T: Fantastic! Now that you have understood ‘What is Pi?’, we shall see in the next class something interesting about Irrational Numbers.

Philosophy of Math | Number System

Have you ever wondered from where did the Number Systems get their names – Natural, Whole, Integer, etc? Well, the below classroom discussion will throw some light on the origin of their names.

Teacher: Natural numbers are the numbers that represent the count.4 apples, 5 oranges, etc.

Sagar: But, we can even have 1/2 apple, so why don’t we call 1/2 as Natural number?

Teacher: Well, when you have cut an apple into halves, what will you say if I ask you – How many halves to you have?

Sagar: 2 halves.

Teacher: So, you see, the number 2 represents the count of the number of halves. Half or quarters or fractions do not have existence without natural numbers.

Nitya: Sir, why are Whole numbers called so?

Teacher: When man started counting, he found that it was becoming difficult for him to represent ‘nothing’. When man discovered the number ‘zero’, he could represent nothingness by giving it a number (even though there is no count for that). Now, he could start counting from nothing. That might seem ridiculous, how can we start counting from zero? But just give it a thought…I had no books (zero books) to read, then I got 1, then 2, then 3, etc. And there was never a time when I had negative number or fraction of a book at any point of time.
Hence, they added zero to make the counts Whole or Complete and named them Whole numbers(0, 1, 2, 3, ….).

Sid: It’s so interesting, the meaning of the terms Natural & Whole numbers are derived from the meaning of the words – natural and whole. But what do we mean by Integers?

Teacher: The name integer can be said to be derived from the Latin word Integer which means ‘whole’ or ‘untouched’ which means they are ‘complete’. The word entire comes from the same source but converted through French.
Integers are the collection of all natural numbers, zero and negative natural numbers.

Bindya: But does a -5 or -23 really exist?

Teacher: We have positive as well as negative emotions. A knife can be used by a surgeon for a surgery (positive action), but the same knife can be used as a weapon to hurt someone by a thief (negative action). The knife is the same, but according to the positive or negative thought, the action becomes a positive or negative action. Now, consider this for numbers. If you get 5 new friends, your number of friends increase by 5. Suppose you have 12 friends and 5 of them leave town, how many do you have? 7. When 5 of them left, your count reduced. So, for doing calculations in our daily life, we require zero, positive & negative numbers. This whole set of positive and negative natural numbers along with zero was made into a new category called ‘Integers’

Gopal: Under which category does numbers like 1/3, 0.45, etc fall?

Teacher: They come under Rational numbers. ‘Rational’ means ‘logical/sensible’. Rational people means people who behave normally, sensibly. Rational numbers also carry the same meaning. They can be represented as a ratio of two integers, in the form of p/q, where p & q are integers and q is not equal to 0. Rational numbers include all integers, plus, even portions (logical or sensible) of integers. e.g. Dividing a cake into 6 equal parts needed to be represented using a number. So, they represented using 1/6 which means 1 cake is divided equally into 6 parts. Or for dividing 9 apples equally among 7 persons, they wrote 9/7 which meant 9 apples equally divided among 7 people.

These numbers are logical and sensible. Hence they named it Rational numbers.

Alex: So, are there any numbers that behave irrationally, without any sense?

Teacher: Yes, just like human beings who behave weirdly, illogically, without any sense…there too are numbers that seem to behave that way. They are called Irrational numbers. Let’s try to understand them. What is Multiplication?

Gopal: Multiplication is ‘repetitive addition’. When we add the number 4 three times (4 + 4 + 4), we get 12. So, we write it as 4 x 3 = 12.

Teacher: Good explanation Gopal. So, multiplication is nothing but addition…adding the same number many times. So, if we add the number 4 four times, we get 4 x 4, i.e. 16. We call it square of the number 4. When we multiply 2 twice, we get its square, which is 4. So, can we say that any number multiplied by itself will result in its square?

Students: Yes.

Teacher: So, 1.5 when added 1.5 number of times will give me its square?

Students: (Wondering)

Teacher: Let’s try to understand. 1.5 x 1 will be 1.5, which means 1.5 remains as it is. A half (i.e. 0.5 in decimal form, 50% in percentage, 1/2 in fraction) of 1.5 will be 0.75. Let us add this half to the one we get 1.5 multiplied 1 and a half times, i.e. square of 1.5 which will come to 2.25 (1.5 + 0.75).

Amar: Now, I understand, we can add any number that many number of times, i.e. 3.2 can be added 3.2 number of times (three times + 0.2 times the number 3.2). This will result in its square. But Sir, in that case, can we say the reverse thing…for any given number, will there always be another number which when multiplied by itself will result in the given number?

Teacher: Excellent question Amar. What do you think about it? Is it possible that for every number there will be another number which when squared will result in the given number? If that is so, you are saying that every number is a square, be it 3 or 3.13 or 3.0008.

Amar: (after thinking for some time) Yes, it is very well possible. Because, any number can be said to be the repeated addition of some other number. When we add 1 twice, we get 2. When we add 1.05 twice we get 2.1. To get 2.0001, we add 1.00005 twice.

So, any number is the repeated addition of some other number.
Hence, a number is always the result of multiplication of two other numbers.
Hence, we can also say that a number will be the result of square another number.
Therefore, we can conclude that any number will be the square of some other number. And it is that some other number that we call as the square root of the given number.

Am I correct?

Teacher: (Applauds) Superb! This is how we need to analyse things. Mathematics is not just calculations, Mathematics is logical ways of thinking.
For the sake of others who didn’t get what Amar gave out in one breath, I’ll repeat his lines again. I want you all to think over it. If you do so, you will understand the meaning in it.

Sagar: Sir, why don’t you just explain the meaning to us?

Teacher: When a child does not have teeth, its mother makes a paste of the food and gives it to the child so that the child can easily digest it. But will she continue the same process even after the child has grown up? Will it look good? Is it good for the child?

Sagar: No. If the mother does so, the child’s teeth & gums will not get developed. It will not generate saliva while chewing. The food will go inside, but will affect the digestion and thereby the child’s growth. I’m getting your point. If you explain everything to us, it will retard our own growth and thinking-ability.

Teacher: (Smilingly) I’m glad you understood. Anyways, these irrational numbers exist for sure, but they are hard to express in numbers. They might look illogical or strange when we express in terms of numbers. If we take the root (square root, cube root, etc) of any number which is not a perfect square or cube, we will get an irrational number. e.g. Square root of 2 will give the value in decimals as 1.41…the numbers after the decimals will go unending and non-recurring. You can go on writing the decimals forever. How can something like this look logical?

Amit: Yes, they don’t seem logical at all! In the case of rational numbers, when we divided numbers like 10 by 3, we got 3.3333….But that three was recurring. Here, forget about rational, they don’t look even Real to me.

Teacher: Talking of Reality, did you know that there exists a set called ‘Real numbers’? Mathematicians formed a bigger group or set of numbers which they named as Real numbers. Real numbers include Rational & Irrational numbers. Again, the name ‘Real’ is derived from the meaning of Real.
Real numbers are those numbers that can be expressed on a Number line. Here lies the ‘reality in irrational numbers’. They might look unreal because they behave irrationally, but that’s only because we are trying to express them in numbers (digits).

Amit: Sir, I didn’t get you.

Teacher: Can you express the sound of a monkey screeching, in words?

Amit: No Sir, not exactly. I can use alphabets and write a word that might sound similar to the monkey’s cry but it cannot be exactly the same sound.

Teacher: What do you think is the reason for this?

Amit: I guess, it’s the challenge of the language.

Teacher: Exactly. Language is a medium of expression. There are some sounds in a particular language that we cannot express in words or even sounds through other language. For example, there is a letter in the South Indian language Malayalam & Tamil, which is written in English as ‘zh’ but the pronunciation is totally different. And this letter does not exist in any other language, but Malayalam & Tamil. Hence, others find it difficult to pronounce the sound. That is the challenge of the other languages.
You can also find words in languages which do not have exact meanings in other languages. Like how you guys call each other – ‘dude’. You cannot give the exact meaning of ‘dude’ in any other language.
In the same way, irrational numbers are real numbers, but when we try to express in the language of digits, we are unable to show the exact value of the number. However, try using the language of Geometry and see if you can express irrational numbers?

Sagar: Yes Sir, we can express square root of 2 using geometry and plot it on a number line. All we have to do is, draw a right-angled isosceles triangle which has got the perpendicular lengths as 1 unit. Pythagoras rule says that the length of the hypotenuse of that triangle would be root 2 units. Now, we can take this measurement of hypotenuse on a compass and draw an arc with centre as 0 on the number line. The point where it hits the number line would be root 2 units away from 0. This way, we can show root 2 on a number line.

Teacher: Yes, this way you can represent an irrational number on a number line. Anything that can be represented on a number line is a Real number. So, that is the reality in irrational numbers. We also saw that irrational numbers cannot be expressed completely in numbers, but a geometric representation is possible.

Shall we stop here? So, how was the class today?

Students: Very nice Sir.

Sam: The discussions and interaction helped me understand Math better.

Nitya: I felt as if we were learning the Philosophy of Mathematics.

Teacher: You are right Nitya. It was indeed the Philosophy of Mathematics that we were discussing. If we understand the Philosophy of any subject, how it works, why it is like that, etc; then understanding the subject will be very easy and the subject will become very interesting.
I’m glad you all enjoyed. I too enjoyed this discussion. We shall have more discussions in the upcoming class. Till then, you guys have a nice time. Bye!