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!