Posts from the ‘Number Systems’ Category


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.


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.