Archive for July, 2012

Felicitation ceremony of Raman Sir and book-release of ‘The Teacher who Taught us to Think’


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.