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.