Dr C K Raju in his lecture delivered on the topic of- Towards equity in Mathematics Education says, “Why is math regarded as a difficult subject? It is the theologification of mathematics that makes it hard to learn…..it is taught that a point is not a dot that one sees on paper but something mysteriously different leaving the child befuddled. So to make math teaching easy is de-theologification of math.” The argumentative math now ruling the kingdom of school education, works on the basis of logic. But the kids are no creatures of logic. They are the creatures of emotion. Logic works on the principal of no fixed goal posts. This perhaps provides the mathematicians with a pretext to overlook defining the basic calculating skills. Once the goal posts are not in place a mathematician goes places without being challenged. So p/q is a number. It means 12/4 is also a number. And 13/4 is also a number. What does 13/4 expressed as 31/4 prove or change. Therefore if p/q is a number then how do we define division? Either p/q is a number (answer) or p/q is a question.

As far as understand it, mathematics does not solve any of nature’s problems. It is sentiment of curiosity that makes a mathematician to propose riddles and then offers to solve them. But my curiosity can’t rest until I discover the reason behind the obsession of European mathematicians with the alleged number√2. They have been moving mountains to prove that √2 is not a rational number. Yet the fact remains that root 2 is a rational length. I can go to market and purchase root 2 meters of cloth very easily.

I will now analyse the following problem proposed by a mathematician for class 10:

Problem: Let a and b be positive integers. Show that √2 always lies between a/b and(a+2b)/(a+b).

The value of expression a/b depends upon the value of the denominator. If denominator is less than the numerator, the value can be large very large. But if the denominator is larger than the numerator, the value can’t attain the value 1.

Next, any student who has even elementary knowledge of algebra will prima facie conclude that the expression (a+2b)/(a+b) can only attain a value less than 2 for any positive integer. It so happens that this expression can be written as 1+b/(a+b). In the expression b/(a+b) since the numerator is less than denominator, it follows that it can attain a value of less than 1. Or in other words the total value will be 1 and a fraction whose value is less than 1.

So we are faced with two situations. One, if b>a the field will have asweep 0<x<2. So the field is expanding but is restricted and can’t breach the firewall 2. Therefore possibilities of finding (1.41) are bright.

In case b<a, then the expression a/b can have an exceedingly large value but the expression b/(a+b) will shrink in value but not reach 0. In this case the field is collapsing from infinity but can’t breach 1. So the two situations create two fire walls. That is the paradox.

But can √3 be found in the field when the field is expanding? It can’t be!

Why!

*Mathematics in its present form has graduated to becoming a subject of faith than a subject of grasping the choreography of numbers. It deifies number line and has as such become a religion which has priests and followers. It doesn’t produce wealth but consumes it like all religions do. In its present form it demoralises young students. Parents of course become panicky.*

Now I shall reproduce the solution of the above discussed problem as given in the book.

Solution. We do not know whether a/b < (a+2b)/(a+b) or a/b > (a+2b)/(a+b)

Therefore, to compare these two numbers, let us compute a/b – (a+2b)/(a+b)

We have,

a/b – (a+2b)/( a+b ) = (a^2-2b^2)/(b(a+b))

Therefore a/b – (a+2b)/(a+b) > 0

(a^2-2b^2)/(b(a+b)) > 0

a2 – 2b2 > 0

a2 > 2b2

a > √2b

Similarly we can prove that

a < √2b

I need not to move any further and type out the two technical pages. Instead we shall take up the two cases separately.

If a > √2b means a is > 1.41 b. It therefore follows if we take a = 1.5b, the original expressions can be computed.

Therefore a/b = 1.5b/b = 1.5

And (a+2b)/(a+b) = (1.5b+2b)/(1.5b+b) = 3.5b/2.5b = 1.4

So between 1.5 and 1.4, 1.41 exists.

And again if we put a= 1.2b means a <√2 we can get the following values

a/b = 1.2 & (a+2b)/(a+b) = 1.45

Means 1.41 or√2 exists between 1.2 & 1.45

Now the question is: √2 is the diagonal of a unit square. What is the geometrical interpretation of this question?

End of Chapter 4