Sunday, September 22, 2013

The secret behind the value of "e" (Euler's number) which is 2.718...

For a long time since my pre-degree  (the current +2) days I was puzzled by the peculiar value of 2.718.. for "e". It is after about 10 years of my graduation, that the puzzle got cleared.  I thought I should uncover the "secret" and let others get benefited out of it.

I will start with the basics. Let us plot two graphs, one for y = 2x and the other for y = 3x . We will see shortly why we do this. We arbitrarily take some x values say from 1 to 10 and plot the graphs.

Next we simply take the ratio of the slope at any point on the graph to the corresponding ordinate. I would request you to refresh your memory on what is an ordinate and abscissa. Ordinate is the y value for a corresponding x value (abscissa). For the y = 2x graph we have a value of 1024 as ordinate for an abscissa of 10. 
Similarly for y = 3x we have a value of 59049 as the ordinate for an abscissa of 10.

Now if we calculate the slope at this point on y = 2x       
 we get a value of 709.7 and on
y = 3x
 we get a value of 64872.

 If we take the ratio of the slope to the corresponding ordinate for each graph we get values of 0.69 (709.7/1024) and 1.09 (64872/59049) respectively for 2 and 3 .

As we watch very closely we arrive at an important finding here. There should be some point between 2 and 3 which when raised to x should give a value of 1 (which is between 0.69 and 1.09). This golden number is the value "e", ie. 2.718.. If we plot the graph for  2.718x we find that the slope/ordinate ratio is equal to one. Slope is nothing but the differential of any point on the graph. Thus slope/ordinate is the same as writing (dy/dx) / y. In our case 
 y = ex. Also please recollect that the slope (differential) of ex is equal to ex
Hence the ratio of differential of ex to ex is always 1 !

1 comment:

Public said...

Excellent.. quite useful.