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

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 2x and 3x .
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).

y = ex. Also please recollect that the slope (differential) of ex is equal to ex

**This golden number is the value "e", ie. 2.718..**If we plot the graph for**2.718****x**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 casey = ex. Also please recollect that the slope (differential) of ex is equal to ex

**Hence the ratio of differential of e****x**to**e****x**is always 1 !
## 1 comment:

Excellent.. quite useful.

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