LET’S TALK ‘E’ In 1864 Benjamin Peirce had his picture taken standing in front of a blackboard on which he had written the formula i-i = (ep).
In his lectures he would say to his students: – Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important. In the intriguing world of complex mathematical symbols, sometimes, one comes across an unusually simple symbol, the letter ‘e’. Inspite of the simplicity, a sort of mysticism surrounds ‘e’, which would definitely urge someone having mathematically inclined neurons or even otherwise, to learn further more about it. Though a relative newcomer in the mathematics scene, it can be considered an asset to the world of mathematics, which has found application in scores of places. ‘E’, commonly known as the base of natural logarithm, a number similar to pie, first comes into mathematics in a minor way.
This was in 1618 in an appendix to Napier’s work on logs. The next possible occurrence of ‘e’ was in the relation between the area under the rectangular hyperbola YX = 1 and the logarithm. Of course, ‘e’ is such that this area from 1 to e, is equal to 1. This is the property that makes ‘e’ the base of natural logs. Next, Huygens made yet another advance and defined what we would refer to, as an exponential curve having the form Y = Ka^x. Again out of this, comes the logarithm to the base 10 of ‘e’.
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Surprisingly, though all this work on logs had come so close to recognising it, it was the study of compound interest that ‘e’ was first discovered. Jacob Bernoulli, in a problem of compound interest tried to find the limit (1+1/X) ^X, as X tends to infinity. Using binomial theorem he found, the limit had to lie between 2 and 3. This could be considered as the first approximation found to ‘e’and also it was the first time that a number was defined by a limiting process. At last ‘e’ had a name and was recognized. So much of our mathematical notation is due to Euler, that it is not at all surprising that the notation for ‘e’ is also due to him.
Euler showed that E = 1+1/1! +1/2! +1/3! +1/4! +… Also that ‘e’ is the limit of (1+1/N) ^N as N tends to infinity. He also gave the continued fraction expansion of ‘e’, in particular he gave, If the continued fraction (e-1) /2 were to follow the pattern shown in the first few terms 6, 10, 14, … then it will never terminate so (e-1) /2 (and therefore ‘e’) cannot be rational. This was the first attempt to prove that ‘e’ is not rational, which aws later buitteressed by other proofs. It was Charles Hermite who proved that ‘e’ is not an algebraic number.
The open question still is whether E^E is algebraic, though no mathematician would seriously agree that it is.’ E’ thus belongs to the family of ‘Transcendental numbers’. Hence ‘e’ can never be the root of any polynomial equation with integral coefficients. Interestingly there are many more transcendental numbers than algebraic ones and ‘e’ was the first number to be proved transcendental by Charles Hermite in 1873. Having learnt what ‘e’ and how it was discovered, a naturally following question springs up in one’s cashew, where and hoe does it find application.’ E’ plays a key role in the description of phenomena such as radioactive decay, population growth and in the calculation of compound interest.
If one were to be asked how much money one would end up with, under C. I. , if under simple interest one earns 100% interest. The answer would be ‘e’ times the original amount. Thus we get a physical meaning for the number ‘e’ as, the factor by which a bank account earning continually C. I.
Here’s a good example from engineering. An aerospace engineer might want to graph the lift of a wing versus the size of the wing, and he wants to show everything from an insect wing all the way up to a jumbo jet. A fly’s wing is maybe 0.1 inch long, and a jumbo jet wing might be 1000 inches long (about 80 ft). It would be pretty tough to put more than one insect on that graph – ...
will increase, if without compounding it would have doubled. The number e is also used in applications involving exponential growth and decay. Entities that are subject to exponential growth increase at a rate that is proportional to the amount of substance in the entity; entities subject to exponential decay decrease at a rate proportional to the amount of substance in the entity. All these pieces of information about ‘e’ make us realize that the world of ‘e’ is not as simple as its name, but it is this complexity in simplicity which makes the world of ‘e’s o very fascinating and above all, lends mathematics its richness..