The base of natural logarithm, e, continues to interest people with some knowledge in maths. This is an irrational number, which means we can never pin-point its location on the line of real numbers with no error. (Pi, the ratio of circumference to diameter of any circle, is another such elusive number.) Logarithm with base e is called natural logarithm because of a reason. It is, to me, like this:
Remember compound interest schemes banks speak about? You deposit some money at first and after say, quarter of an year, an interest on your deposit is calculated for this period with some rate of interest. This amount of interest is compounded to your initial deposit and for the next quarter, you derive interest for this compounded amount. This addition continues till the deposit is with the bank.
If we reduce this period of calculation of interest to smaller intervals, to months, to days, to seconds, we approach an exponential rate of growth. From nowhere, irrespective of rate of interest, the factor e comes into picture, when the period is infinitesimally small. The ratio of amounts in your account at times 2t and t is equal to e raised to the rate of interest.
When the rate of growth of something is equal, not just proportional, to the amount of the thing, e appears. That is how a colony of bacteria grows, with very small periods and with characteristics very close to e. That is how temperature of a hot small metallic ball drops. That is how a radioactive element decays. A lot of naturally occurring growths and decays closely follow the e- based exponential curve. The title of 'natural' came this way, probably.
It is still a mystery how this number walks into these situations with complete authority and exuding confidence.
Remember compound interest schemes banks speak about? You deposit some money at first and after say, quarter of an year, an interest on your deposit is calculated for this period with some rate of interest. This amount of interest is compounded to your initial deposit and for the next quarter, you derive interest for this compounded amount. This addition continues till the deposit is with the bank.
If we reduce this period of calculation of interest to smaller intervals, to months, to days, to seconds, we approach an exponential rate of growth. From nowhere, irrespective of rate of interest, the factor e comes into picture, when the period is infinitesimally small. The ratio of amounts in your account at times 2t and t is equal to e raised to the rate of interest.
When the rate of growth of something is equal, not just proportional, to the amount of the thing, e appears. That is how a colony of bacteria grows, with very small periods and with characteristics very close to e. That is how temperature of a hot small metallic ball drops. That is how a radioactive element decays. A lot of naturally occurring growths and decays closely follow the e- based exponential curve. The title of 'natural' came this way, probably.
It is still a mystery how this number walks into these situations with complete authority and exuding confidence.
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