How does "e" (2.718) help apply to applications/implications in real life?

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How does &quot;e&quot; (2.718) help apply to applications/implications in real life?

possetzvjm · Accepted Answer

There aren&#039;t many common real-world applications for Euler&#039;s number, e. Instead, it frequently surfaces in issues with growth, like population models. It also shows up quite frequently in physics.Regarding issues with growth, picture going to a bank with one dollar, one pound, or any other amount of money you have. The bank gives you a yearly interest rate of 100%. This implies that you will have $2 the following year. Such a kind bank.Let&#039;s say they offer you 50% every six months rather than 100 every year. You&#039;ll have 1.5 dollars in 6 months, and in another 6 months you&#039;ll have 1.5+50%of  1.5=2.25Now, they give you 25% interest once every 3 months. If you still have 1 dollar in the bank, now you will haveIn three months:  1+25%of  1=1+1/4=1.25In another three months:  1.25+25%of  1.25=1+1/4+1/4⋅(1+1/4)=(1+1/4)(1+1/4)=(1+1/4)2Yet again:  (1+1/4)2+1/4⋅(1+1/4)2=(1+1/4)(1+1/4)2=(1+1/4)3If we repeat the process, at the end of the year you will have (1+1/4)4 dollars.If we take a general case, say you get 100/n% interest every 12/n months and you begin with 1 dollar, at the end of the year you will have(1+1n)n dollars.As a result, we could see that getting a smaller interest over shorter periods of time was advantageous. Let&#039;s confirm this; let f(n) define how much money you get after one year with 100/n% interest over 12/n months:f(1)=2f(2)=2.25f(3)≈2.37f(4)≈2.44f(5)≈2.49Let&#039;s say your bank offers you interest that increases by an amount n that goes to infinity almost every nanosecond (in fact, much much faster than that). By the end of the year, you&#039;ll have:limn→∞f(n)=limn→∞(1+1n)n=eThis is one of the definitions of e.But since banks don&#039;t operate this way in real life, this isn&#039;t exactly practical. It does, however, give us a pretty accurate picture of how e affects growth.

How does "e" (2.718) help apply to applications/implications in real life?

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