What is e (2.718…) and why does it literally appear everywhere?
In: 756
Suppose you have $1000 and invest it in something that gives you 10% interest once a year.
A year later you have 1000*1.1=$1100
But suppose instead you compound it twice, so you get 5% after 6 months and then another 5% at the end of the year. After six months you have $1000 times 1.05=1050. But at the end of the year you get $1050*1.05=$1102.50, so you got an extra $2.50. This is the same as $1000 times 1.05^2, or 1.05 to the second power.
If you compound it 4 times, once every quarter, you get $1000 times 1.025^4 or $1103.81.
If you compound it every month, then by the end of the year you get $1000*(1.0083^12) = $1104.71.
At this point math dorks started looking at this to figure out the pattern. What if you compound it 100 times? 1000 times? Every time you get a little bit more, but the increase is smaller and smaller so it’s homing in on some sort of hard limit. What if you compound it an infinite number of times so that your investment is always growing at a rate of 10% per year?
And they found e. The answer to continous compounding is after growing for 1 term at 10%, you have $1000*e^0.1 or $1105.71. When you generalize out from there, you get e^(rt) where r is the interest or growth rate and t is the number of time-lengths your investment grows.
Latest Answers