- Good morning class, today we are going to talk about the number "e". Yes I know that Sesame Street would be appalled with the idea that we are learning about a number that is represented as a letter, yet here we are learning about the NUMBER "e".
-First of all lest talk about were the number "e" came from and what it means. Between the years 1727 and 1728 a man named Euler started using e to represent the mathematical constant of (1 + 1/m)m. In this equation as the value for m increases (all the way to infinity) the output of this equation gets closer and closer the the value 2.71.
- e is the base of natural logarithms. (just an FYI)
-This is an extremely valuable concept to understand especially is the financial world of compound interest in this particular case it is used for continuous compound interest.
- The formula that we use to calculate continuously compounded numbers at a particular rate for an amount of time is A
= Pert. where A is the amount you either have or want to have, P is your principle balance, e is the number e, r is the interest rate, and t is the amount of time in years that the account is compounding continuiously.
- So let us say that you wanted to find out how long it would take to triple an investment you are making. Your original investment was $2000 that is being continuously compounded at 8.5% interest. To find out how long it is going to take to take to triple this investment we will set our equation up as 6000
= 2000e(.085)t
-So we just need to solve for t. First lets divide 6000 by 2000. this gives us 3. Now earlier we mentioned that e is the base of a natural logarithm. So now we are going to take as ln3
= lne(.085)t which now gives us ln3=.085t. So, to get the value of t we are now going to divide ln3/.085. this gives us 12.925 as the value of t. therefor it will take 12.925 years to triple your investment of $2000 at a 8.5% intrust rate that is continuously compounding.
Hey Shannon,
ReplyDeleteI really liked how you started off talking about Seasame Street - it was a cute way to introduce "e" as a number rather than a letter. Also using the A=Pe^rt as a real world example of "e" was a great idea, as "e" is something that people should know for later in life when trying to find which banks have good interest rates. I think it would have been a little easier to understand the part where you solved for "t" if you had put it in a table on excel and then pasting it in to your post. Overall i really liked the way you explained the number "e".
- Sarah
I like that you chose the number e for your post. This concept is particularly confusing and you presented it well! The examples were helpful.
ReplyDeleteshannon,
ReplyDeletei like the sesame street intro, they would be appalled! your lesson is good and i like how you give some background about the number before starting your lesson. your calculations are good and i like that you sort of showed how e and ln x are connected. i would have gone into a little more detail on discussing the natural log since you used it to solve your example problem, but other than that, nice job!
professor little