What are the mathematical properties of e?

Question

A few simple examples would be great. As I'm only in high school, nothing too complicated, please! Thanks a lot!

PS: I'm not asking this for homework, I'm just interested....

Answer 1

Here's a good jumping-off point:

Answer 2

Let's say that you invest $1 in an account that has an interest rate of 100% after 1 year (unrealistic, but bear with me). Obviously without compounding you'll have $2 at the end of the year.

You should already know about compounding though, where the interest gained throughout the year gains interest in of itself. You can think of it as splitting the year into smaller periods where at the end of each period you take the money you earned plus the original $1 and re-invest it for the next period. The formula for the amount of money that you'd have at the end of the entire year would be...

--- (1+1/x)^x

...where x is the number of periods that you're breaking it up into (without any compounding, x is 1 and the money you'd wind up with would be 2 as noted beforehand.

If x= 2, (1+1/2)^2 = 2.25
If x= 4, (1+1/4)^4 = 2.44
If x= 365, (1+1/365)^365 = 2.714
If x= a billion, (1+1/billion)^billion = 2.718

Notice that the answer rises quickly at first, but then slows down, to the point where the difference between compounding the interest 365 times (every day) isn't much from compounding a billion times (or many times each second). Basically, as x approaches infinity, the answer will approach e.

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Another property of e involves factorals (if you see something like 5!, it means 5 factoral, or 5*4*3*2*1). 0! is considered 1.

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ... (if you continued this forever).

Answer 3

1. e is the limit as n approaches infinity of ( 1 + 1/n)^n.
That means if you compute (1 + 1/n)^n for larger
and larger values of n, the closer you will get to e.

2. e is an irrational number. That means it
cannot be expressed as a fraction, or a terminating
or repeating decimal. Thus one can never write
down its exact value!

3. Even more, e is a transcendental number.
That means it is not a root of any polynomial
equation with integer coefficients.

Thus, to paraphrase the immortal Spock:
"e is a transcendental figure without resolution."

Answer 4

This is more Calculus-based.
If you graph the curve y =1/x, if you find the area under the curve from x =1 to x = e, the area will be 1.

e^x can be approximated as
1+ x +x^2/2! +x^3/3! +x^4/4! +x^5/5! +...

if x is 1, you can approximate e as
1 +1 +1/2! +1/3! +1/4! +1/5! +....

Answer 5

There are various ways to represent e. One of them is:

1/1! + 2/2! + 3/3! + 4/4! + 5/5!...+n/n!

It can also be represented as the limit of:

(1 + 1/n) ^ n

In practical terms, e is also the amount of money you will have if you deposit one monetary unit in a bank account that accrues 100% interest, and the interest is compounded continuously, instead of periodically. For example:

$1.00 deposited at 100% APR for 1 year,

* if compounded annually, becomes $2.00
* if compounded biannually, becomes $2.25
* if compounded quarterly, becomes $2.44
* if compounded monthly, becomes $2.61
* if compounded weekly, becomes $2.69
* if compounded daily, becomes $2.71
* if compounded continuously, becomes $e (~$2.72)

The constant is most commonly used in logarithms.

Answer 6

Jess has a great answer. Some examples:
e is used to calculate continuous compounding of interest
e is used to calculate population growth or decay
e is used in calculus proofs.
You'll probably see it in physics or chemistry class associated with radioactive decay. You'll see it in algebra 2 associated with interest calculations.

Answer 7

Well, e like pi is one of very important numbers in math. Here are some properties.

e = limit of (1+1/n)^n as n approaches infinity.

derivative of e^x = e^x, i.e. the slope of this function is equal to the function itself.

There are many more... Perhaps one of the more famous ones that connects e, pi and i, imaginary 1, which is a complex number equal to sqrt(-1):

e^(i*pi) = -1

Answer 8

e is named for Euler.

e is exactly equal to the lim [1 + (1/x)]^x as x approaches infinity.

Or you can look at it like a sum:

e is equal to the summation of [1/(k!)] where k starts at zero and runs to infinity.

e is approximately 2.1783.

Hope this helps.

=)

Answer 9

e is the natural number... we say this because...

the equation e=(1+1/n)^n...(one plus one-over-n...all raised to the nth power).. has a limit to its value ...
no matter how big you make "N"... the value of e "approaches 2.71.....
(as n increases... the decimal changes, but the place that changes is so far from the decimal point, that its value is not significant)

Answer 10

there a lot of reference of that "e", but usually its "exponent"
u can see it in some calculator with e...

10e10, its says 10 to the power of 10....
that is
10x10x10x10x10x10x10x10x10x10=10,000,000,000.00