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Until very recently I had no clue what "e" really stood for, though I was good at maths in college. I took it as the result of some complicated infinite series.

Until very recently I had no clue what "e" really stood for, though I was good at maths in college. I took it as the result of some complicated infinite series. Until that is I came across this[1] wonderful article. That site is a treasure trove of very good insightful articles. Can't recommend enough. [1] https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ https://betterexplained.com/articles/an-intuitive-guide-to-e...

rrauenza 5y ago
I was going to post this Mathologer video (https://www.youtube.com/watch?v=CaasbfdJdJg https://www.youtube.com/watch?v=CaasbfdJdJg) based on e being "the most irrational number" -- but I misremembered. It's the golden ratio, phi. But its still a really interesting video!
- Smithalicious 5y ago
  There's something ironic about calling the "most irrational number" the "golden ratio"
gnatman 5y ago
I like this explanation from Numberphile as well[1] [1]https://www.youtube.com/watch?v=AuA2EAgAegE https://www.youtube.com/watch?v=AuA2EAgAegE
jstx1 5y ago
I mean it is an infinite polynomial. Sometimes it’s a lot more useful to think of it that way and not in terms of compounding. Or you can think of the function exp() represented by the Taylor series and e just happens to be exp(1). But the particular number itself isn’t as important as the function.
- dxbydt 5y ago
  <<< this is important <<< this is not important To be fair, neither is important. You don't have to reach for a transcendental if you want infinite polynomials, you can manufacture one for integers, rationals, irrationals...the familiar example - f(x) = 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + ... f(1) = 2
  - jstx1 5y ago
    Oops, I deleted that part before you posted the comment; didn't think it was worded well. I guess my point was that the specific value of 2.718... isn't as important as a lot of pop-math sources make it out to be.
boulos 5y ago
Hmm. Did you end up studying differential equations? I feel like the most clear statement from that article is "it's the base of exponential growth", but that (for a math inclined audience) the best way to show that is via equations like dy/dt = \alpha y.
huachimingo 5y ago
Another good way to see the limit definition is by solving the discrete case of that diff. equation (y' = y, y(0)=1). Replace dy/dx with ∆y/∆x, so ∆y = y(x+∆x) - y(x) Once you solve that (use some induction and numerical methods), see what happens when ∆x appraches to 0 and x=1. For example try with ∆x=1 and then generalize that case to ∆x=n.
- hota_mazi 5y ago
  You didn't justify why you picked y' = y, though, and I'm having a hard time connecting that differential equation with the one that follows "so" in your message. Could you break it down further?
Gunax 5y ago
My favourite use of e is determining how many (independent) trials it would take to encounter an event of some likelihood. Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials? Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is: = 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%) This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!
kevin_thibedeau 5y ago
I can recommend "e: The Story of a Number". It has a good explanation of transcendentals along with an informative dive into the history and significance of e.
User23 5y ago
The defining characteristic of the exponential function is that d/dx e^x = e^x. The series construction makes this really obvious, because each element of the series is the first derivative of the next element, ad infinitum. And it wasn't as if this series was somehow luckily discovered, it was constructed from the defining characteristic. Visual Complex Analysis, by Tristan Needham, covers this lucidly.
chobytes 5y ago
I think I find viewing at the result of some series to be more satisfying. After seeing series solutions to differential equations, the number e kind of faded into the background of the numerical process that gives us e.
foxes 5y ago
Where did you see e as a series before seeing it as the limit of some growth? I remember seeing the intuitive explanations in school, before anything else.
zarzavat 5y ago
IMO the most elegant definition of pi and e comes from differential equations. e is the base of the exponential function, the solution of y = y' pi arises from sine and cosine, which are the solutions to y'' = -y

rrauenza 5y ago

I was going to post this Mathologer video (https://www.youtube.com/watch?v=CaasbfdJdJg https://www.youtube.com/watch?v=CaasbfdJdJg) based on e being "the most irrational number" -- but I misremembered. It's the golden ratio, phi. But its still a really interesting video!

Smithalicious 5y ago
There's something ironic about calling the "most irrational number" the "golden ratio"

gnatman 5y ago

I like this explanation from Numberphile as well[1] [1]https://www.youtube.com/watch?v=AuA2EAgAegE https://www.youtube.com/watch?v=AuA2EAgAegE

jstx1 5y ago

I mean it is an infinite polynomial. Sometimes it’s a lot more useful to think of it that way and not in terms of compounding. Or you can think of the function exp() represented by the Taylor series and e just happens to be exp(1). But the particular number itself isn’t as important as the function.

dxbydt 5y ago
<<< this is important <<< this is not important To be fair, neither is important. You don't have to reach for a transcendental if you want infinite polynomials, you can manufacture one for integers, rationals, irrationals...the familiar example - f(x) = 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + ... f(1) = 2
- jstx1 5y ago
  Oops, I deleted that part before you posted the comment; didn't think it was worded well. I guess my point was that the specific value of 2.718... isn't as important as a lot of pop-math sources make it out to be.

boulos 5y ago

Hmm. Did you end up studying differential equations? I feel like the most clear statement from that article is "it's the base of exponential growth", but that (for a math inclined audience) the best way to show that is via equations like dy/dt = \alpha y.

huachimingo 5y ago

Another good way to see the limit definition is by solving the discrete case of that diff. equation (y' = y, y(0)=1). Replace dy/dx with ∆y/∆x, so ∆y = y(x+∆x) - y(x) Once you solve that (use some induction and numerical methods), see what happens when ∆x appraches to 0 and x=1. For example try with ∆x=1 and then generalize that case to ∆x=n.

hota_mazi 5y ago
You didn't justify why you picked y' = y, though, and I'm having a hard time connecting that differential equation with the one that follows "so" in your message. Could you break it down further?

Gunax 5y ago

My favourite use of e is determining how many (independent) trials it would take to encounter an event of some likelihood. Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials? Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is: = 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%) This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!

kevin_thibedeau 5y ago

I can recommend "e: The Story of a Number". It has a good explanation of transcendentals along with an informative dive into the history and significance of e.

User23 5y ago

The defining characteristic of the exponential function is that d/dx e^x = e^x. The series construction makes this really obvious, because each element of the series is the first derivative of the next element, ad infinitum. And it wasn't as if this series was somehow luckily discovered, it was constructed from the defining characteristic. Visual Complex Analysis, by Tristan Needham, covers this lucidly.

chobytes 5y ago

I think I find viewing at the result of some series to be more satisfying. After seeing series solutions to differential equations, the number e kind of faded into the background of the numerical process that gives us e.

foxes 5y ago

Where did you see e as a series before seeing it as the limit of some growth? I remember seeing the intuitive explanations in school, before anything else.

zarzavat 5y ago

IMO the most elegant definition of pi and e comes from differential equations. e is the base of the exponential function, the solution of y = y' pi arises from sine and cosine, which are the solutions to y'' = -y