e is the base of the natural logarithms.
It is a transcendental.
Curve ball coming:
Although that are far fewer primes than natural numbers, Euclid proved that there are infinitaly many of them.
What does this mean?
How the fuck should I know?
I barely passed Grade 12.
If you’re an optimist, you haven’t been paying attention.
Geenius,
While this is true for the first 12 digits or so, this pattern stops after that and becomes pretty random looking
e =
2.718281828459045235360287471352662497757247093699959574966…
TheDude
What it means is that when you deal with infinities, things get really strange.
Consider this one; suppose you have a hotel with an infinite number of rooms. You also have an infinite number of guests staying in those rooms. Your hotel is full.
A new guest comes in and you need to make room for them. Call the person in room 1 and have them move to room 2. The person in room 2 moves to room 3 and so on; for every room, the person in room n moves to room n+1. You put the new guest in room 1. You have just added one person to your full hotel.
Now, suppose an infinite number of guests show up. Now what? Easy. Instead of n+1 have each guest move to room n*2. The person in room 1 moves to 2, the person in 2 moves to 4 and so on. Every odd numbered room in your hotel is now empty and, since there are an infinite number of odd numbers, you can now add an infinite number of guests to your full hotel.
“You can’t run away forever; but there’s nothing wrong with getting a good head start.” — Jim Steinman
Dennis Matheson — Dennis@mountaindiver.com
Hike, Dive, Ski, Climb — www.mountaindiver.com
And to tie the two together:
Say in infinite number of people check into the hotel with an infinite number of rooms (all empty), and each get assigned an individual room, so that now the hotel’s full. Then they all go out, get drunk, and nobody can remember the correct rooms, so they all pick one randomly. Everybody goes back to some random room, and each room gets exactly one person in it.
What’s the probability that no one is in the correct room?
1/e.
Please, please, PLEASE post the proof of that being the probability! I’m going crazy trying it on my own.
Well, the cook at that hotel isn’t very accomplished, so for breakfast it’s either 2 scrambled or 1/ez.
Ray (Is that with French pi’s?)
JWK says:
My dictionary defines ‘rational number’ as a number capable of being expressed as an integer or quotient of integers. It defines a rational algebraic expression as one having no variable that appears in an irreducible radical or with a fractional exponent.
But WallyM7 says:
My dictionary defines ‘transcendental’ as not capable of being determined by an combination of a finite number of equations with rational integral coefficients.
So that it is not sufficient that pi not “repeat itself” in order to be pi, right?
But are there numbers that infinitely “repeat themselves” but are not rational? And are there numbers/quantities that are transcendental but do infinitely “repeat themselves” and/or are not irrational?
Ray (If I have repeated myself here, does that necessarily prove I’m rational? And can I transcend at more than one level? And if at first I don’t curse right, should I recurse again? And how does aleph null get into the picture.)
Here’s a link for the proof of 1/e:
http://www.unc.edu/~rowlett/Math148/notes/derange.html
As for the irrationals and transcendentals, a number is rational if and only if it infinitely repeats itself. So if it infinitely repeats itself, it can’t be irrational–same for transcendentals, and all transcendentals are irrational.
And as for aleph null, that’s the number of rooms in the hotel.
Whoever asked about e, it is the base for natural logarithms.
A logarithm is the exponent you use to raise a base to a given number. “Common” logarithms are base 10. For example, the logarithm of 10 is 1 since 10[SUP]1[/SUP] is 10. The logarithm of 100 is 2 (10[SUP]2[/SUP], 1000 is 3 (10[SUP]3[/SUP]) and so on.
Logarithms were commonly used to change multiplication to addition. Suppose you needed to multiply 10 by 100. Instead of multiplying you could convert the two number to their logarithms (1 and 2) add them together (giving 3) and take the exponent to get the result (10[SUP]3[/SUP] = 1000)
The logarithm for any number can be determined. For example, the logarithm of 17 is 1.2304 since 10[SUP]1.2304[/SUP] is 17.
These are common logarithms which are calculated on a base of 10. Natural logarithms are on a base of e, where e is the number described above (2.718281828…) Natural logarithms show up quite a bit in many fields; I am familiar with them from both calculus and engineering where they occur quite often.
“You can’t run away forever; but there’s nothing wrong with getting a good head start.” — Jim Steinman
Dennis Matheson — Dennis@mountaindiver.com
Hike, Dive, Ski, Climb — www.mountaindiver.com
The question remains what is so “natural” about e and its logarithm? I’ve never figured this out. The only “natural” thing I can think of is that in calculus, the function e[sup]x[/sup] is its own derivative. This doesn’t work for other numbers a, where you invariably get a factor of ln a in the derivative.
But that can’t be the whole story, can it?
It’s that, plus the fact that ln x is the integral of 1/x, plus the fact that ln turns up when doing trig with complex numbers and all sorts of other lovely things.
Here’s one of the best:
The natural logarithm of -1 is pi*i.
With calculators and computers, base-10 logarithms have become entirely pointless – but natural logarithms have a permanent place in math.
John W. Kennedy
“Compact is becoming contract; man only earns and pays.”
– Charles Williams
IIRC, it had a cool effect like:
For t = e[sup]x[/sup]
(d/dt)e[sup]x[/sup] = e[sup]x[/sup]
In other words, the natural exponential function has a rate of change equal to it’s value. That’s a pretty neat property.
I knew that felt wrong. It’s been awhile since I’ve actually used that stuff. It should have been
(d/dx)e[sup]x[/sup] = e[sup]x[/sup]
Neat indeed. And it’s exactly what I said, too. 