witherflower on 7/4/2010 at 09:04
I don't know if you have seen this, but I found it extremely interesting and also terrifying. I'd like to hear your thoughts and opinions. To me the idea of population growth and energy consumption was a little too abstract to grasp until I saw this. And it really opened my eyes to how important education and a little more than basic understanding of mathematics is.
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http://www.youtube.com/user/sablesanctum?feature=mhw5#p/a/f/1/F-QA2rkpBSY)
Noidypoos on 7/4/2010 at 12:07
e is one of those numbers that has pure and applied mathematical significance, to the extent that it seems almost mystical. Euler's identity, the symmetry of f(x) = e^x = f'(x), logarithms, and not to forget its potential for modelling growth, applications in electronics and physics, and engineering are some examples of where is appears.
Trying to find out why e is so important is a thorny philosophical issue along the same lines of, "Why is pi equal to 3.14159...?", or "Why is there no regular pattern to the distribution of the primes?"
The answer to this is probably that e, along with other important constants, are defined by the fundamental axioms of our number systems. Certainly, if the definition of the derivative was changed, f'(x) = e^x may no longer hold. But what's important to note is that Descartes, Fermat, Newton and Leibniz were not thinking in terms of e alone when they were developing the calculus, it's more a case that e spookily presented itself with this property when (was it Euler?) derived it using series expansion.
The bottom line is it's just a really important ass number because of the way it has either personally announced itself, or that people have found good use for. You could argue that field elements like 1 and 0 are far more important, and certainly have far, far more widespread use, but they're so fundamental, so ubiquitous that they seem almost a given- e mystifies because it seems like such an arbitrary measure. Who chose it? Why? Was it always there? Did we invent it? Did we discover it?
smoke weed everyday
Brian The Dog on 7/4/2010 at 12:39
e is only important as it's the base number for which f(x) = e^x = f'(x), i.e. the gradient of the curve at that x-value is equal to the f-value itself. Logarithms etc would still be around, people just like playing around with natural logarithms as they're used to it and it makes calculus a whole lot easier.
What is spooky is how it appears in Euler's equation in the form: e^{i*pi} = 1, as this equation contains:
- e, Napier's Constant.
- i, the complex number
- pi, the ratio of a circle's circumference to its diameter
- 1, the most basic rational Real number
The Institute of Physics held a poll of their members and they voted this to be the most profound equation in mathematics.
d0om on 7/4/2010 at 13:17
uhm, isn't e^(i*Pi) = -1?
Since e^(i*Angle) = Cos(Angle) + i.Sin(Angle) (from the taylor series expansion)
so when Angle = Pi, Cos(Pi) = -1, and Sin(Pi) = 0, so e^(i*Pi)= -1.
if you want it to be 1 then you need to do e^(i*0) :p
Brian The Dog on 7/4/2010 at 14:32
Yup, this is what happens when I post in my lunchbreak :p Good spot D0om.
Edit - Just to clarify, it was me that got it wrong, not the highly prestigious IoP! I'd be kicked out if I claimed they get that wrong...
Noidypoos on 7/4/2010 at 17:21
Also note Euler's identity is written: e^(iπ) + 1 = 0, which then includes 0, uniting the 5 most important numbers ever together.
witherflower on 7/4/2010 at 17:57
Welcome to the twilight zone, boys and girls... This certainly spiraled off into an unpredictable direction. Allthough very interesting insights into the mysteries of mathematics, I was thinking more along the lines of thoughts regarding the problems of global population increase and energy consumption.
DDL on 7/4/2010 at 18:08
Well..presumably we just get more and more plentiful, then hit a resource crash, almost everybody dies, and then we start again.
Ideally using nitro-injection cars with machineguns bolted on.
witherflower on 7/4/2010 at 18:26
Quote Posted by DDL
... Ideally using nitro-injection cars with machineguns bolted on.
Hmmm... Nice. Perhaps we should start doing that right now. You know, give it a boost.:ebil:
dj_ivocha on 7/4/2010 at 21:26
Assuming the average human weighs 70kg, the current world population growth is 1%, the mass of all matter in the entire universe is 8*10<sup>52</sup>kg and the population growth remains constant, then the combined mass of all humans will reach the mass of the universe in about 9500 years.
So in 9500 years, there will be no stars, no planets, no black holes, no dust - just humans floating around in the void. :D
