20 Jul 06
Originally posted by XanthosNZDoes the e^(ix) = cosx + isinx observation get you there? x=pi isn't the only point on the line where cosx=-1 and sinx=0. What about where x=3pi? cos(3pi) + isin(3pi) also =-1. so does e^(ipi) = e^(3ipi)?
This relationship is normally stated as e^i*pi = -1.
You can find a good basic explanation here:
http://www.math.toronto.edu/mathnet/questionCorner/epii.html
20 Jul 06
Originally posted by deriver69because it works
One answer is because it works, but if you think of factorials as being the number of ways of arranging that many objects, there is only one way of arranging zero objects.
Ahh, the obverse side of the coin which reads "We Haven't the Foggiest." Sounds conclusive enough to fool 'em everytime!
20 Jul 06
1 edit
Originally posted by deriver69Um, there are no ways to arrange zero objects. You are into semantics if you state that because there is no way to arrange zero objects then that "no way" can be re-written as "one way"
One answer is because it works, but if you think of factorials as being the number of ways of arranging that many objects, there is only one way of arranging zero objects.
The only truth is to state there are zero ways to arrange zero objects.
0!=1 should be re-visited.
24 Jul 06
Originally posted by FreakyKBHSince Infinity - 1 cannot be a finite number (no finite number plus 1 can equal infinity), then:
Howzabout a whole lot of zero's? Surely they cannot amount to nothing.
Infinity - 1 = Infinity
And one could subtract infinity from both sides, then multiply by -1, to conclude that 1 = 0. Now, with a lot more zeros you can begin to see how they will amount to quite a lot.
24 Jul 06
Originally posted by PalynkaOh, thanks Palynka!
0! is part of a larger axiom that states that empty products are equal to one, just as empty sums are equal to zero.
Why 0! equals 1 has bothered me for years... No one has been able to give a good explanation and you made me understand it right away.
Well done!
24 Jul 06
Originally posted by uzlessThe only way of arranging zero giraffes on my kitchen table is the way I have zero giraffes on my kitchen table at the moment. There is no other way of arranging zero giraffes on the table.
Um, there are no ways to arrange zero objects. You are into semantics if you state that because there is no way to arrange zero objects then that "no way" can be re-written as "one way"
The only truth is to state there are zero ways to arrange zero objects.
0!=1 should be re-visited.
On another point mentioned earlier, 1 not being a prime number is consistent with the definition of prime numbers being numbers with only 2 factors (1 has one factor).
24 Jul 06
1 edit
Originally posted by deriver69Well said.
The only way of arranging zero giraffes on my kitchen table is the way I have zero giraffes on my kitchen table at the moment. There is no other way of arranging zero giraffes on the table.
On another point mentioned earlier, 1 not being a prime number is consistent with the definition of prime numbers being numbers with only 2 factors (1 has one factor).
However, 1 could have an infinite number of factors: each 1.