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Vorticity wrote:Since this is a brand new forum to which I myself am new, I thought it fitting that I should start by contributing this inevitable and indispensable thread to the mathematics section. Each and every forum of this sort is guaranteed to have this as an acrimonious 30+ page thread eventually, so we might as well do it now. So let's have at it...
I assert that the repeating decimal 0.9999999... is exactly, rigorously, and indisputably equal to 1.
I submit that anyone in disagreement with this is, in fact, a nematode.
Thoughts?
Isn't it just infinitesimally close to 1, but not actually 1.
RPizzle wrote:Isn't it just infinitesimally close to 1, but not actually 1.
In the set of numbers known as the Reals, to which (regardless of whether or not they are distinct) both the numbers 0.9~ and 1 belong, there is no such thing as two numbers which are "infinitesimally close" to one another.
In other words, if a and b are both real numbers, then either a = b exactly, or the difference between them is some finite, non-zero, real number. Since we can name no finite, non-zero, real number corresponding to the difference between 0.9~ and 1, the only alternative is that they are equal.
Vorticity wrote:Since this is a brand new forum to which I myself am new, I thought it fitting that I should start by contributing this inevitable and indispensable thread to the mathematics section. Each and every forum of this sort is guaranteed to have this as an acrimonious 30+ page thread eventually, so we might as well do it now. So let's have at it...
I assert that the repeating decimal 0.9999999... is exactly, rigorously, and indisputably equal to 1.
I submit that anyone in disagreement with this is, in fact, a nematode.
Thoughts?
Shaker wrote:Isn't it just infinitesimally close to 1, but not actually 1.
That's what I'd have thought, but no mathematician I - I have to take my shoes and socks off to do sums with numbers bigger than 10 in. I can't see how 0.9999999999 ... can be equal to 1, only an infinitesimally closer and closer approach to it.
Vorticity wrote:I submit that anyone in disagreement with this is, in fact, a nematode.
RPizzle wrote:I'm no math major, so I can't really put up much of a counter-argument. I am likely a nematode.
Spearthrower wrote:RPizzle wrote:I'm no math major, so I can't really put up much of a counter-argument. I am likely a nematode.
Once I find a digestive tract, you can take the anus end!
RPizzle wrote:0.9*(1+1/10+1/100+1/1000...) If you were to graph these individually you'd end up with an asymptote so not equal to 1 no matter how many iterations you go.
Spearthrower wrote:But it can't be 1 because it's got a 0 at the beginning!
Spearthrower wrote:Once I find a digestive tract, you can take the anus end!
Spearthrower wrote:Can you explain without numbers?
Vorticity wrote:Spearthrower wrote:But it can't be 1 because it's got a 0 at the beginning!
Inherent in this statement is the assumption that a given real number can have only one decimal expansion, and hence that 0.9~ and 1 must be distinct since their first digits are distinct. In fact, this is false. Every real number with a finite decimal expansion (such as 1) has at least two decimal expansions. Examples:
3.1 is the same as 3.0999999~
4 is the same as 3.999999~
1 is the same as 0.999999~
Vorticity wrote:Spearthrower wrote:Can you explain without numbers?
Hmmm.
No, I guess not. I mean, not without numbers in it anywhere at all. I'd have to use at least the numbers 1 and 9.
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