I am not sure if you are just joking around. This would be the appropriate thread to say untrue math equalities.
That one was. The first one wasn't.
We can write 1/3 as a decimal, but this is not an exact number
It is an exact number (or did you mean that the decimal isn't an exact number until it is shown that repeating decimals occupy a specific point on the real number line?). I'm not sure if Dedekind was the first to show this but I know his method is the one used more often as it has the advantage of defining real numbers. And I can't get decimal expansions out of my head but that's for the other way around.
. The vinculum bar denotes infinite repetition. Thus .3 with a vinculum bar over the three multiplied by 3 will indeed yield .9 with a vinculum bar over the 9.
I was thinking about this. That's "equal" to one, or at least you can set it up so that this is true, but I came down on the side that you cannot write it as is (without limits) and have that be true. But it's 5 in the morning so I'm not sure of anything at the moment. My idea (avoiding Dedekind cuts) was more like this:
Given an infinite sequence
and any rational number
r, then
for some sequence. If one then defines a null sequence as above only with 0 instead of
r, then the limit of a sequence is
r iff
r -
= a null sequence.
but saying this is equal is a different proof. That proof needs calculus. Such that lim 1= 1 with infinite progression in whatever equation which = lim 1. I enjoy this concept because it illustrates the infinite within the finite.
That's not exactly the thing I was thinking of by you could be simplifying or I could be wrong. Or it could be notation. By progression do you mean sequence?
once we establish that then we can recognize that .9 with a vinculum bar over the 9 has lim 1. Which we can then say = 1.
Can we translate this to say that the sequence converges to one or are you referring to something different?