Thursday, October 11, 2007
Why QED
""Not a proof" so why the "QED"?" says Rich.
Well actually, it is not far off a proof as it happens. Consider the number 0.11111... (= 1/9). What does this mean? Well, 0.1111... is shorthand for
1/10 + 1/102 + 1/103 + .... = 1/9
Clearly
9 . (1/10 + 1/102 + 1/103 + ....) = 9/9
and therefore by the distributive law
9/10 + 9/102 + 9/103 + .... = 9/9
or 0.999.... = 9/9.
QED
Well not quite. I have assumed
1/10 + 1/102 + 1/103 + .... = 1/9
But that is trivially easy to show.
Well actually, it is not far off a proof as it happens. Consider the number 0.11111... (= 1/9). What does this mean? Well, 0.1111... is shorthand for
1/10 + 1/102 + 1/103 + .... = 1/9
Clearly
9 . (1/10 + 1/102 + 1/103 + ....) = 9/9
and therefore by the distributive law
9/10 + 9/102 + 9/103 + .... = 9/9
or 0.999.... = 9/9.
QED
Well not quite. I have assumed
1/10 + 1/102 + 1/103 + .... = 1/9
But that is trivially easy to show.
