| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1781.1 | YES | TROOA::RITCHE | From the desk of Allen Ritche... | Sat Aug 14 1993 12:36 | 26 |
| _
> .tech about infinitely repeating numbers... Now is 9.9 equal to 10, or
> is it infitesimally smaller?
>
The two numbers represent exactly the same value assuming base 10 of course.
The second number is equal to 1*10 + 0 = 10
while the first is
9 + .9 + .09 + .009 + .0009 + ........ an infinite geometric series whose sum
is exactly 10 (no more, no less).
i.e. S=a/(1-r) = 9/(1-.1) = 9/.9 = 10.0000000000000000
It is interesting that some numbers do not have unique decimal representations
e.g. 10.000000000000000... is equal to 9.9999999999999999... but obviously
not identical in form.
I wonder if this anomoly only occurs for rational numbers. And what is
the description of all other real numbers that have more than one
decimal expansion.
Allen
|
| 1781.2 | | 3D::ROTH | Geometry is the real life! | Mon Aug 16 1993 07:39 | 7 |
| One can avoid that ambiguity (equivalence of .99999... and 1.0 and
the like) if one expresses real numbers in terms of continued
fraction expansions.
Than that kind of stuff doesn't occur.
- Jim
|
| 1781.3 | another approach | HERON::BUCHANAN | The was not found. | Wed Aug 25 1993 06:07 | 3 |
| Also, see 1302.*, (which I no longer understand! :-( )
Andrew.
|
| 1781.4 | | HANNAH::OSMAN | see HANNAH::IGLOO$:[OSMAN]ERIC.VT240 | Wed Aug 25 1993 12:04 | 22 |
|
One way to prove that 9.999... is 10:
1) start with: 9.9999... = x
2) mul by 10: 99.9999... = 10x
3) subtract 1 from 2: 90 = 9x
4) divide by 9 10 = x
Does this help ?
/Eric
|