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1.)
The simple common man proof:
Remember
that 1/3 = 0.333...
Remember that 2/3 = 0.666...
Also
1/3 + 2/3 = 3/3, which equals 1 .
Now
if we add the decimals here 0.333... + 0.666..., we get 0.999...
(repeating threes added to repeating sixes give repeating nines)
But WHOA! 1/3 + 2/3
= 1, so the 0.999... must ALSO equal 1. It's just that
simple.
back
2.)
The Algebraic justification proof:
To
begin this more "sophisticated" proof, we will
set the variable "x" equal to 0.999999999...
(x = 0.99999999999...)
Then
we will use the multiplication property of equality to create
a new equation.
To
do this we will multiply both sides of
x = 0.999999999... by 10. This will give us:
(10x = 9.99999999...).
Now
we will arrange these two equations one underneath the other,
and we will subtract them.
10x = 9.999999999...
- x = 0.999999999...
9x = 9.0000000000... notice
the repeating nines all drop out
Now
what is the only number that can be multiplied by nine to MAKE
nine?
Well
one, of course.
So
x = 1
BUT
we DEFINED "x
"at the beginning to be equal to 0.99999999....
Therefor
0.999999999... must also be equal to 1.
Assignment:
Go forth and see if you can win some cash by getting someone
to bet that you can't
prove 0.999... = 1