what you do:
You go up to some unsuspecting
person who thinks you are not mathematically sophisticated.
You say, "Did
you know that zero point nine repeating is really equal to one?"(0.999...
They will look at you
and say, "It is very CLOSE to one, but it is not EXACTLY
EQUAL to one."
At which point you
will say, "I bet you that it IS
exactly equal to one?"
Now most folks will
bet you, because most folks have not seen the PROOFS
that you have in your back pocket. You learned them here
and have printed them out, and or memorized them for just such
a money making occasion as this.
offer two proofs:
simple common man proof, is easy enough
for a baby to understand.
algebraic justification proof, is for
those who are a bit more difficult to convince.
Copyright 1999-2006 themathlab.com
The simple common man proof:
that 1/3 = 0.333...
Remember that 2/3 = 0.666...
1/3 + 2/3 = 3/3, which equals 1 .
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
The Algebraic justification proof:
begin this more "sophisticated" proof, we will
set the variable "x" equal to 0.999999999...
(x = 0.99999999999...)
we will use the multiplication property of equality to create
a new equation.
do this we will multiply both sides of
x = 0.999999999... by 10. This will give us:
(10x = 9.99999999...).
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
what is the only number that can be multiplied by nine to MAKE
one, of course.
x = 1
we DEFINED "x
"at the beginning to be equal to 0.99999999....
0.999999999... must also be equal to 1.
Go forth and see if you can win some cash by getting someone
to bet that you can't
prove 0.999... = 1