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... = 1)

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.

We offer two proofs:

1. The simple common man proof, is easy enough for a baby to understand.
   
2. The algebraic justification proof, is for those who are a bit more difficult to convince.

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.

 

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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