We all know that 1/3 = 0.3333333333333333?

Question

Multiplying both sides of the equation

1/3 = 0.3333333333333333333 by 3, we get 1 = 0.9999.... How can this be?

Does 1 really equal 0.99999999999999?

Answer 1

Consider that any two distinct (different) real numbers must have another number in between them. However, there is no number between 1 and 0.9999.... so  1 and 0.9999... are not different numbers. They must be the same.

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Let me explain this further for you guys. Rounding off is where you are wrong.

Suppose someone gives you $1000, but says: "Now, don't spend it all, because I'm going to go off and find the largest integer, and after I find it I'm going to want you to give me $1 back." How much money has he really given you?  

On the one hand, you might say: "He's given me $999, because he's going to come back later and get $1."

But on the other hand, He's given me $1000, because

he's never going to come back!  because he will never find that largest integer!

It's only when you realize that in this instance, 'later' is the same as 'never', that you can see that you get to keep the whole $1000. In the same way, it's only when you really understand that the expansion of 0.999999... never ends that you realize that it's not really a little below 1' at all.

I hope this helps.

Answer 2

You should take a class in Discrete Mathematics.  Then take a few Engineering classes where you have to be accurate to the 100th decimal place.  Then you'll have everything figured out.

Answer 3

Dude - round off error - you aren't using all the significant figures.  If your calculator does this, trash it and get a better one.

Answer 4

x = 0.99999....

10x = 9.99999.....

Subtracting the two and you get

9x = 9

or

x = 1

So: .99999..... = x = 1

Answer 5

0.333333 really means the infinity 3, and we cannot infer that 1 is equal to 0.99999999999999999999999999999999999999... and so on

Answer 6

Er, this could be a philosophical or a calculator accuracy topic. choose!

Answer 7

close enough for government purposes

Answer 8

"Does 1 really equal 0.99999999999999?"

Yes, it does.  And it's not just an approximation.  What you wrote is the actual proof for this fun fact of math.  Good stuff.

Answer 9

Suer we can.

Do you not infer that .333...is 1/3 without question?

Throw out your Kunkulaters books, you're on the wrong page.

0.009... is a convergent series whose sum is 1

If it is less than one, you should be able to add some fraction to it to make it equal one, right?

But the difference between 1 and 0.999... is 0.

0.999...converges to 1  expressed simply as 0.999... = 1.