Is 0.999... really equal to 1?

[Dooga Blackrazor]

0.999... = 1 (Why?)

Sorry to put such a basic question on here, but it's not for homework so I figured I'd post here.

On these forums, I've saw the issue of .99... = 1 brought up before; however, I recently discovered it in Math class.

My teacher said it equals one because it is being rounded; however, it actually doesn't equal one. I understand what she means; however, for some reason, I recall seeing a formula that mathematically proved .99... = 1 without rounding. Perhaps I am seeing things.

 

[whozum]

That formula (or the closest thing to it, there is no formula) is in those fifty other threads.

 

[Icebreaker]

If your teacher said that 0.999... is only approximately 1, then she is wrong.

 

[quantumdude]

My teacher said it equals one because it is being rounded; however, it actually doesn't equal one.

Your teacher is wrong.

Quickie demonstration:

$$\frac{1}{3}=0.\bar{3}$$

$$3\left(\frac{1}{3}\right)=3(0.\bar{3})$$

$$1=0.\bar{9}$$

And if your teacher still thinks that $0.\bar{9}=1$, then ask him/her to try to find a real number between the two. It can't be done.

 

[Integral]

This is not a new topic

 

[HallsofIvy]

Not only is this not a new topic, it's a regular topic!

My only objection to (1/3)= 0.33333... so 1= 0.999... is that the same people who object to 1= 0.9999... would also object to 3(0.3333...)= 0.999...- and they have a point. Proving one is equivalent to proving the other.

The real point is that, by definition of a "base 10 number system", 0.999... means the infinite series .9+ .09+ .009+... which is a geometric series whose sum is 1.

By the way, what grade is this teacher? And who is his/her principal/college president?!