Why is 0.999 the same as 1?

Question

and please explain in a math for dummies format.

thanks

Answer 1

1/3 *3 is one right? right
and 1/3 is equal to .3333... right? right
and .33333...*3 =.99999..... right? right

we have just proved that 1=.999999...

there is no rounding, it is a fact.
crazy but true.

Click link for more examples and a longer explanation of exactly why this is true:
http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
this will take you step by step as to why.

(its .9999..... that is equal to one .
.999 is not 1, is .999=.999)
But I know what you meant.

Answer 2

0.999 is not the same as 1.
1-0.999=0.001, that is, the two numbers differ by one one-thousandth.

However, 0.999... (in which the sequence of "9" digits is infinitely long) *IS* the same as 1! There are a number of ways to show this equality, some sophomoric, some very rigorous. Here is one of the sophomoric ones:

0.999...
= (9*0.999...)/9
= ((10 - 1)*0.999...)/9
= (10*0.999... - 0.999...)/9
= (9.999... - 0.999...)/9
= (9 + 0.999... - 0.999...)/9
= 9/9
= 1

Underlying this "proof" there remains a great deal to be proven about the validity of adding and multiplying infinite decimal fractions. Most of that "missing piece" can be proven using theorems regarding absolute convergence of infinite series of numbers (I know, I've already lost you, but I only have so much time and space to write an answer.).

But those same convergence techniques can be used to show directly that 0.999... = 1, thus eliminating the need for the above "proof" altogether. That is why I say the "proof" is sophomoric. While the proof is "true", it sort of misses the point: what is really needed is a proof about infinite series convergence.

ASIDE TO PAM: Great answer! If it comes to a vote, you've got mine.

Answer 3

There's no doubt that this equality is one of the weirder things in
mathematics, and it _is_ intuitive to think: No matter how many 9's
you add, you'll never get all the way to 1.

But that's how it seems if you think about moving _toward_ 1. What if
you think about moving _away_ from 1?

That is, if you start at 1, and try to move away from 1 and toward
0.99999..., how far do you have to go to get to 0.99999... ? Any step
you try to take will be too far, so you can't really move at all -
which means that to move from 1 to 0.99999..., you have to stay at 1.

Which means they must be the same thing!

Here's another way to think about it. When you write something like

0.35

that's really the same as 35/100,

0.35 = 35 / 100

right? Well, you can turn that into a repeating decimal by dividing by
99 instead of 100:
__
0.35353535... = 0.35 = 35 / 99

Play around with some other fractions, like 2/9, 415/999, and so on,
to convince yourself that this is true. (A calculator would be
helpful.)

In general, when we have N repeating digits, the corresponding
fraction is

(the digits) / (10^N - 1)

Again, some examples can help make this clear:
_
0.1 = 1/9
__
0.12 = 12/99
___
0.123 = 123/999

and so on.

So, here's something to consider: What fraction corresponds to
_
0.9 = ?

It has to be something over 9, right?
_
0.9 = ? / 9

The _only_ thing it could possibly be is
_
0.9 = 9 / 9

right? But that's the same as 1.

Ultimately, though, this probably won't _really_ make sense until you
come to grips with what it means for a decimal to repeat _forever_,
instead of just for a r-e-a-l-l-y l-o-n-g t-i-m-e.

When you think of 0.999... as being 'a little below 1', it's because
in your mind, you've stopped expanding it; that is, instead of

0.999999...

you're _really_ thinking of

0.999...999

which is not the same thing. You're absolutely right that 0.999...999
is a little below 1, but 0.999999... doesn't fall short of 1 _until_
you stop expanding it. But you never stop expanding it, so it never
falls short of 1.

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, you might say: "He's given me $1000, because
he's _never_ going to come back!"

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.

Answer 4

It all has to do with rounding. You need to know at which position you are rounding to (tenth, hundreth, thousandth, etc.). If the number to the right of the rounding position is 5 or higher, you round up. For example, if we were rounding 5.76 to the nearest tenth, the rounded number would be 5.8.

In your case, 0.999 is equal to 1 no matter how you round it. If you were to round it to the nearest whole number, you would look at the first 9, and round up to 1. If we were rounding to the nearest tenth, you look at the second 9, and once again, you must round it to 1. If we were rounding to the nearest hundredth, you would look at the third 9 and once again the answer is 1.

Answer 5

i don't think 0.999 is the same as 1. but it is close to 1 (in my opinion)

0.999 can be proven equal to 1 mathematically:

fraction proof (the simpliest)

0.333=1/3
3x0.333=3x1/3
0.999=1

algebraic proof

x=0.999
10x=9.999
10x-x=9.999-0.999
9x=9
x=1

hope this helps

Answer 6

0.999 is not equal to 1. people usually round up 0.999 to 1 though

Answer 7

who or what tells you that it is ? A computer will tell you that it is the integer representation of 0.999. Or any value from 0.5

Answer 8

you need to round off...
0.999 if you round off is 1

Answer 9

Because you need to round 0.999 up.

Answer 10

If we round of to two decimal places then =1.00