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2=1.999...infinity! What are your thoughts?

S

sjjordan

macrumors 6502
Original poster
Jun 10, 2003
296
1
United States
I'm having a great discussion with a friend about the following...

1. if x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.

2. Using your calculators on this one. 1/9=0.111...?. 2/9=0.222....?. 3/9=0.333...?. 4,5,6,7,8 follow the same pattern. What, then is 9/9? It is 0.999...infinity.

Do you think those are valid? I think they are and thinking in infinite terms 1.999...infinity is equal to 2.

My friend thinks he can add any number to infinity because he just can.

Food for thought. Any you want to add?
 
I won't get into proofs here, but I am certain someone on this board may feel the urge.

Basically 2.00000000 isn't equal to 1.999999999 to infinity... they are very close, but not equal. I think your first argument has a flaw, but I think you are breaking some proof law I don't know about.

If your calculator is indicating anything different there might be a flaw in its logic.
 
you have to be careful when doing arithmetics with infinite decimals...

that said, i think mathematically, 2 is equal to 1.99999.... depends on the definition of equal. i believe a little more technical way to say two numbers, say N1 and N2, are equal is to show that no matter how small of a number you choose, you can make the difference of N1 and N2 smaller than that number. and in this case, by extending the decimals far enough, you can make the difference between 2 and 1.9999... smaller than any possible number.

math is kinda convoluted... it's been a while since i've had to think about this stuff. (i majored in math in college, but that was more than 5 years ago.) any active mathematicians out there?
 
Re: 2=1.999...infinity! What are your thoughts?

Originally posted by sjjordan
I'm having a great discussion with a friend about the following...

1. if x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.
Click to expand...

x=1.99... is an irrational number, a number that cannot be expressed as a fraction. Thus cannot be used as a true value for x.

On number 2 you 1/9 = .11.. ets, they are rational numbers. Of course 9/9 =1 The division you use shows the standard conversion that happens in division. It looks cool though and did get my rather slow brain working again. Thanks, Now to ponder the meaning of life.
 
You're saying that 2 == 1.999...
That is incorrect (but approximately true).
What is more correct is to define 1.999... as being (2 - 1/n), where n tends to infinity.

1/n does approach a limit (0), so
2 - 1/n approaches 2, for an arbitrarily large n.
(You could write it as:
2 - 1/n --> 2, n --> infinity)

You're discussing limits here, which is 1st year university mathematics.

The definition of a limit is:
"For any |1/n - 0| arbitrarily close to 0, there exists such an n"

(arbitrarily close ~= as close as you like)

-matt
 
Originally posted by jxyama
i think mathematically, 2 is equal to 1.99999.... depends on the definition of equal.
...
by extending the decimals far enough, you can make the difference between 2 and 1.9999... smaller than any possible number.
...
math is kinda convoluted...
Click to expand...

definition of equal? smaller than any possible number? math is convoluted?

1, equal means equal.
2, impossible, there is always something smaller.
3, math is the language of pure and universal logic.
 
You falsey assume that 10x - x = 9x.

While this may work for real/complex number, it won't for infinite numbers.

If you multiply both 10x and x by the same "size" infinity then you will get:

x times infinity = 199999...99999.0
10x time inifinity = 1999999...99990.0

You can't say that x infinity minus 10x infinity is 18 infinity because as you get to infinity the 10x will be just a bit smaller.

I guess this doesn't make any sense, but infinite numbers never do :)
 
Originally posted by mangoduck
definition of equal? smaller than any possible number? math is convoluted?

1, equal means equal.
2, impossible, there is always something smaller.
3, math is the language of pure and universal logic.
Click to expand...

1. what does "equal means equal" mean? that doesn't define anything!

2. many definitions within math use limits. the point is, the difference can be made smaller than any arbituary (but fixed) number. i didn't say that the difference is the smallest number.

3. you left out an obvious but important adjective: math is the language of pure and universal human logic.
 
Math itself is not a problem. However, decimals are only measured in powers of 10, therefor fractions that do not divide into powers of 10 cannot always be displayed completely accurately in decimals. I'm pretty sure that there's no actual way to get to .99999_, as all multiplications, additions, etc. of decimals/fractions would come out to 1 instead of .99999_



Edited for grammar
 
1.99999999999 does not equal 2.

otherwise it would be 2.

calculators just do that crap because they can't handle big numbers.

imagine your calculator trying to do the sine of .4444444444444.

it would burst into flames.
 
math major chiming in

1.999... is the same number as 2, just written differently. Take any positive number and the absolute difference between 2 and 1.999... (that is, |2 - 1.999...|) will be less than that number, which can be true only if they are the same number. Some numbers have more than one decimal expansion.

Put it differently, The decimal expansion 1.999... represents a limit of a sequence of numbers that each can be represented as a finite decimal expansion. 2 is also a limit of that sequence, and since a convergent sequence can only have one limit, the two numbers are the same.
 
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Re: math major chiming in

Originally posted by betta
1.999... is the same number as 2, just written differently. Take any positive number and the absolute difference between 2 and 1.999... (that is, |2 - 1.999...|) will be less than that number, which can be true only if they are the same number. Some numbers have more than one decimal expansion.

Put it differently, The decimal expansion 1.999... represents a limit of a sequence of numbers that each can be represented as a finite decimal expansion. 2 is also a limit of that sequence, and since a convergent sequence can only have one limit, the two numbers are the same.
Click to expand...

As I studied math, I will only say that all the guys who said the above are right.
 
Aah, but 1 = 2!
How?

Well using the rules for algebra do the same thing to both sides (divide by zero).

1 = 2

: 1/0 = 2/0

so therefore infintity = infinity
 
1.99999... equals 2

The notation 1.9999... is defined as the limit of the series. This limit is 2. Period.

Please google for "p-adic numbers" or "p-adic expansion".
 
i believe the explanation that those two numbers are the "same" just different representations, as posted by others. that's something i remember from my days in math.

all this discussion about 2 = 1.999... reminded me of one of the most important things i learned in physics...

i understand that some of us, not having studied math vigorously, would be uncomfortable that the number "two" can have more than one representation...

the story i remember is the question of "is light wave or particle?" people have problems because they consider wave and particle to be two distinct things and can't imagine things that are both... when in fact, light is both, in the sense it will display wave like properties under certain conditions and particle like properties under another.

we are most comfortable with the number "two" represented as 2. however, that doesn't mean the number "two" is restricted to being represented by the symbol "2"... no?
 
I did a science fair project on the particle-wave theory of light. I had a pretty nifty demonstration with flashlights, a cookie sheet, and a cardboard box with a few holes cut into it. Funny thing was, I just did it to be lazy since it was such an easy project and I hated doing science fair stuff and it somehow impressed my teacher and I got stuck going to the state science fair and wasting 5 days bored out of my mind.
 
Originally posted by jxyama
i believe the explanation that those two numbers are the "same" just different representations, as posted by others. that's something i remember from my days in math.

all this discussion about 2 = 1.999... reminded me of one of the most important things i learned in physics...

i understand that some of us, not having studied math vigorously, would be uncomfortable that the number "two" can have more than one representation...

the story i remember is the question of "is light wave or particle?" people have problems because they consider wave and particle to be two distinct things and can't imagine things that are both... when in fact, light is both, in the sense it will display wave like properties under certain conditions and particle like properties under another.

we are most comfortable with the number "two" represented as 2. however, that doesn't mean the number "two" is restricted to being represented by the symbol "2"... no?
Click to expand...

But light acting as a particle or wave doesn't mean that it is a particle and a wave, but just that we are unable to describe light properly using these crude terms.

Similarly, I agree the limit of 1.999... is 2. But 1.999... is an irrational number, 2.0 is not. So under some systems of math, they may be considered equivalent. But other terminologies have been developed to describe the differences between these two numbers. Ancient humans probably counted 1, 2, many. So in their view, 10 deer and 12 deer are the same. But we have better systems for counting, and so they are different. Likewise with 1.999... and 2.
 
Originally posted by Dros
But light acting as a particle or wave doesn't mean that it is a particle and a wave, but just that we are unable to describe light properly using these crude terms.

Similarly, I agree the limit of 1.999... is 2. But 1.999... is an irrational number, 2.0 is not. So under some systems of math, they may be considered equivalent. But other terminologies have been developed to describe the differences between these two numbers. Ancient humans probably counted 1, 2, many. So in their view, 10 deer and 12 deer are the same. But we have better systems for counting, and so they are different. Likewise with 1.999... and 2.
Click to expand...

No. To make it easy, just look at 0,999... = 9/9 = 1. So it's the same, there's no difference, just forget about it.

If you want to know more, look here:
http://mathquest.com/dr.math/faq/faq.0.9999.html
 
Originally posted by Dros
But light acting as a particle or wave doesn't mean that it is a particle and a wave, but just that we are unable to describe light properly using these crude terms.
Click to expand...

exactly. so 1.999... may not look like "2" that we are used to but it's equal to two. the symbol "2" is perhaps too crude to adaquately describe all the properties of the number "two"?

math can't change its definitions... either 1.999... is equal to 2 or it is not. "equivalent" doesn't really make much sense. what does it mean? it's equal sometimes, but not all the time? that's too ambiguous for math...

who says 1.99999... is not a rational number? 0.1111... is a rational number because it's 1/9. so it's 1 + 9*0.1111..., sounds perfectly rational to me.
 
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