Got a tip for us?
Let us know
Become a MacRumors Supporter for $50/year with no ads, ability to filter front page stories, and private forums.
2=1.999...infinity! What are your thoughts?
- Thread starter sjjordan
- Start date
- Sort by reaction score
i live in a retirement area in northern california near silicon valley and a lot of the locals i know were in computer science and data processing...and there are tons of them i can ask so i will see who i can find...one of them is bound to have stayed current with math to know the answer to this...SO WHY do i pick retired computer scientists you say??
these days i am a computer techie and EVERY software side computer techie i have ever met in my life over the age of 50 has a math degree or were math majors in college and may have dropped out for one reason or another
...however, i do have one friend who is exactly 50 and he is a programmer who does not have a math degree...but an actual computer science degree back when nobody offered the degree as a stand alone subject and when just about nobody in the world had a grasp of why programming may be important or if programs/software could be a commodity...when he studied for his phd at MIT back in 1974, the school yanked the specific program he was in thinking there was really no need for his brand of programming and specialty on the grad level since it was hard enough just to find any entry level computer scientists to fill a program and keep it going financially...my friend had to leave MIT after 7 years of intense computer science study (being awarded an MS but not a phd) and re-entered graduate school at stanford later on in the 70s and got his phd there in 1982...he was very ahead of his time studying programming and computer science for programming's sake without a strong math background
what is fascinating is that MIT and Stanford had access to intel processors that did not officially exist yet and my friend really thought nothing of it since he did not, and still does not follow, commercial and industrial trends of computers...he still uses a 486 at home if that is any indication of what he thinks about endless comsumerism and having to keep up with the jonses
You are right
1.99.. does equal 2
1.99.. = 1.99..
1.99.. = 1+ .99..
1.99.. = 1 + 9 * .11..
1.99.. = 1 + 9 * 1/9
1.99.. = 1 + 1
1.99.. = 2
Wow,
That's a great way to logically express it.
Thanks!
May we all be enlightened to a fact that has no significance in our lives whatsoever.
That's a great way to logically express it.
Thanks!
May we all be enlightened to a fact that has no significance in our lives whatsoever.
Re: Re: 2=1.999...infinity! What are your thoughts?
I am sure someone has said this:
The answer to life is 42.
I am sure someone has said this:
The answer to life is 42.
for many, accepting that 1/9 truly = .11... is as difficult as accepting that 1.99... = 2; therefore, i'm not sure how the "proof" above is helpful. nevertheless 1/9 does = .11..., and 1.99... does = 2.
I would imagine it would be ok, just like simplifying variables, 9x divide by 9 gives you x. your not actually performing a math function with it. So its infinite characteristics remain unchanged just simplified.
Re: 2=1.999...infinity! What are your thoughts?
I just want to add that this is some of the most brain damaged logic ever.
1. Christ.
2. Super christ. Since when did calculators get infinite precision you dolt? I sincerely hope you are still in middle school or life is going to be tough.
I just want to add that this is some of the most brain damaged logic ever.
1. Christ.
2. Super christ. Since when did calculators get infinite precision you dolt? I sincerely hope you are still in middle school or life is going to be tough.
This math is insane.
You are saying .999 ... is the same is 9 * 1/9. That's not true. You are introducing some kinda laymen haze with this 1 + business.
9 * 1/9 = 1.
1 CLEARLY does not equal 0.99999...
This math is nothing short of ridiculous.
Let's take out the 1+ which does nothing.
.9999... = .9999...
.9999... = 9 * 1/9
.9999... = 1
That's what you are saying. Clearly that's nonsense.
You are replacing .9999... with 9 * .1111, which equals 1. You clearly cannot replace .9999... with 1.
I hope this ends the discussion.
ahhh, but you CAN replace .9999... with 1. That's the whole point, johnnowak. Think of it this way:
If .9999... is different than 1, then 1-.9999... would have to be something other than zero. But because .9999... means a decimal with an infinite number of nines after it, the only possible value for 1-.9999... would be .0000000....1 in other words, an infinite number of zeroes..... followed by a one. Clearly, this is absurd, and therefore 1-.999999... must be zero.
P.S.- Do not take this as a mathematical proof! This is just a way to explain the issue in a way that people can understand.
Furthermore: BY DEFINITION, 1/3 is equal to .3333bar, 2/3 is equal to .6666bar, and 3/3, or 1, is equal to .9999bar. BY DEFINITION. This is simply the case. It cannot be helped. I am truly sorry.
If .9999... is different than 1, then 1-.9999... would have to be something other than zero. But because .9999... means a decimal with an infinite number of nines after it, the only possible value for 1-.9999... would be .0000000....1 in other words, an infinite number of zeroes..... followed by a one. Clearly, this is absurd, and therefore 1-.999999... must be zero.
P.S.- Do not take this as a mathematical proof! This is just a way to explain the issue in a way that people can understand.
Furthermore: BY DEFINITION, 1/3 is equal to .3333bar, 2/3 is equal to .6666bar, and 3/3, or 1, is equal to .9999bar. BY DEFINITION. This is simply the case. It cannot be helped. I am truly sorry.
You cannot replace .9999... with 1!
1 - .9999.. = 1/infinity
NOT ZERO.
You can have your own feelings about what is absurd, but I'm guessing you didn't exactly max out on math classes in college.
You thinking 1 minus a number less than 1 is 0 is what's absurd to me.
1 - .9999.. = 1/infinity
NOT ZERO.
You can have your own feelings about what is absurd, but I'm guessing you didn't exactly max out on math classes in college.
You thinking 1 minus a number less than 1 is 0 is what's absurd to me.
Johnnowak,
I am currently in college, and I am going to MAJOR IN MATHEMATICS. I am really sorry to repeat myself, but BY DEFINITION: 1 is equal to .9999bar. As much as you would like this not to be the case, it still is. There is nothing you can do to change that FACT. Infinity is indeed confusing, but that is no reason to start throwing insults, especially if you are ENTIRELY WRONG. It will just make you look stupid. I'm sure you are a very intelligent human being, but at this point you need to cut your losses and realize that you are wrong. Sorry.
And yes, thinking 1 minus a number less than one is equal to zero WOULD be absurd. But .99999bar IS NOT LESS THAN 1, IT IS 1!!!!
I am currently in college, and I am going to MAJOR IN MATHEMATICS. I am really sorry to repeat myself, but BY DEFINITION: 1 is equal to .9999bar. As much as you would like this not to be the case, it still is. There is nothing you can do to change that FACT. Infinity is indeed confusing, but that is no reason to start throwing insults, especially if you are ENTIRELY WRONG. It will just make you look stupid. I'm sure you are a very intelligent human being, but at this point you need to cut your losses and realize that you are wrong. Sorry.
And yes, thinking 1 minus a number less than one is equal to zero WOULD be absurd. But .99999bar IS NOT LESS THAN 1, IT IS 1!!!!
It is true that 0.9 + 0.09 + 0.009 etc does equal 1. However the method in the original post for proving this is nonsense.
I do apologize though for assuming you were talking crazy talk. I'm a math major as well, and have often gotten into some interesting discussions when infinity came up. I feel that the way they work conceptually and mathematically are different, at least going on my intuitive sense of what infinity is.
But yes, if there is an idiot in this discussion, it would be me. Got a bit carried away.
I do apologize though for assuming you were talking crazy talk. I'm a math major as well, and have often gotten into some interesting discussions when infinity came up.
I feel that the way they work conceptually and mathematically are different, at least going on my intuitive sense of what infinity is.But yes, if there is an idiot in this discussion, it would be me. Got a bit carried away.
I must admit that when it comes to infinity, I have trouble wrapping my mind around things. I don't think infinity is an easy thing for the human mind to comprehend. I would also agree that there is a difference between mathematical and conceptual reasoning, and my initial post was indeed more conceptual than mathematical. I felt that I could explain it better if I pointed out conceptually why it makes sense instead of going into a mathematical proof. Also, sorry I got a bit preachy there at the end!
You're a fine human being. I haven't dealt with infinity for awhile, so I instantly snapped into conceptual reasoning mode instead of recalling basic math facts.
Next time you're in New York we'll go out for coffee.
I haven't dealt with infinity for awhile, so I instantly snapped into conceptual reasoning mode instead of recalling basic math facts.Next time you're in New York we'll go out for coffee.
haha, Good idea. But if we're going out for COFFEE, we should do it in my home city of Seattle! The single most coffee obsessed city in the entire world. : )
hmm...
alright... lets see if i cant explain this somewhat
x = 1.999...
x = 1 + .999...
now lets just work with .999...
in order to complete this problem properly we must understand that
1/(1-x) = 1 + x + x^2 + x^3 + ...
proof:
s = 1 + x + x^2 + x^3 + ...
multiply both sides by x
sx = x + x^2 + x^3 + ...
now subtract the bottom line from the top line
s - sx = 1
s (1 - x) = 1
1 / (1 - x) = s
therefor
1 / (1 - x) = 1 + x + x^2 + x^3 + ...
alright, now that we have that we can apply it to this problem
lets break down the decimal expansion of .999...
.999... = (9/10) + (9/10^2) + (9/10^3) + ...
then we can factor out 9/10 from the right side
.999... = (9/10) (1 + (1/10) + (1/10^2) + (1/10^3) + ...)
we can now apply our understanding of the equation:
1 / (1 - x) = 1 + x + x^2 + x^3 + ...
1 / (1 - (1/10)) = (1 + (1/10) + (1/10^2) + (1/10^3) + ...)
and finally we can subsitute in to get:
.999... = (9/10)(1 / (1 - (1/10)))
if we simplify this we get
.999... = (9/10)(10/9)
therefor
.999... = 1
---------------
and in the case of x = 1.999...
it would seem that we could do the following now that we know
.999... = 1
x = 1 + .999...
x = 1 + 1
x = 2
1.999... = 2
-----------------
is this a bit more convincing?
alright... lets see if i cant explain this somewhat
x = 1.999...
x = 1 + .999...
now lets just work with .999...
in order to complete this problem properly we must understand that
1/(1-x) = 1 + x + x^2 + x^3 + ...
proof:
s = 1 + x + x^2 + x^3 + ...
multiply both sides by x
sx = x + x^2 + x^3 + ...
now subtract the bottom line from the top line
s - sx = 1
s (1 - x) = 1
1 / (1 - x) = s
therefor
1 / (1 - x) = 1 + x + x^2 + x^3 + ...
alright, now that we have that we can apply it to this problem
lets break down the decimal expansion of .999...
.999... = (9/10) + (9/10^2) + (9/10^3) + ...
then we can factor out 9/10 from the right side
.999... = (9/10) (1 + (1/10) + (1/10^2) + (1/10^3) + ...)
we can now apply our understanding of the equation:
1 / (1 - x) = 1 + x + x^2 + x^3 + ...
1 / (1 - (1/10)) = (1 + (1/10) + (1/10^2) + (1/10^3) + ...)
and finally we can subsitute in to get:
.999... = (9/10)(1 / (1 - (1/10)))
if we simplify this we get
.999... = (9/10)(10/9)
therefor
.999... = 1
---------------
and in the case of x = 1.999...
it would seem that we could do the following now that we know
.999... = 1
x = 1 + .999...
x = 1 + 1
x = 2
1.999... = 2
-----------------
is this a bit more convincing?
johnnowak said:
1/3 = .333333333333...
2/3 = .666666666666...
1/3 + 2/3 = .999999999999...
1/3 + 2/3 = 3/3 = 1
.999999999999... = 1
I add my confirmation to those above. Yes, .999... is exactly the same number as 1. They are two notations for the same value, just as all of these are notations for the number 13:
13
15 (base 8)
XIII
9+4
13
15 (base 8)
XIII
9+4
Doctor Q said:I add my confirmation to those above. Yes, .999... is exactly the same number as 1. They are two notations for the same value, just as all of these are notations for the number 13:
13
15 (base 8)
XIII
9+4
and, of course, 12.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 (to infinity and beyond!)
I think people get bogged down in the semantics of the whole thing.
1/∞ approaches but never reaches 0 - its an infinitely small number...so, for standard math, you have to use the limit of 0 to solve the problem and have a usable answer.
If your solving proofs, you have forms for representing this, so I'd say 1.999...∞ approximately equals 2, etc. in most other cases.
D
1/∞ approaches but never reaches 0 - its an infinitely small number...so, for standard math, you have to use the limit of 0 to solve the problem and have a usable answer.
If your solving proofs, you have forms for representing this, so I'd say 1.999...∞ approximately equals 2, etc. in most other cases.
D
Jeez the last 1.99999∞ pages just make me feel stupider than I thought I was!
It reminds me of a time i was surveying a property at work with a lazer distance measurer. All was going well until we got to a bath room with mirror on every wall the lazer went crazy and we recorded the rooms area as ∞. Needless to say the valuation was a little on the high side!
It reminds me of a time i was surveying a property at work with a lazer distance measurer. All was going well until we got to a bath room with mirror on every wall the lazer went crazy and we recorded the rooms area as ∞. Needless to say the valuation was a little on the high side!