It would be interesting to know whether Green or Yellow is what the eye actually sees.
As I understand things, the reason that television used red GREEN and blue instead of yellow, was because phosphors that glow green were much cheaper than phosphors that glow yellow.
I don't know much about human vision. So I don't actually know whether the eye sees yellow or whether it sees green, or whether color perception works entirely differently.
The whole purpose of the elipsis notation is so that you can express any rational number in decimal form.
(Rational number being any number which can be expressed as the ration of two integers.)
A rational number such as 1/3 could not be expressed exactly in decimal form without the invention of the elipsis notation. That is the whole point of the special notation. (Or writing a horizontal bar over the trailing repeating digits.)
How can you write the statement I quoted above, inclucing the elipsis, when the whole purpose of that notation is to provide a way so that anyrational number could be written conveniently in decimal form?
0.999999.... is also a rational number. Expressable as the ratio of two integers. 1/1. See all of the proofs I have given in this topic. See the references I have linked. Google for it yourself (although I included a Google set of results in one of my replies in this thread).
If 1/3 is exactly 0.33333.... (the whole purpose of the invention of the special notation) then it stands to reason that 3/3 is exactly expressable as 0.99999....
And note that I did not write some long string of 9's and stop. I used a recognized mathematical notation to express that the nines are infinite. Not part way to infinity. Not somewhere along the way, in your mind where you are thinking of some "last 9". But all the way to infinity. Stop thinking of "one the way" to infinity, and actually get there. Stop thinking of what a function approaches, and think of what it is when it finally gets there at infintiy. (Sort of like the guy who will never run into the wall because he will always have to get halfway there first, and it will take an infinite number of halfways, therefore he will never hit the wall. Therefore the repeating nines will never equal 1.)
So you do not agree that 10 times 0.99999... is 9.999....?
Then there is not much else to say.
The whole problem lies between 1 and 0.999...
The first number is definatly a unit, whereas the last number is constantly being defined.
Not true. The last number has an infinite number of nines. Not some continually expanding number. Quit thinking about getting closer and closer to infinity, and just get there. Sort of like the man who will never crash into the wall because he will always just get closer and closer to it. Just get to infinity. The purpose of the elipsis is to indicate an infinite number of repititions, not an ever expanding number. But an Infinite number. Complete.
The correct way to subtract the two...
Do you really expect me to buy that? The correct subtraction is...
Term 1: 9 + 0.9 + 0.09 + 0.009.....
Term 2: 0 + 0.9 + 0.09 + 0.009.....
Now take term 1 minus term 2. The infinite nines drop off. What is so difficult to grasp about that? Is that your pre-formed conclusion that 0.9999... is not 1 getting in your way? Refusal to face a simple conclusion.
If you don't agree that 10 times 0.999... is 9.9999... then there really isn't any more to say.
I can accept that Yellow might be a primary color and Green is a mixture, based on an explanation of how our eyes work.
But let me repeat. I am NOT, NOT, NOT talking about subtractive colors. CMYK. I am talking about ADDITIVE colors with LIGHT. RGB.
Equal parts of RED LIGHT, ADDED, to equal parts of GREEN LIGHT give you YELLOW light.
Get the subtractive colors thing out of your head. I'm talking about additive colors.
Let me repeat that. I'm talking about additive colors. If you add equal parts of red light, and equal parts of green light, you get yellow light. And I did say add, not subtract.
Now, is there any valid reason why Yellow is considered a primary color instead of green? You can mix equal amounts of red light and green light to get yellow light. So Yellow is a mixture, and R, G and B are the primary colors. Why is this not valid.
I would most definitely be convinced by an explanation that has something to do with our eyes responding only to R, Y and B light, instead of R, G and B.
Let me just add a note here, for those who might be confused. I am talking about an ADDITIVE model of colors, involving the use of light. Please don't consider other color models or subtractive colors, because these are not what I'm talking about.
Let's talk only about the medium of light, rather than printing inks.
Red, Green and Blue can produce any color like Yellow.
Red, Yellow and Blue can produce any color, like Green.
Question: is there some fundamental reason that people think of Yellow as a primary color and Green as a mixture of Yellow and Blue? Isn't it equally valid to think of Red, Green and Blue as primary colors, and Yellow as a mixture of Red and Green as I described above? Or am I missing something that somehow makes "Yellow" primary and "Green" not primary? Why can't Green be primary, and Yellow be the mixture color?
Well, if I can't assign a rational number like 1 (which is 0.99999....) or 1/3 (which is 0.33333...) then I how I can even assign an irrational, such as...
Let x = pi
or... Let x = sqrt( 2 )
At least rational numbers can be expressed as the ratio of two integers (1/1) or (1/3). Irrational numbers cannot.
After googling, there seems to be plenty of argument that 0.9999... is equal to 1. Therefore, 0.33333... is equal to 1/3.
I'll use the same proofs...
If there is a difference between 0.33333... and 1/3, then there must be a number in between the two. Please identify this number.
Would you agree that 0.333333... is equal to 0.3 + 0.033333...?
Is it also equal to 0.33 + 0.0033333...?
Is it also equal to 0.333 + 0.00033333...?
But no matter how small a fraction I add, I get a number larger than 1/3.
0.3 + 0.1 + 0.033333.... > 1/3
0.33 + 0.01 + 0.00333333.... > 1/3
0.333 + 0.001 + 0.00033333..... > 1/3
So no matter how small a fraction I add to 0.33333... I get a number larger than 1/3.
How could LED light ever be anything near close to natural light? In fact, if anything, it would be much further away from natural light than even incandescent or fluorescent.
At least incandescent emits a range of color frequencies.
Aren't LED's highly monochromatic? (i.e. an extremely narrow slit of color on the spectrum?)
So even if you combine a red, green and yellow LED to get "white" light, you really have three very narrow slits of color on a spectroscope. Not a "rainbow" of color, as in natural sunlight. Take your R+G+B "white" light from three LED's and shine it through a prism. Wouldn't you just get three narrow slits, one red, one green, one blue on the wall? Not a rainbow?
Green is a mix of blue and yellow. Green is a secondary color, not a primary.
Maybe I'm missing something. Why is it any less valid for me to say...
Yellow is a mix of Green and Red. Yellow is a secondary color, not a primary.
You get pure yellow with equal amounts of red and green. Just try a color like RGB( 255, 255, 0 ). You get orange with 2 parts red to 1 part green. RGB( 255, 127, 0 ).
So what makes Yellow a primary color. In fact, a color wheel from HSB color notation would put red at hue 0.0 and 1.0, green at 0.3333 and blue at 0.6666. So R, G, and B would seem like obvious primarys.
(Yes, I was taught red, yellow, blue primary colors in grade school as well. But green phosphors in crt's are much cheaper than yellow phosphors. Any why is Green any less correct as a primary color?)
Here's another enlightening argument from
Burger
. I never met anybody who thought 0.999... greater than 1. So, if it's not equal to 1, it is less than 1. Let's think of the average of 0.999... and 1. As an average of any two numbers, it's greater than 0.999... but is less than 1. Can we determine its decimal expansion? Say, what is its integer part. Since it's less than 1 but greater than 0 < 0.999..., its integer part is bound to be 0. What about its first decimal digit. Since 0.9 < 0.999..., that digit must be 9. And the second one? Since 0.99 < 0.999..., the second digit must also be 9. And so on. It appears like the average of 0.999... and 1 is 0.999... If the latter is denoted as X, (X + 1)/2 = X. X + 1 = 2X. X = 1. The conclusion can't be helped.
No matter how small a fraction 0.0000.......001 you add, you get a number greater than 1.
As I have already stated here. If 0.99999... is less than 1, then please state a number in between the two. For any number A < B, then there exists a number Z such that A < Z < B. Please state Z for me.
there is always going to be a remainder - even with an infinite number of decimals.
Um, no.
When you think you have a remainder, then there is simply NOT an infinite number of digits. You have stopped at less than infinity. You have a 0.3333 that is NOT recurring. It is 0.333 recurring that we are talking about here. You are talking about 0.3333 with some fixed finite number of 3's.
When you have some remainder, you have not finished yet. Add an additional infinite number of 3's and keep dividing. Then you have 1/3.
This is like looking at an expansion of some function (cos or sin) or value such as pi. There are various expansions that will give you the value of pi. But only when you add an infinite number of terms to the expansion. Now, in implementing a calculator, you can do, say the first fifty terms of the expansion and have a good enough result. But not the real thing. No calculator does an infinite number of steps when you calculate, say, the cosine of some value. But at the same time, you don't see it stated that "the following series is approximately equal to pi", it is stated that "the following series is equal to pi". You just need to do the infinite number of steps. Sorry, I don't have an example of one handy.
Unless somewhere along the line you make the error of assuming that 0.9999... = 1, you cannot arrive at 9x = 9.
I never made such an assumption. Let's review my proof.
Let x = 0.99999.....
Now, don't you agree that 10x would be 9.999999..... ?
So far I am not assuming what you have said.
Now is it true that 9.9999.... minus 0.9999.... would be exactly 9? An infinite number of nines minus an infinite number of nines is zero.
But there is an even simpler proof that someone else mentioned. If there is a difference between 1 and 0.9999... then there must be some number in between these two. Would you be so kind as to tell me what what number is?
Slashdot Mirror
It would be interesting to know whether Green or Yellow is what the eye actually sees.
As I understand things, the reason that television used red GREEN and blue instead of yellow, was because phosphors that glow green were much cheaper than phosphors that glow yellow.
I don't know much about human vision. So I don't actually know whether the eye sees yellow or whether it sees green, or whether color perception works entirely differently.
because 1/3 is not equal to 0.3333...
Let me add one more thing to my other reply.
The whole purpose of the elipsis notation is so that you can express any rational number in decimal form.
(Rational number being any number which can be expressed as the ration of two integers.)
A rational number such as 1/3 could not be expressed exactly in decimal form without the invention of the elipsis notation. That is the whole point of the special notation. (Or writing a horizontal bar over the trailing repeating digits.)
How can you write the statement I quoted above, inclucing the elipsis, when the whole purpose of that notation is to provide a way so that any rational number could be written conveniently in decimal form?
0.999999.... is also a rational number. Expressable as the ratio of two integers. 1/1. See all of the proofs I have given in this topic. See the references I have linked. Google for it yourself (although I included a Google set of results in one of my replies in this thread).
If 1/3 is exactly 0.33333.... (the whole purpose of the invention of the special notation) then it stands to reason that 3/3 is exactly expressable as 0.99999....
And note that I did not write some long string of 9's and stop. I used a recognized mathematical notation to express that the nines are infinite. Not part way to infinity. Not somewhere along the way, in your mind where you are thinking of some "last 9". But all the way to infinity. Stop thinking of "one the way" to infinity, and actually get there. Stop thinking of what a function approaches, and think of what it is when it finally gets there at infintiy. (Sort of like the guy who will never run into the wall because he will always have to get halfway there first, and it will take an infinite number of halfways, therefore he will never hit the wall. Therefore the repeating nines will never equal 1.)
because 1/3 is not equal to 0.3333
But 1/3 is 0.3333... exactly. (Notice I put on elipsis after the last 3.) I could even correctly restate it that 1/3 is exactly equal to 0.3...
Too bad there isn't also a way to write a horizontal bar over the 3 on Slashdot.
So you do not agree that 10 times 0.99999... is 9.999....?
..... .....
Then there is not much else to say.
The whole problem lies between 1 and 0.999... The first number is definatly a unit, whereas the last number is constantly being defined.
Not true. The last number has an infinite number of nines. Not some continually expanding number. Quit thinking about getting closer and closer to infinity, and just get there. Sort of like the man who will never crash into the wall because he will always just get closer and closer to it. Just get to infinity. The purpose of the elipsis is to indicate an infinite number of repititions, not an ever expanding number. But an Infinite number. Complete.
The correct way to subtract the two...
Do you really expect me to buy that? The correct subtraction is...
Term 1: 9 + 0.9 + 0.09 + 0.009
Term 2: 0 + 0.9 + 0.09 + 0.009
Now take term 1 minus term 2. The infinite nines drop off. What is so difficult to grasp about that? Is that your pre-formed conclusion that 0.9999... is not 1 getting in your way? Refusal to face a simple conclusion.
If you don't agree that 10 times 0.999... is 9.9999... then there really isn't any more to say.
I can accept that Yellow might be a primary color and Green is a mixture, based on an explanation of how our eyes work.
But let me repeat. I am NOT, NOT, NOT talking about subtractive colors. CMYK. I am talking about ADDITIVE colors with LIGHT. RGB.
Equal parts of RED LIGHT, ADDED, to equal parts of GREEN LIGHT give you YELLOW light.
Get the subtractive colors thing out of your head. I'm talking about additive colors.
Let me repeat that. I'm talking about additive colors. If you add equal parts of red light, and equal parts of green light, you get yellow light. And I did say add, not subtract.
Now, is there any valid reason why Yellow is considered a primary color instead of green? You can mix equal amounts of red light and green light to get yellow light. So Yellow is a mixture, and R, G and B are the primary colors. Why is this not valid.
I would most definitely be convinced by an explanation that has something to do with our eyes responding only to R, Y and B light, instead of R, G and B.
Let me just add a note here, for those who might be confused. I am talking about an ADDITIVE model of colors, involving the use of light. Please don't consider other color models or subtractive colors, because these are not what I'm talking about.
my bad you are right 0.333... is rational and so is 0.999.... however i still think 0.999... does not equal 1. yet this is still a matter of debate.
0.999... is indeed rational. Expressable as the ratio of two integers. 1/1.
1. first you apply rational analysis to an irrational number but whatever.
0.9999... is not an irrational number. It is rational. That is, it can be expressed as the ratio of two integers.
For instance, 0.333333... is rational, because it is expressed as 1/3.
0.99999.... is rational because it is expressed as 1/1.
1/3 = 0.33333.....
Multiply both sides by 3, you get...
3/3 = 0.999999......
I got that.
Let's talk only about the medium of light, rather than printing inks.
Red, Green and Blue can produce any color like Yellow.
Red, Yellow and Blue can produce any color, like Green.
Question: is there some fundamental reason that people think of Yellow as a primary color and Green as a mixture of Yellow and Blue? Isn't it equally valid to think of Red, Green and Blue as primary colors, and Yellow as a mixture of Red and Green as I described above? Or am I missing something that somehow makes "Yellow" primary and "Green" not primary? Why can't Green be primary, and Yellow be the mixture color?
A non-complete can never be assigned
What is a non-complete? Like pi, sqrt(2), or 1/3. So you're saying that 1/3 cannot be assigned.
The whole point of the elipsis is to have a way to express a number such as 0.9999... which most definitely can be assigned.
Also see this.
Well, if I can't assign a rational number like 1 (which is 0.99999....) or 1/3 (which is 0.33333...) then I how I can even assign an irrational, such as...
Let x = pi
or... Let x = sqrt( 2 )
At least rational numbers can be expressed as the ratio of two integers (1/1) or (1/3). Irrational numbers cannot.
See these other proofs .
Please show me these mathematical texts.
Please see this .
After googling, there seems to be plenty of argument that 0.9999... is equal to 1. Therefore, 0.33333... is equal to 1/3.
I'll use the same proofs...
If there is a difference between 0.33333... and 1/3, then there must be a number in between the two. Please identify this number.
Would you agree that 0.333333... is equal to 0.3 + 0.033333...?
Is it also equal to 0.33 + 0.0033333...?
Is it also equal to 0.333 + 0.00033333...?
But no matter how small a fraction I add, I get a number larger than 1/3.
0.3 + 0.1 + 0.033333.... > 1/3
0.33 + 0.01 + 0.00333333.... > 1/3
0.333 + 0.001 + 0.00033333..... > 1/3
So no matter how small a fraction I add to 0.33333... I get a number larger than 1/3.
How could LED light ever be anything near close to natural light? In fact, if anything, it would be much further away from natural light than even incandescent or fluorescent.
At least incandescent emits a range of color frequencies.
Aren't LED's highly monochromatic? (i.e. an extremely narrow slit of color on the spectrum?)
So even if you combine a red, green and yellow LED to get "white" light, you really have three very narrow slits of color on a spectroscope. Not a "rainbow" of color, as in natural sunlight. Take your R+G+B "white" light from three LED's and shine it through a prism. Wouldn't you just get three narrow slits, one red, one green, one blue on the wall? Not a rainbow?
Green is a mix of blue and yellow. Green is a secondary color, not a primary.
Maybe I'm missing something. Why is it any less valid for me to say...
Yellow is a mix of Green and Red. Yellow is a secondary color, not a primary.
You get pure yellow with equal amounts of red and green. Just try a color like RGB( 255, 255, 0 ). You get orange with 2 parts red to 1 part green. RGB( 255, 127, 0 ).
So what makes Yellow a primary color. In fact, a color wheel from HSB color notation would put red at hue 0.0 and 1.0, green at 0.3333 and blue at 0.6666. So R, G, and B would seem like obvious primarys.
(Yes, I was taught red, yellow, blue primary colors in grade school as well. But green phosphors in crt's are much cheaper than yellow phosphors. Any why is Green any less correct as a primary color?)
Please see these other proofs .
Please see these other proofs .
Please see these other proofs .
Easy as can be. The average of those two numbers falls between them.
So what is that number?
From this web page, I got the following text....
Here's another enlightening argument from Burger . I never met anybody who thought 0.999... greater than 1. So, if it's not equal to 1, it is less than 1. Let's think of the average of 0.999... and 1. As an average of any two numbers, it's greater than 0.999... but is less than 1. Can we determine its decimal expansion? Say, what is its integer part. Since it's less than 1 but greater than 0 < 0.999..., its integer part is bound to be 0. What about its first decimal digit. Since 0.9 < 0.999..., that digit must be 9. And the second one? Since 0.99 < 0.999..., the second digit must also be 9. And so on. It appears like the average of 0.999... and 1 is 0.999... If the latter is denoted as X, (X + 1)/2 = X. X + 1 = 2X. X = 1. The conclusion can't be helped.
Please see these other proofs .
Please see these other proofs .
Here is another good proof after some Googling . It goes like this....
Do we agree that 0.99999..... is equal to 0.9 + 0.099999...?
If so, then do we also agree that 0.99999.... is equal to 0.99 + 0.00999999....?
Then we find that no matter small a fraction we add to 0.99999... that we get a number greater than 1.
0.9 + 0.1 + 0.099999... > 1
0.99 + 0.01 + 0.009999... > 1
0.999 + 0.001 + 0.0009999.... > 1
No matter how small a fraction 0.0000.......001 you add, you get a number greater than 1.
As I have already stated here. If 0.99999... is less than 1, then please state a number in between the two. For any number A < B, then there exists a number Z such that A < Z < B. Please state Z for me.
there is always going to be a remainder - even with an infinite number of decimals.
Um, no.
When you think you have a remainder, then there is simply NOT an infinite number of digits. You have stopped at less than infinity. You have a 0.3333 that is NOT recurring. It is 0.333 recurring that we are talking about here. You are talking about 0.3333 with some fixed finite number of 3's.
When you have some remainder, you have not finished yet. Add an additional infinite number of 3's and keep dividing. Then you have 1/3.
This is like looking at an expansion of some function (cos or sin) or value such as pi. There are various expansions that will give you the value of pi. But only when you add an infinite number of terms to the expansion. Now, in implementing a calculator, you can do, say the first fifty terms of the expansion and have a good enough result. But not the real thing. No calculator does an infinite number of steps when you calculate, say, the cosine of some value. But at the same time, you don't see it stated that "the following series is approximately equal to pi", it is stated that "the following series is equal to pi". You just need to do the infinite number of steps. Sorry, I don't have an example of one handy.
How does a mathematically incorrect assumption become insightful here on Slashdot?
This is not a mathematically incorrect assumption.
Let me try that again, this time, previewing first :_)
If A < B, then there must be some other number Z such that A < Z < B.
If 0.9999... is less than 1, then please name me a number that falls in between the two.
If A
If 0.99999... is truly less than 1, then please name me a number that falls in between these two.
Unless somewhere along the line you make the error of assuming that 0.9999... = 1, you cannot arrive at 9x = 9.
I never made such an assumption. Let's review my proof.
Let x = 0.99999.....
Now, don't you agree that 10x would be 9.999999..... ?
So far I am not assuming what you have said.
Now is it true that 9.9999.... minus 0.9999.... would be exactly 9? An infinite number of nines minus an infinite number of nines is zero.
But there is an even simpler proof that someone else mentioned. If there is a difference between 1 and 0.9999... then there must be some number in between these two. Would you be so kind as to tell me what what number is?