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0.9 repeater = 1

Vorpal said:
[...]

If they're shown something like this:
[qimg]http://media-2.web.britannica.com/eb-media/79/26979-004-CF3F4DA2.gif[/qimg]
I suspect that most people won't have any trouble "seeing" that 1/2+1/4+1/8+... = 1, and that the reaction of "but... infinity!" will be far less common. You can see the entire thing at once; the contrary reaction is irrelevant.

Yet this question is exactly alike. In fact the above is a geometric proof that the binary .111~ = 1. Cases like these are called geometric series, and one can make make a pretty picture for the decimal case of 9/10 + 9/100 + 9/1000 + ... just as well, although it won't be as nicely symmetrical as this one.

[...]
Click to expand...

This is great. Thanks for this.

Could I make a request to you or anyone who is able and willing?

I would like to see a graphical representation, despite it's possible asymmetry of 9/10 + 9/100 + 9/1000 + ...

Does someone have the ability to do this in a logical and clean looking way, visually? I would love to see it
 
Also,

Would it be fair to say the following:

If someone denies that .999... = 1, then they must also, to stay logically consistent, deny that say,

1/2 does not equal exactly .500... ?

Because it seems to me at that point, it becomes a physics argument.

If someone is arguing that upon some "terminal magnification" of anything, you will reach a perhaps jagged line, instead of a perfect division, then that's not a math argument, but more in the realm of physics, right?

And it's true that in real life, you cannot divide an apple into infinite tiny pieces of the apple. But pure mathematics tells us, that were that apple somehow divided into infinity pieces, each piece would be equal to zero units of the whole apple.

edit: and that's just bizarre to think about
 
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kevinquinnyo said:
This is great. Thanks for this.

Could I make a request to you or anyone who is able and willing?

I would like to see a graphical representation, despite it's possible asymmetry of 9/10 + 9/100 + 9/1000 + ...

Does someone have the ability to do this in a logical and clean looking way, visually? I would love to see it
Click to expand...

I guess what I'm really asking, is, geometrically, what is the most logical way of drawing this out on a sheet of paper?
 
kevinquinnyo said:
I guess what I'm really asking, is, geometrically, what is the most logical way of drawing this out on a sheet of paper?
Click to expand...

Since 10 cannot really be easily converted to a square I would advise using a rectangle of 2x5, where the individual blocks also have a 2:5 ratio.

Eta: Mmm, I'm not sure this is the right solution. Let me Photoshop something...

Eta 2: See attachments:

Eta 3: You want the ratio of the individual blocks to be the same as the ratio of total surface, so not like what I have created.
 

Attachments

  • 2x5-1.jpg
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  • 2x5-2.jpg
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kevinquinnyo said:
I guess what I'm really asking, is, geometrically, what is the most logical way of drawing this out on a sheet of paper?
Click to expand...
Take a piece of coarsely ruled graph paper, and draw a line around a 10x10 square. Let that square represent 1.0.

Let the top 9 rows of that 10x10 square represent 0.9. Cross them out, because our 0.9 has accounted for them. That leaves the bottom row, with 10 grid squares.

Let the first 9 grid squares of that bottom row represent 0.09. Cross them out too, because our 0.99=0.9+0.09 has accounted for them. That leaves one grid square in the bottom right corner.

Using a microscope, a ruler, and a very fine-tipped pen, mark a 10x10 grid on that remaining grid square. Repeating the process above will account for 0.9999=0.9+0.09+0.009+0.0009 of the original square. Now mark an even finer grid on the one remaining hundredth of the original bottom right square.

This process can be repeated as long as desired. If we regard the process of going from a 10x10 square to a 1x1 square as one step of the process, then each step adds two more digits to the right of the decimal point. Every point within the interior of the original square will be crossed out by some finite step of this process. (The bottom right point won't ever be crossed out by any finite step, but its singleton set has measure zero so we can disregard it. ;))
 
W.D.Clinger said:
Take a piece of coarsely ruled graph paper, and draw a line around a 10x10 square. Let that square represent 1.0.

Let the top 9 rows of that 10x10 square represent 0.9. Cross them out, because our 0.9 has accounted for them. That leaves the bottom row, with 10 grid squares.

Let the first 9 grid squares of that bottom row represent 0.09. Cross them out too, because our 0.99=0.9+0.09 has accounted for them. That leaves one grid square in the bottom right corner.

Using a microscope, a ruler, and a very fine-tipped pen, mark a 10x10 grid on that remaining grid square. Repeating the process above will account for 0.9999=0.9+0.09+0.009+0.0009 of the original square. Now mark an even finer grid on the one remaining hundredth of the original bottom right square.

This process can be repeated as long as desired. If we regard the process of going from a 10x10 square to a 1x1 square as one step of the process, then each step adds two more digits to the right of the decimal point. Every point within the interior of the original square will be crossed out by some finite step of this process. (The bottom right point won't ever be crossed out by any finite step, but its singleton set has measure zero so we can disregard it. ;))
Click to expand...

Or this ;)
 
I have not looked at this whole thread, so if this is redundant, sorry.
I see a problem here with the understanding of what .999... means. The ... indicates an infinity of nines, so it must be that .999... = 1 as follows.

Here is a simple demonstration:
Consider that 1/9 = .111...
so 9(1/9) = 9(.111...) = .999.... But we know that 9(1/9) = 1. There can be no other answer.
 
W.D.Clinger said:
Take a piece of coarsely ruled graph paper, and draw a line around a 10x10 square. Let that square represent 1.0.

Let the top 9 rows of that 10x10 square represent 0.9. Cross them out, because our 0.9 has accounted for them. That leaves the bottom row, with 10 grid squares.

Let the first 9 grid squares of that bottom row represent 0.09. Cross them out too, because our 0.99=0.9+0.09 has accounted for them. That leaves one grid square in the bottom right corner.

Using a microscope, a ruler, and a very fine-tipped pen, mark a 10x10 grid on that remaining grid square. Repeating the process above will account for 0.9999=0.9+0.09+0.009+0.0009 of the original square. Now mark an even finer grid on the one remaining hundredth of the original bottom right square.

This process can be repeated as long as desired. If we regard the process of going from a 10x10 square to a 1x1 square as one step of the process, then each step adds two more digits to the right of the decimal point. Every point within the interior of the original square will be crossed out by some finite step of this process. (The bottom right point won't ever be crossed out by any finite step, but its singleton set has measure zero so we can disregard it. ;))
Click to expand...

Awesome
 
Okay, following the insight of W.D, I have creatd an image that represents, geometrically/visually, that .999... = 1.

I know, I'm a dork.

Enjoy:

9repeaterfinal.png
 
No matter how many times this process is repeated, the total are = 1
Click to expand...

Uh ... I thought that you had to do this an infinite number of times? As long as you give me any finite number of times, i could tell you exactly how much smaller than 1 your result would be. Right?

(But then, these graphs don't really help my understanding all that much, because I still have to wrap my head around the idea of doing something an infinite number of times.)
 
Rasmus said:
Uh ... I thought that you had to do this an infinite number of times? As long as you give me any finite number of times, i could tell you exactly how much smaller than 1 your result would be. Right?

(But then, these graphs don't really help my understanding all that much, because I still have to wrap my head around the idea of doing something an infinite number of times.)
Click to expand...

Don't think of infinity, think of trends.

It's like how you get that original 0.333...

3 into one won't go. So include the next digit to get 10
3 into 10 goes three times, carry the one to the next digit to get 10
3 into 10 goes three times, carry the one to the next digit to get 10
3 into 10 goes three times, carry the one to the next digit to get 10
...

How many times do you have to do this before you realise it's never going to stop repeating?

Same thing with the difference between 0.999.... and 1.000...

To 1 DP the difference is 0.1
To 2 DP the difference is 0.01
To 3 DP the difference is 0.001

How many times do you need to do this before you realise that as you increase the number of decimal places you're considering the difference between the two numbers is approaching 0.000...
 
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Ocelot said:
Don't think of infinity, think of trends.
Click to expand...

Oh, I get it don't worry. I just don't think the image is helpful to me, personally - so my comments should be taken carefully and with a large grain of salt.
 
Rasmus said:
Oh, I get it don't worry. I just don't think the image is helpful to me, personally - so my comments should be taken carefully and with a large grain of salt.
Click to expand...

The picture was supposed to show that, no matter how many times you zoom in, the box is and was 1 from the beginning, so .999... is another way of saying 1.

I guess it's supposed to show that the failure of the viewer to magnify or physically write a number isn't a failure of the math.

If it weren't 1, but slightly less, then there would be a very small notch in the corner.
 
Perpetual Student said:
Those people are wrong!
Click to expand...
In many different ways. This guy stated in his introduction post that he isn't very good at math and physics, and then proceeded to spout nonsense in every single physics thread on that forum. I tried correcting him a few times, but he stood by his nonsense; he wouldn't learn.

Another guy on that forum is trying to prove that gravity is an illusion because E=0. E=0 is the basis of his 'theory'. At the same time, he maintains that Einstein was almost right by stating E=mc2. The square is wrong: E=mc is what it should be.
When I asked him if that meant that c=0, he said that light and energy are also illusions.

Right now he is arguing that things fall because of some Archimedes-like mechanism in the atmosphere. Also, the vacuum of space is pressing down on the atmosphere.

All this is presented in the form of word salad.
 
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Consider what we mean by .333... when we say it equals 1/3. We are taking the ratio of two natural numbers (in this case 1/3) and translating it to a series of fractions where the denominators are ascending powers of ten. First .3 means 3/10. Then .33 means 3/10 + 3/100. Continuing this way, .333 means 3/10 + 3/100 + 3/1,000. Each time we add a fraction in the form 3/10^n we get closer to 1/3. But the expression .333... means we never stop; i. e. we keep adding fractions an infinite number of times, so .333... = 1/3. If that is not convincing enough try to find a number that is between 1/3 and .333...!
 
Perpetual Student said:
Consider what we mean by .333... when we say it equals 1/3. We are taking the ratio of two natural numbers (in this case 1/3) and translating it to a series of fractions where the denominators are ascending powers of ten. First .3 means 3/10. Then .33 means 3/10 + 3/100. Continuing this way, .333 means 3/10 + 3/100 + 3/1,000. Each time we add a fraction in the form 3/10^n we get closer to 1/3. But the expression .333... means we never stop; i. e. we keep adding fractions an infinite number of times, so .333... = 1/3. If that is not convincing enough try to find a number that is between 1/3 and .333...!
Click to expand...


What's weird is that if someone denies that .999... = 1, but they also say that exactly 1 can never be achieved, just an approximation, then they aren't logically wrong. Whether that's true or not I leave to the physicists and mathematicians.

But if someone says, yes exactly 1 is a real number, that is perfectly 1, but if you divide it by 3, those thirds are merely approximations, then it's logically inconsistent.

Because the 1 is a perfect third of the number 3, by their own reasoning.
 
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