Looking at the graph at the bottom of the first link, is always how I had pictured .999... in my mind. That is how I can conceive .999... becoming 1. The resolution of the infinite 9's becomes so small that it becomes 1.
My question is, why does .999... become 1, and .333... never be anything more that .333... ?
Or how about this.
In the equation 2+1-1=2 You can reverse the last 2 calculations, and still get the same answer 2-1+1=2, but here you can't:
Yes, I took algebra, but rarely have to use it. So it is very rusty. That's about as far as I got, as I had a rough childhood, and had to drop out and start working. So I do appreciate everything, that everyone is saying to me here. It's not like I don't think about mathematical concept's all the time. It is my favorite subject. I use math immensely in my work, and my hobbies, but not at calculus levels. I have been thinking lately of going to college, to get an engineering degree, it's never too late.
I invented parts of trigonometry out of necessity, because I am a master sheet metal layout mechanic. At one time, I needed to figure out what the other sides, and angles of a triangle would be, if I only knew one angle, and one length.
I had invented some formulas, but I can't seem to find them. I later learned that it was similar to sin, and cos.
Sheet metal fabrication is all about triangulation, and after doing for so many years, you gain a different perspective on the world. There is nothing I couldn't break down into a 2D pattern. I could cover you in metal.
Now, about the formula. If that formula works for 10x, shouldn't it work for 2x? also shouldn't the result be similar when we use .333...?
Why does .999... resolve to a 1, and .333... never resolve?
In 0.999 we don't lose the 9/1000. All that happens is that the 1000ths column becomes 0 because the digits are shifted up one column (in base 10). when there are an infinite number of 9s we don't put a zero at the end, but the essential operation (shifting the digits up a column) can still happen. This is part and parcel of the number system.
In wikipedia, what I am saying is considered wrong, but worthy of mention.
quote:Some argue that, in the second step of the equation given above, 10x is 9.999...0 and not 9.999... but this is not the case: the right-hand side does not terminate (it is recurring) and so there is no end to which a zero can be appended.
You cannot prove infinity. I am 40 years old, and for the first half of my life, I believed infinity could exist. For the second half, I started to doubt it. There is a possibility that infinity does not exist.
It sounds like you are confusing concept with the universe. Infinity is a mathematical concept, just as a point and a line are concepts. A line is defined as having two dimensions. It has no width. You won't find actual lines out in the world that have no width. They don't exist. As far as I know at this point, well, at least for the known universe matter is assummed to be finite. There is a very large number of electrons but not an infinite number.
But we are talking about mathematics. Do you accept that between 0 and 100 there are an infinite number of numbers (you can call them points as well) and that between .99999 and 1 there are also and infinite number of numbers? Infinite is not a number it means that no matter how far you count, or how many times you divide the line you can still keep counting, still keep dividing, there is no end to it. Now in the universe there may be a limit to how small you can make a division but we are talking mathematics here, pure concept.
ABE: I'm sure they prove infinity in number theory every day. Some one here can do this better than I but the proof is along the lines of pick some number as large as you want, call it n. You claim there is no larger number. But by adding 1 to it we have n+1>1. If you say that is the largest well we make another number by adding 1 to that.
Do you accept that between 0 and 100 there are an infinite number of numbers
I do not accept it, or not accept it. I think about the possibilities.
Infinite is not a number it means that no matter how far you count, or how many times you divide the line you can still keep counting, still keep dividing, there is no end to it. Now in the universe there may be a limit to how small you can make a division but we are talking mathematics here, pure concept.
A concept that has no end. So if the last number of a infinite numver has to change, then we have a problem houston.
So if the last number of a infinite numver has to change, then we have a problem houston.
Two errors here. First there is no such thing as an infinite number. There are infinite sets, infinite series, etc. But infinity is not a number.
Two the last number of an infinite number doesn't make sense. If you are referring to the last digit of a decimal representation of pi for example there is no last digit. There is not a last digit to .999... If there were a last digit it wouldn't be an infinitely repeating decimal, it wouldn't be an infinite series, it would be finite, that is to say countable.
So there is no last digit, so the last digit, which never existed in the first place, can't change. There is no problem. The calculus works. You can cross a room.
Why does .999... resolve to a 1, and .333... never resolve?
They both resolve.
.9 is not equal to 1. .99 is not equal to 1 but it is closer. .999 is not equal to 1 but it is closer yet. .9999 is not equal to 1 but it is still closer. .999999999999999999 is not equal to 1 but it is very close indeed.
.9... (the ... means an endless row of 9's. We need to be VERY careful with our notation) IS equal to 1 the difference is now zero.
.3 is not equal to 1/3 .33 is not equal to 1/3 but is closer. .333 is not equal fo 1/3 but it is pretty close.
.3... is equal to 1/3
You may think there is a diffence because the first case is a whole number (1) and the second is a fraction (1/3). That doesn't make a difference to what we are talking about.