Strictly speaking "decimal" notation refers only to base 10. (deci-, one tenth; Latin root decimus, tenth.) Just call base 10 "decimal" and base 3 "ternary".
In decimal, 3 x 0.333... = 0.999
In ternary, 3 x 0.1 = 1
1 in ternary = 1 in decimal = 0.999... in decimal (= 0.111... in binary, = 0.FFF... in hexadecimal, etc.)
0.1111... Is not 1/9. It is an infinitesimal less.
At a party, there'll be people who already know each other and people who don't.
What is the minimum number of people required at the party to be 100% sure that there will EITHER be three people who already knew each other OR three people who didn't know each other?
Put another way: you connect dots with either a red or a blue line. How many dots do you need there to be in order to be 100% sure that there will be either a red OR a blue triangle?
6
Nope. 0.999... = 1. It is 1. It is another way of writing 1, just as 2-1 or 5/5 are. They all sit at 1; they all are 1.
0.0000...1 on the other hand is not just meaningless, it is impossible to define. The "..." means an infinite number of zeroes. Infinite. Zeroes which never end. There is no last 1, because there is no last zero for it to go after.
If you say that 0.0..01 is impossible to define, then how can you be so confident with (0.99..)^2=0.99...9800...01
A rational number expressed in another base will still be either terminating or repeating, no?
And you can prove that .1111... is 1/9, the same way he proved the number was rational earlier.
x = .11111...
100x = 11.111111...
99x = 11.111111-1.111111 (as they both have infinite decimals, the decimals cancel).
99x = 11
x = 1/9
Nope. 0.999... = 1. It is 1. It is another way of writing 1, just as 2-1 or 5/5 are. They all sit at 1; they all are 1.
0.0000...1 on the other hand is not just meaningless, it is impossible to define. The "..." means an infinite number of zeroes. Infinite. Zeroes which never end. There is no last 1, because there is no last zero for it to go after.
If you say that 0.0..01 is impossible to define, then how can you be so confident with (0.99..)^2=0.99...9800...01
Very simple. See that 0.999...9800...01? Anything past that first set of dots is meaningless, since there is no last 9 behind which the 800whatever goes. So simply, 0.9999...^2 = 0.999... = 1.
The problem here is how you think about infinities. You can't ask what infinity plus 1 is, it makes no sense. Much as you can't say "there's an infinite chain of 9s, after which there's an 8 then infinite chain of zeroes..." There is no "after" the infinite 9s, they're infinite.
Same for the 0.000...01, there is no last zero, so there is no 1 "after it". There's an infinite number of zeroes, and no such thing as an "after". And just because a series (or representation) is infinite does not mean it cannot hold a well-defined finite (even rational) value. 1.000... is also 1, is it not? (With nothing "after" the zeroes - there is no "after".)
0.333... is 1/3, wouldn't you say?
Also please don't argue that infinity plus one or infinity plus infinity exists or what have you. I know the basic properties of infinite sets at the very least (set theory's very irritating) and can say that can of worms is not to be opened to argue this.
Also please don't argue that infinity plus one or infinity plus infinity exists or what have you.
Meanwhile, in some other number system, infinity + 1 is crying...
Anyway. Understanding 0.9999... = 1 requires some understanding of the real number system (read: the numbers you usually find) itself, that there is no infinitesimal number (read: number that is so small but still more than 0) in the real number system. And that's handled in a fairly advanced topic (called real analysis). (You can create a system containing infinitesimals, and they will work differently from the usual numbers that you know, but they do work.)
I refuse to go anywhere near hyperreals and their ilk. Why would you even need those to prove 0.999... = 1. That's like using a steamroller to iron clothes. Plus that particular steamroller is extremely hard to use.
Yeah, 0.999... = 1 so it's not a big deal when you start with x=0.999... and end up with x=1.
1/9 = 0.1111...
4/9 = 0.4444...
7/9 = 0.7777...
Guess what 9/9 equals?
.99999?
0.999... = 1. It is 1.
This is not correct. You might say, "For many practical purposes, 0.999... may be treated as if it were the same as 1."
If the value "1" represents a set boundary in the context of analytic functions, then what you wrote is horrifically wrong.
As a mathematican, I promise you that is correct.
You're thinking as .999... and 1 as being two different numbers that happen to have the same value. They're not. They're the exact same number written in different notation.
No 0.9999... is different from 1.
For most people the hardest thing about learning is THINKING you already know.
For those who agree with FancyKnight, once you accept that you might be wrong and read the explanations with an open mind, you might figure it. LongIslandMark, and a few others, explained it quite well.
This is a mathematical truth, not really open to debate.
As a mathematican, I promise you that is correct.
You're thinking as .999... and 1 as being two different numbers that happen to have the same value. They're not. They're the exact same number written in different notation.
You did not indicate a base when writing them down, so the default assumption is that they are expressed in the same base. Further, in all mainstream mathematical praxis, absence of an annotated base implies base 10. In any base, the two numbers are unequal.
They are in the same base, and they represent the same number.
1.000 = 1.0 = 1.000... = 1/1 = 1
0.999... = 1.
There is no difference. When you subtract the two, you get 0.000... = 0. There is no 1 at the end, because there is no end. That's what "an infinite number of nines/zeroes" means. They are the same number.
Yes, they are in base 10, and yes, .999... is equal to 1 even in base 10. (Similar thing: .111... is equal to 1 in base 2. Also, .999... is not equal to 1 in any base greater than 10, and .999... is not even defined in base less than 10.)
I also want to bring the concept of whether 1 is a prime number, or whether 0 is even too here... These three concepts are some of the most controversial subjects of mathematics (when shown to the general population), because the society thinks they know the answers.
EDIT: Clearly beaten by Remellion.
Yeah, 0.999... = 1 so it's not a big deal when you start with x=0.999... and end up with x=1.
1/9 = 0.1111...
4/9 = 0.4444...
7/9 = 0.7777...
Guess what 9/9 equals?