0.999... = 1
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0.999... = 1: Prove that 0.999 (repeating to infinity) isn't equal to 1.
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
0.333... = 1/3
0.666... = 2/3
0.999... = 3/3 = 1
All numbers that aren't equal to each other will have a number(s) that comes in between.
x < x+1 --- The number that could come in between of x and x+1 could be x+1/2.
So, give a number that comes in between 0.999... and 1.
You (as Con) will try to prove 0.999... (repeating to infinity) IS NOT EQUAL TO 1.
This is a debate on truism...
Sorry I made a few errors as I didn't have the time to proof-read such a long argument.
1. I said "You all" but it should have been just You.
2. I said 0.999... is rational but it isn't, it is in fact IRrational.
3. I sid wither, it should have been either.
There may be a few more I didn't catch either but in that case just assume the most logical thing I would have probably said.
I hope you'll understand.
As to the number in between them, that is true as well.
Once you get to rational numbers (decimals), any 2 numbers have an infinite number of numbers between.
1 and 2 have 1.1, 1.2, 1.2
1.1 and 1.2 have 1.11, 1.12, 1.13
1.12 and 1.13 have 1.111, 1.112, 1.113 and so on.
If all numbers must have an infinite number of numbers between them. This leads to the conclusion that if there are no numbers between them, there is no difference between them.
Repeating decimals are rational numbers.
Whole integers can be represented as fractions or decimals.
"All numbers that aren't equal to each other will have a number(s) that comes in between."
You made this up. No math textbook has ever listed this as a fact of math that I'm aware of, and I'm pretty good at math.
it's simple. 0.99999... is an irrational number, and 1 is a rational number. 0.99999... is a decimal and 1 is a whole integer. By definition these cannot be the same number as they belong to mutually exclusive categories. I could draw a Venn diagram in Microsoft Paint if that would help you understand the concept.
"It will become infinitely small".
You say as if its value is changing in time. But it isnt. That is how we describe it as we try to determine its value. But when we examine it as a whole, rather then what it is gradually becoming, we look at what it ultimately is. And ultimately it ends up at 1.
Ah, okay. That makes much more sense now. Thank you.
As someone who knows calculus, I find this question interesting. The answer I learned is that, while 0.999... is not equal to 1, there is no functional difference. 0.999... can get infinitely close to 1, but it will never equal one. However, the difference between 0.999... and 1 will also become infinitely small, so the difference is incalculable. So while they are not technically equal, they may as well be. (Unless you start dealing with quantum theory, but I'm not going there.)
It's not a mystery for people who actually understand the math.
A preeminent determining factor is the degree of one's education.
I would recommend taking a look at this article...
So does that mean that this debate is all merely based on opinion?
I'm not a Math guy.
But still, I fell like something that simple should have a definite proven answer in Math, unless this is one of those pesky unsolved Math mysteries?
You’d be surprised how many people disagree with it
But isn't that a Mathematical fact?