What is .9~?
1.
A number equal to 1 to anyone who understands math.
Topic should not even be a debate, however, it is understandibly quite hard to accept by some.
1.) All sequential repeating decimals can be expressed as the repeating series divided by a 9 per each digit.
For example, 3/7 = .428571~
Thus, 428571/999999 = .428571~
Therefore, 428571/999999 = 3/7
When we apply this rule to .9~:
9 is the repeating sequence. We put it over an identical number of 9's, which is one 9, thus giving us 9/9, which is 1.
2.) All rational numbers can be expressed as a/b with a and b as integers. .9~ is a rational number. Thusly, it can be expressed as a/b. 9.~'s a/b expression is 1/1. To all those who disagree, I challenge you to find another possible way that .9~ can be expressed as a/b.
3.) For any two numbers, there is an infinite number of other numbers that fit between them.
Example:
1.08, and 1.09.
1.081, 1.0801, 1.08001...
There are no numbers that can fit between .9~ and 1. thus, they must be equal.
Demonstrations:
1.)
Define x as .9~
x = .9~
10x = 9.9~
10x - x = 9
9x = 9
x = 1
Thus, .9~ = x = 1
2.)
1/3 = .3~
1/3 X 3 = 1
.3~ X 3 = .9~
If .3~ and 1/3 are equal, identical operations on them result in an equal product. Thus, .9~ = 1.
ARGUMENTS:
".9~ is not equal to 1, it gets closer to 1 with each 9 added but never reaches it"
Reply: .9~ is a number. This means it has value. The value of .9~ doesn't "get closer" to anything. It is a number, and has value.
"If .9~ equals 1, then doesn't 1.9~ equal 2, 2.9~ equal 3, 3.9~ = 4, and so forth?"
Reply: Yes.
"There is no point at which as certain number of 9's makes .9~ equal 1."
Reply:
Of course there isn't. If there were a point at which a certain number of nines "made" .9~ equal 1, then it would not be .9~ that equaled 1, just the decimal with enough nines to reach that point.
A "certain number of nines" directly contradicts the concept of an infinite string of nines. Approaching the question from this angle is entirely illogical.
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