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.

See