What is .9~?


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.


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.



Define x as .9~

x = .9~

10x = 9.9~

10x - x = 9

9x = 9

x = 1

Thus, .9~ = x = 1


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.


".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."


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 math, .9, 1, equal, controversy, debate


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