Is the number 1.999999... considered rational or irrational?

Question

I know that any rational is the ratio of two integers.
I also know that an irrational number is supposed to be an infinite nonrepeating decimal.

So, where does that leave the number 1.999999999...?
If it can be written as the ratio of two integers, please show me which two.
Thanks.

If it's not rational and it's not irrational, then what would it be classified as?

Answer 1

It's rational and it's 2.

I think you answered your own question, 1.9999... is a repeating decimal, so it is a rational.

I think some of the confusion comes from the fact that some repeating decimals, like .33333..., come out of repeated long division. But there isn't any long division that will give you 1.99999...

The thing to remember is that either you accept that 1.9999... represents a number, or you don't. And if you accept that it represents a number, then 2 is the only value it can take that won't break all of your rules of arithmetic.

Answer 2

1.9999999999.... is equal to 2.

Edit: Don't listen to people that are telling you irrational. There are multiple proofs to show that 1.9999999....is equal to 2. One is to express it as 1 + 9/10 + 9/100 + 9/1000.... There is a formula to determine the sum of a geometric series when the ratio is less than 1, which in this case is true. So we have 1 + 9/10 / (9/10) which evaluates to 1+1 which is 2.

Answer 3

It is not only rational, but also an integer - specifically, 2. If you want that as a ratio, it is 2/1.

Answer 4

Irrational

Answer 5

A rational number is something that can be expressed as a fraction such as 1/2 or 7/13 ... there is no equivalant fraction that equates to 1.9999999999 ... so therefore it is irrational.

Answer 6

It's rational. It equals 1 + 9/9, or 2.

Answer 7

Here is a simple proof that 1.9999... = 2

Let n = 1.999....
10n = 19.999... (multiplied both sides by 10)

Now subtract n from 10n

10n - n = 19.999... - 1.999....

Hopefully you can see that all the trailing nines will cancel out and you'll be left with

9n = 18

So n = 2

Since we started with n = 1.999... and ended with n = 2,

1.999... = 2

2 is clearly a rational number.

By the way, you can use a similar method to find a ratio equivalent to any repeating decimal.

Answer 8

The number is irrational, because if you have .3333 . . ., which is equal to 1/3, times 6, you get 1.9999 . . . , and that equals 2, because 1/3 times 6 equals 2, which is rational.

Answer 9

Its rational. Your question has the answer. It is a repeating decimal, therefore it must be rational. In calculus, this can be shown to equal 2 using the range 0->infinity

Answer 10

it's 2. Rational.