Is .9999...infinite a rational number?

Answer 1

the reason this confuses people is that most people believe that every number has exactly one decimal representation. But that is a false assumption.

0.9999.....represents the same number as 1.0000.........

Yes, 0.9999...... IS the number 1, believe it or not!

And if you are (understandably) skeptical, here's an argument that might convince you:

First of all, a basic property of the number line is that in between any two numbers, we can always fit another number.

Consider the numbers .9 and 1. Can you think of a number that "fits" in between? Sure you can. One such number is
.99

Ok, can you fit a number between .99 and 1? Sure, you can fit .999 between them.

Keep doing this. Can you fit a number between
.9999999999999999999999999 and 1?

Sure, just append another 9 at the end.

But what about
.9999..................................... and 1 ?

There is no number you can fit in between the two. You can't tack another 9 onto the "end" of the decimal, because there is no end. And you can't increase any of the decimal places, because that would give you a number GREATER than 1.

So if there is no number in between .9999......... and 1, then these two must be the same number.

This disturbs people more nowadays than in the past, I think, because we live in an age of digital readouts and expect decimals to never go weird on us.

For example, .49999999999.......... and .5 are the same number also, but this didn't bother anybody when we all wrote 1/2 ;)

Every number has a unique fractional representation in lowest terms. But the decimal representation of a number need not be unique. The decimals are only represenatations: they are not the numbers themselves. (There is only one number 1, even if there are two decimals representing it.)

Answer 2

The answer to your question is yes.
.9999... is a rational number because it is 9/9.

Here's an explanation:

Definition of rational number:
Any number which can be expressed as a ratio of two integers. In other words, any number that can be written as a fraction.

1/9 is a rational number and is equivalent to .1 repeating infinitely...

4/9 is a rational number and is equivalent to .4 repeating infinitely...

9/9 is a rational number and is equivalent to .9 repeating infinitely....


See the following tutorial:
http://mathforum.org/dr.math/faq/faq.integers.html
which explains what rational numbers are and discusses the example of .9999...

Answer 3

0.9999999999999999... can be used as rational numbers!

Vedic Mathematics is just that.

If you want to find (any number of 9999999...)^2 a Vedic Mathematics manner of ding it is

first fix number of digits 'n', which may include infinity also.

(99999<-- n times-->99999)^2 =

99999<-- n times-->99998 , 00000<-- n times-->00001.

Answer is dead accurate!

You can do it for

99^2= 9801
999^2= 998001
9999^2= 99980001
and so on... any number of digits can be squared.

People can extend these number applying manners to endless needs.!

If we can fix a need (no matter the need is too huge) an irrational number become rational.

Answer 4

yeah. the definition of a rational number is that it either terminates or that it's repeating. .999999.... is repeating. so it's a rational number. ;)

Answer 5

Yes it is.


.999999..... = 1.


How?

Let x = .999999........

10x = 9.99999999

10x-x = 9.9999999... - 0.99999999

9x = 9

x = 1