Can anyone explain this: 0.999... is a Rational Number. In Fact 0.9... is an element of Z. Give evidence?

Question

Can anyone explain this:
0.999... is a Rational Number. In Fact 0.9... is an element of Z. Give evidence to support this statement.

Answer 1

It is mathematically proved that 0.9999.... = 1. You can find plenty of information in the Wikipedia article I give you below. If you find it hard to believe, then think of that:

What's the result 1 - 0.999... = ??

I've talked about this with friends and most of them say it's 0 point many many zeroes and an 1 in the end. BUT if you put an one in the end, then the result doesn't have infinite digits, as the initial number does. So the result of this subtraction is zero. What seems to confuse most people is that they understand infinite as too many rather than as unending. Of course this is not a scientific proof, just an intuitive one to make you feel why the case is so.



For the real mathematical evidence, check out the article below, where you will find plenty of useful information.

Answer 2

Let x = 0.99999...... ---------------------eqn 1 or

10x = 9.999..... or

10x = 9 + 0.999...... or

10x = 9 + x [from eqn 1] or

10x - x = 9 or

9x = 9 or x = 1 hence

x ∈ N or Z, set of natural number every member of which is rational.

Answer 3

Rational number can be a number which is non terminating(non ending) but repeating.(that is the same number or same pattern of numbers are present). Here 0.999.... is non terminating but repeating.

Answer 4

It is about notation and definition.



0.9999... is, BY DEFINITION, the number to which the series 9/10+9/100+9/1000+... converges, so it is equal to 1.

Answer 5

Let x = 0.9999999999...

10x = 9.999999999....



Subtracting the two equations gives 9x = 9, so x = 1.