Question

Is 0.99999999... a rational number?

Answer 1

yes.., it is

Answer 2

So can u express it in the form a/b where a and b are integers

Answer 3

one

Answer 4

just for fun
rational numbers
irrational number

Answer 5

But 0.9999999999999... must very close to one...... how is it one

Answer 6

there must be a difference of 0.000000000...0001 between 1 and 0.999999...

Answer 7

right?

Answer 8

i don't think its rational
0.11111..... = 1/9 is rational, but if u multiply it by 9 to get 0.9999....
u get 9/9 = 1 instead and not 0.99999.....
so can't be expressed as ratio of integers, so irrational.....

Answer 9

Agree with @hartnn

Answer 10

\[x=0.999...\]\[10x=9.999...\]\[10x-x=9.999...-0.999...\]\[9x=9\]\[x=\frac 99=1\]

Answer 11

isn't a number that has a sequence of digits repeats infinitely is a rational number?

Answer 12

\[\frac{1}{3}=0.3r\]
\[3\times \frac{1}{3}=3 \times 0.3r\]
\[1=0.9r\]
where 0.3r means 0.3 recurring etc.

Answer 13

So 0.99999999 recurring =1 which is rational.

Answer 14

0.99... = 1
Why?
http://www.khanacademy.org/math/vi-hart/v/9-999----reasons-that--999------1

Answer 15

agree with @UnkleRhaukus and @Traxter

Answer 16

repeating decimals are rational

Answer 17

So I guess 0.99... is actually a natural number as well?

Answer 18

But i must add 0.00000000000000000000000000000000000000000000000000000000...1 to 0.9999999999999999999999999999999999999999999... to get 1 right?

Answer 19

no. watch the video I linked :) A lot of people can't believe 0.99... = 1 because they can't believe that infinity is infinite, and they think that there is one last digit.

Answer 20

Ok

Answer 21

But my net is too slow

Answer 22

@sauravshakya It's good that you're questioning the reasoning behind this. Have a look at my proof above and you'll see why this is though. Well done for having an inquisitive mind and not just taking things as they are!

Answer 23

Well, I'll try to explain it.

If x = 0.99...
then 10x = 9.99..

10x-x = 9
x = 1

You must understand what happens here. The number got shifted. If it were a rational number, we couldn't do this, because:
y = 0.9999999
10y = 9.9999990

But because the decimals repeat infinitely, we can do this, and it all works out.

Answer 24

I mean, if the decimals weren't repeating infinitely (instead of "if it were a rational number"). Sorry

Answer 25

@sauravshakya my answer is no

Answer 26

Can u pLZ explain why no

Answer 27

because it is repeating

Answer 28

Then it must be rational

Answer 29

@pourushgupta you are wrong!!!

Answer 30

@sauravshakya it is rational no.

Answer 31

because it is terminating..

Answer 32

@sauravshakya where you are......lol

Answer 33

@hartnn I think in 1/9 = 0.1111111....... the right hand side is just a approximate representation of the left. As right will be equal to left only when 0.11111.... series is infinite.

Answer 34

0.999999..... and 1.0000000..... are both representations of 1

Answer 35

yes, i considered ..... as infinite only ..... like 0.99999999...till infinity.
which actually is 1.

Answer 36

In a similar was we can say that 00.99999..... is an approximate representation of a rational number. which can be found using procedure given above.

Answer 37

so 0.99999..... is rational I think.

Answer 38

and a natural number!

Answer 39

@hartnn... approximately. The rational number it represent is natural also

Answer 40

I think 0.9999999999999999... can be written as 9/10+9/100+9/1000+9/10000+...

Answer 41

where first term is 9/10 and common ratio is 1/10

Answer 42

yup, that sum to infinity =1

Answer 43

So, Sum=(9/10)/(1-1/10)=1

Answer 44

yes, an elegant solution to problem