Question

0.142871428714287....

convert this into rational number

Answer 1

ok so I know this but I want the answer from some one else.... (any user)

Answer 2

I have a confusion regarding this method... I will post after some one solves this

Answer 3

\[10^5a=14287.1428714287...\]\[(10^5a-a)=14287\]\[a=\frac{14287}{99999}\]

Answer 4

that is the same method I would have used :)

Answer 5

Ok so mukushla has a right process ...

Answer 6

Now : my question is

0.9999999....

convert this into rational number

Answer 7

.99999... = 1

Answer 8

right but why so?

Answer 9

because it is infinitely long series of 9's after the decimal point - do you want a proof of this?

Answer 10

0.99999... = x

10x = 9.999...
10x-x = 9.999.. - 0.999... = 0
9x = 9
x-9/9 = 1

Answer 11

there are many proofd

Answer 12

0.9999...=x
10x=9.999...
10x-x=9.999...-.9999=9
9x=9
x=1

Answer 13

yes asnaseer sir...

I have to do that proof in tomorrow's exam

Though must say that "in the question we are provided to discuss with your teacher"

Answer 14

e.g. 1/3 = 0.3333333...
therefore 3(1/3) = 0.9999...
therefore 1 = 0.999...

Answer 15

Turing take a look at the question above

Answer 16

the trick is to find the point where the decimal starts repeating and miltiply
first to get the decimal at the beginning of the part that repeats, then the get the decimal at the end of the repeating part

Answer 17

agent47 ... check my comments

Answer 18

huh? I'm lost

Answer 19

0.3333... should be 0 (rounding off)
x = 0.333...
10x = 3.333
10x-x = 3.333... - 0.333...
9x = 3
x =3/9 = 1/3

Answer 20

since 0.999 = 1 (rounded off)

so why not then 0.333... = 0 ?

Answer 21

rounding off is a different process - it does not produce identities

Answer 22

it is a wat of getting approximations to a given number

Answer 23

*way

Answer 24

why 0.333.... is 1/3 and why 0.999... is 1 ?

@asnaseer and @TuringTest is this trick of which you both are referring to ?

0.3 = 3/10

0.33... = 1/3

3(0.33...) =3(1/3)

0.999 ..= 1
"hence proved?"

Answer 25

my method is different than asnaseesr's more common sense approach. Though I think his will be tricky from more ugly numbers\[x=4.56\overline{248}\]first multiply to get the decimal at the beginning of the repeating part\[100x=456.\overline{248}\]then again to get it after the repeating part\[100000x=456248.\overline{248}\]now subtract the two\[100000x-100x=456248.\cancel{\overline{248}}-456.\cancel{\overline{248}}\]\[999900x=4455972\]

Answer 26

I think asking why 0.333...=1/3 is equivalent to asking why is 1=1

Answer 27

:P I didn't think so of my question .... @asnaseer ... I got it now...

thanks @TuringTest and @asnaseer

Answer 28

yw :)

Answer 29

welcome

Answer 30

Though "if" I will ask that prove 1 = 1 .... what will you think of me? and of your answer?

Answer 31

and BTW @TuringTest - I use the same method as you and @mukushla did, I showed that there are also other ways of proving these identities :)

Answer 32

yes I noticed, and appreciated your more down-to-earth approach :)

Answer 33

If I recall correctly, the proof of 1=1 is not trivial :)

Answer 34

I am lucky to have you both answering my question.... thanks anjum and max.. again

but I will ask again "Though "if" I will ask that prove 1 = 1 .... what will you think of me? and of your answer? "

Answer 35

ok so what's your opinion max.?

Answer 36

asnaseer seems to know more about that than I
I would have said it's trivial...

Answer 37

it seems more axiomatic to me
peano's first: 1 is a number

Answer 38

agreed.

Answer 39

not sure what theorem is required to convince someone that a number is equal to itself, but that would complete the proof

Answer 40

reflexivity I guess

Answer 41

"Euclid's axiom"

Answer 42

sorry - I confused it with the proof of 1+1=2 which is here: http://humor.beecy.net/misc/principia/

Answer 43

1 + 1 = 2
1 = 2-1 =1

hence proved... :P thanks anjum

Answer 44

:D

Answer 45

haha the russel/whitehead principia does not think that proof is sufficient @mathslover
I think Russel would call it a circular argument

Answer 46

:P

Answer 47

he called most things circular arguments - I guess he enjoyed eating \(\pi\) too much :D

Answer 48

:)

Answer 49

apple pi ? :P

Answer 50

peeps ... m i wasting my time for exams? "yes"

so thanks guys

bbye

Answer 51

"closed"

Answer 52

wow. look at the number of the math lovers here

Answer 53

So, actually 0.999999999... is not close to 1.......... its equal to 1?

Answer 54

@mukushla

Answer 55

yes

Answer 56

Thanx........ I think I got it..

Answer 57

welcome