Does 0.9 recuring = 1?

[the_smurflord]

HAving an argument in the office atm.

0.9 recuring is 0.999999999999999999999999 with an infinate number of 9s after it.

As the difference between 0.9 recuring and 1 is infinately small, does that mean that 0.9 rec = 1 or not?

Discuss.

Trolls need not reply, unless they have super-cooled helmets.

 

[Cap'n Sissyfoo]

Only if you are rounding it up otherwise 0.9 rec = 0.9 rec and 1.0 = 1.0

 

[the_smurflord]

but how can then not be equal when the difference between them is infinately small?

If something is infinately small it must be zero, surely?

 

[Addlcove]

nope, when something is infinelty small it still IS... something that is zero isn´t.

 

[Cap'n Sissyfoo]

The way I see it, even if it is infinitely small then it still exists hence it can't equal nothing.

For example, if you had 10 gajillion eggs but one egg was rotten you would be incorrect in saying that 100% of the eggs were fine...because they aren't. ALL 10 gajillion eggs would have to be fine and healthy before you could say that 100% of the eggs were fine.

Most people would round up 0.9 rec to 1.0 because it is easier to write 0.9 and then 500 pages of 9s but strictly speaking it is still 0.9 rec.

 

[the_smurflord]

but the point is that we're are not talking about something that's very very small, we're talking infinately small.

If I was to right out 1 - .9 rec it would be

0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...........

I can always write more 0s, and there would have to be an infinate number of 0s before you could put the 1, therefore the 1 could never ever be placed. Therefore it is 0.

 

[Sarum TheBlack]

No.

It's 0.9rec. Even infinitely small isn't zero. Practically, it is of course, but mathmatically, it's not.

 

[Easy]

If X tends to zero, (1-X) tends to 1. ^_^

 

[Cap'n Sissyfoo]

The number can be as infinitely small as you want it to be but unless it the number = 0 it can't be 0.

You could argue that to all intents and purposes 0.9 rec is effectively equal to 1 but mathimatically speaking, it isn't and never will be.

 

[old.Psi]

There was a formula or theory type thing we were told in school that (in theory) proved all recurring numbers were whole numbers...

It started with something like '66.66666 rec - 6.66666 rec = 60' and then went on from there if anyone wants to figure it out.

 

[Kharok Svark]

If 2 to the power of 2 is 4 - 2 x 2

and 2 to the power of 3 is 8 - 2 x 2 x 2

back to 2 to the power of 2 is 4 - 2 x 2 x 2 / 2

the 2 to the powere of 1 is 2 - 2 x 2 x 2 / 2 / 2

therefore 2 to the power of zero is 1 - 2 x 2 x 2 / 2 / 2 / 2

therefore anything to the power of zero is one

therefore zero to the power of zero is 1

doesn't help fin you 0.0(billions of noughts)01

but hey ...

 

[Cap'n Sissyfoo]

Grr...I hate you Smurfy! I have just been searching around for a more concrete answer to your question and it has left me more confused since tje day I came across pointers.

From what I have read 0.9 rec DOES equal 1.

Let me start off with a more general principle called Geometric progression (GP for short). for example,

1, 2, 4, 8, 16... and
3, 6, 12, 24... etc.

are GP's. We shall call the first term of a GP 'a' and the number we always multiply by 'r' - the common ratio, so in the first series, it is a GP of a = 1, r = 2 and the second one is a GP of a = 3, r = 2.

Now the problem is this... What is the sum of the first n terms? Let us denote it by S

S = a + a × r + a × r2 + a × r3 + .. + a × r(n-1)

but now,

r×S = a × r + a × r2 + ....... + a × rn

Calculating

S - r × S= a - a × rn

S = (a - a×rn) ÷ (1 - r) = a × (1 - rn) ÷ (1 - r)

Now let's consider a special case. If the absolute value of r is less than 1 (exclusively). eg. -1 < r < 1. you can see that rn ® 0 as n ® infinity.

So solving for S and let n ® infinity, we get

S(inf) = a / (1 - r)

Now recall that in 0.9 recurring, it is just an infinite sum of

0.9 + 0.09 + 0.009 + 0.0009 +....

So above formula holds with a = 0.9, r = 0.1 and S(inf) = 1 from the formula, so 0.9 rec = 1

PS the above formula does not hold for r > 1 or r < -1, since the term rn would grow as big as we like, and hence the series diverges.

Ah well, you sorta learn something new everyday...something which shatters your faith in universal constants. Bleh.

 

[- Pathfinder -]

There is no way that 0.9 recuring can be absolute 1

 

[old.Psi]

The thoery we were shown did work, and did make 0.9 to be 1 or whatever, but we all thought it was stupid as 0.9 can't be 1, but the theory proved it was, but it can't be, but it was!

 

[Cap'n Sissyfoo]

I hate to say it but I haven't been able to find anything on the web that can disprove it. I still don't like it though...it just doesn't feel right. :/

 

[klavrynd]

how can two value be the same if they're different, even with an infinitly small difference

 

[Cap'n Sissyfoo]

The problem lies in the fact that an infinite value has been specified. If it was just 0.9 with 100 gazillion gabijillion 9s following then it would never equal one. As it is an infinite number of 9s it kinda clouds the matter. Can't explain it because I don't understand it fully myself but that is what all the math websites I have visited say. Stupid bloody infinite numbers.

The explanation that made the most sense is here...

http://www.nrich.maths.org.uk/mathsf/journalf/aams/q100.html

and here...

http://www.bbc.co.uk/dna/h2g2/alabaster/A479720

Bastards. I hate being wrong.

 

[Easy]

(1-X) = 1 only if x=0 IMO ^_^
_
0.9 = 1 is impossible IMO.

However...

x = 0.9 rec.
100 x = 99.9 rec.

Note that 100x - x = 99x, and that 99.9rec. - 0.9rec. = 99

So we have:

99 x = 99 and thus

x = 1 lol.

There, problem solved ^_^

One more post and I'll have my custom Avatar lol.

 

[Easy]

Oh I see Sissyfoo posted a link on
the solution lol. Oh well

My solution was pulled from a
offline math book

 

[old.LandShark]

Originally posted by Sissyfoo
I hate to say it but I haven't been able to find anything on the web that can disprove it. I still don't like it though...it just doesn't feel right. :/

I think Easy has just beat me to it here, but basically: it's true if you assume that "x tends to infinity" is the same as "x equals infinity", which, well, isn't and cannot be true.
It's true in practical terms; an engineer might say it was true, but not a physicist.

 

[Cadire]

I can remember my old maths teacher posing us the following question (I'm paraphrasing as my schooldays were xx years ago).

If there were an infinite number of chickens, and 1 in 1000 chickens laid a red egg.... would that mean there was also an infinite number of red eggs?

At first glance it seems preposterous! There are undoubtedly less red eggs than normal ones, so how can both be an infinite number?

But if there are an infinite number of chickens, then there must also be an infinite number of red eggs!

Infinity is merely an extropolation of a never ending number. Ergo, there will never be as many red eggs as normal ones, and 0.9 will never = 1.

He actually drew a lot of squiggly lines on the blackboard... but my memory isn't that great

 

[Teh Krypt]

Shouldnt you be working in the office?

And its quite logical really. 1.0 = 1

1.9 = 1.9

1 is close to 2, but it can't possibly be called one or the other

1.9999999999999999999 is 1.(however many 9's I used) .

Its only ever 1 if you round up or add 0.1 to 1.9 ect

 

[Tobold]

because you cannot write 0.999999 (repeat 9 indefinately) it is considered normal among mathmatitions to just write 1.0
to be more accurate, just write <1.0

but no, 0.999999 (repeat 9 indefinately) does not EQUAL 1.0, just it is written as 1.0 to avoid spending eternity writing 9s

for example.
X has been rounded down the nearest hole number, and that number is 4, what is the maximum value of X?

the maximum value is 4.49999999999999999999 rec, but because that cannot be written, the number is taken as 4.5 (even though if it were 4.5, the 'mathimatically nearest' number would be 5, not 4)

 

[old.shotgunstow]

This little bit of maths theory is easily explained.

1/3 = 0.33333333333333333 recurring etc
2/3 = 0.66666666666666666 recurring etc
Therefore 3/3 = 0.999999999 recurring etc
And 3/3 = 1, therefore 0.999999 recurring = 1

Done

 

[Easy]

Originally posted by old.shotgunstow
This little bit of maths theory is easily explained.

1/3 = 0.33333333333333333 recurring etc
2/3 = 0.66666666666666666 recurring etc
Therefore 3/3 = 0.999999999 recurring etc
And 3/3 = 1, therefore 0.999999 recurring = 1

Done

An interesting reverse approach to the
"generating fraction" thing I pulled from
this textbook

 

[old.Dillinja]

I see your point smurf, but no matter how infinite it is, it can never become 1. It will always be 0.9 recurring. It's like saying 9*infinity is the same as 9.00000000000000000000000001*infinity, just different numbers.

 

[Sharma]

/boggle

my head hurts


anyways, no matter how many times the 9 recurred, the 0.0000000 etc till the one appears would forever have to difference

if 0.9999999999999999999999999999 went on forever then there would always be something to make it tick in 1.0

there will always be that 0.0000 .......<forever>... 0001 on the end to make it change to 1.0

 

[Danya]

I liked shotgunstow's one best.

But I think it's fairly clear that 0.9 rec. is equal to 1 perverse as that seems.

 

[old.Tohtori]

Originally posted by old.shotgunstow
This little bit of maths theory is easily explained.

1/3 = 0.33333333333333333 recurring etc
2/3 = 0.66666666666666666 recurring etc
Therefore 3/3 = 0.999999999 recurring etc
And 3/3 = 1, therefore 0.999999 recurring = 1

Done

Then again...

0.333(rec)=3/10

0.666)rec)=6/10

Therefore 0.999(rec)=9/10

9/10 is not 10/10

Therefore 1 is not 0.999(rec)

And since 3/10 can't be divided into smaller fractions, according to basic math, 0.3333(rec) can't be fractioned as 1/3 because it is not accurate enough.


Teh Seel version:

If you have eaten a PIE!! but have left crumbs all over teh PIE!! plate then the amount of PIE!! eaten can not be 100%.

One PIE!! minus crumbs is 0.999999(rec) PIE!!!

ergo... 1 PIE!!! is not 0.9999(rec) PIE!!!

 

[Teh Krypt]

STop it !

Theres no logic at all in the world as we know it to make 0.999999 (rec) = 1 .