# Why does .99999999... = 1?

#### LaVoS

##### Senior member
i could be wrong but look at this... seems right to me...Assign 0.99999... as the variable x.
x = 0.99999...

Multiply x by 10.

10x = 9.99999...

Subtract x from 10x and solve for x.

10x = 9.99999...
-x = -0.99999...
-----------------
9x = 9
x = 9/9
x = 1

Didn't we assign x as 0.99999...? Yes, we did. Then how did the
answer magically change to 1? It's not magic. It's just that
0.99999... was equal to 1 all along.

k? theres that one AND this one...

go to Calculator on your computer (or a normal calculator its up to you)

type in
1/3 (use fractions to decimles o get .33333333...)
then multiply that by 3
and you will get 1.... i thought .33333333333... * 3 = .99999999999...
why? i think that math is screwed up.... cos .9999999999 clearly does not = 1...

#### Static911

##### Diamond Member
ah math!

static911 <--starts running and wailing!

#### Kelvrick

##### Lifer
What grade are you in? In calculator, its an approx because then the repeating .3333 represents 1/3, and 1/3 *3 = 1.

#### Mucman

##### Diamond Member
You have to remember it is 0.9999... not just 0.9999

There is a difference

I remember seeing a good proof of it in the highly technical forum

#### ViperMagic

##### Platinum Member
Becasue you did it wrong thats why

BTW he is a HS Freshman

#### Shantanu

##### Banned
It has to do with computer's general inability to handle vulgar (common) fractions, unless a specific instruction set is written telling it how to handle such fractions. Best not to worry about it too much. Better yet, know the math yourself so you won't be dependent on your calculator so much.

#### Martin

##### Lifer
10x-x = 9x.

x = 0.999999

therefore

9x = 9*0.99999 = 8.9999991

8.9999991 /9 = 0.999999

your method is wrong.

#### geno

##### Lifer

<< type in
1/3 (use fractions to decimles o get .33333333...)
then multiply that by 3
and you will get 1.... i thought .33333333333... * 3 = .99999999999...
why? i think that math is screwed up.... cos .9999999999 clearly does not = 1...
>>

That's all in the code of the Calculator. It retains that you typed in 1/3, and squares that. Type in .333333333333333333333333 (as far as it goes) and square that, watch what it looks like.

#### Martin

##### Lifer
for the second one, most calculators can't handle fractions. Mine can and answer comes to 1. as long as you dont use finite decimals to represent fractions, you wont get anything screwed up.

#### wildwildwes

##### Senior member

<< i could be wrong but look at this... seems right to me...Assign 0.99999... as the variable x.
x = 0.99999...

Multiply x by 10.

10x = 9.99999...

Subtract x from 10x and solve for x.

10x = 9.99999...
-x = -0.99999...
-----------------
9x = 9 <--------- 9x is not 9, it is 8.99991
x = 9/9
x = 1

Didn't we assign x as 0.99999...? Yes, we did. Then how did the
answer magically change to 1? It's not magic. It's just that
0.99999... was equal to 1 all along.

k? theres that one AND this one...

go to Calculator on your computer (or a normal calculator its up to you)

type in
1/3 (use fractions to decimles o get .33333333...)
then multiply that by 3
and you will get 1.... i thought .33333333333... * 3 = .99999999999...
why? i think that math is screwed up.... cos .9999999999 clearly does not = 1...
>>

#### RossGr

##### Diamond Member
From the view point of abstract mathmatics, and the world of Real Analysis.
.99... =1 Where .9... represents an infintly repeating string of 9s.

There are many ways to prove this. Above 3* 1/3= 1 and the algebraic addition is also valid, both point to the truth.

This link shows what equality means to a mathematician.

Did you understand that?

Here is a proof using the fact that a geometric series has a sum.

Here is a proof that uses the concept of equality as stated above.

Now these proof are good and fine if you have sufficient math to understand them. IF you are a high shcool student who has yet to be exposed to high mathematics, Please look at these and attempt to understand them. When you do calculus you will see much more of this type of proof, get used to it.

IF you are of mindset which simply cannot accept the concept then please read every post in the link provided by Viper GTS above. Then feel free to post your opinion.

#### NakaNaka

##### Diamond Member
The Highly Technical Forum scares me

<--- vows to never go into HT ever again

#### MustPost

##### Golden Member
Im not exactly sure why it needed 10 pages. I stopped reading it sometime whenthere were two pages of the thread there.

#### RossGr

##### Diamond Member
BTW No calculator or computer can deal with an infinte string of digits. They all have a fixed word length which limits the accuracy of the result. To see this simply create a loop with a single precision variable and increment it by .1 and display the total each time though the loop. It is surprising how quickly the round off errors show up.
You may ask why is the sum of x = x + .1 being rounded! Good question. The sad fact is that when .1 is converted to binary fraction by your binary computer it become an infinitly repeating base 2 number.

.1 (base 10) = .000110011001100... (base 2) it must be truncated at the computers word length.

#### yoda291

##### Diamond Member
<goes catatonic mumbling "mommy" make the nines stop">

<< 10x = 9.99999...
-x = -0.99999...
-----------------
9x = 9 <--------- 9x is not 9, it is 8.99991
x = 9/9
x = 1
>>

Wild, you have got to be kiddin me. take 9+x-x and trust me, you'll end up with 9.

I got tired of it around page 8 or 9 on the old thread

#### TonTo

##### Banned
what if the way we humans understand mathmatics is fundementaly flawed?

#### RossGr

##### Diamond Member

<< what if the way we humans understand mathmatics is fundementaly flawed? >>

We created it, so I would think we understand it. It is not a case of flawed understanding, it is an artifact of the construction of the Real number line.

#### joohang

##### Lifer
Check out the thread on this at the Highly Technical forum.

#### MF1

##### Senior member
Marty and wild,

LaVos set x=0.99999... meaning that 9 is repeated infinitely.

10x = 9.99999... ; notice "..."

10x - x = 9.99999... - 0.99999...
9x = 9 ; notice that "..." is gone
x = 1

#### b0mbrman

##### Lifer
Proof is too easy.

1/3 = 0.333...
2/3 = 0.666... (mark of the beast :< )
1/3 + 2/3 = 0.333... + 0.666...

0.333... = 0.3 + 0.03 + 0.003 + ...
0.666... = 0.6 + 0.06 + 0.006 + ...
(0.3 + 0.03 + 0.003 + ...) + (0.6 + 0.06 + 0.006 + ...) = 0.999...
0.333... + 0.666... = 0.999...

1/2 + 2/3 = 0.999...

1/2 + 2/3 = 3/3
3/3 = 1

1 = 0.999...

Probably way longer than it had to be but i got there...

#### GoldenBear

##### Banned
In calculus we almost never find the exact number, but rather it's a strong approximation. Hence the concept of limits.

You can also prove the .9999... stuff through the infinite sum equation:

S = a/(1-r) - since to get .9999... you add .9 to .09 to .009 to .0009 and such.

a = .9 and r = .1

.9/(1-.1)

= .9/.9 = 1

I didn't bother reading those links so if this is posted already, eh..

#### Midnight Rambler

##### Diamond Member
I say blame it on the accountants ... one of those rounding errors/discrepancies.

#### Androck99

##### Senior member
Dude b0mberman wtf is this part?

<<
1/2 + 2/3 = 0.999...

1/2 + 2/3 = 3/3
>>

#### SinNisTeR

##### Diamond Member
its impossible, the reason you get that in the calculator is because of the flaw in the calculator. this is so simple all you need is common sense.