0.99999... = 1

[surendar]

1/3 doesn't equal 0.333, it equals 1/3. 1/3 x 3 = 3/3 = 1.

End of story.

Its not END of the Story! Your point ( 1/3 NOT EQUAL TO 0.333..333) can be proven wrong! because it's mathematics.. & I love to play with it!


0.333333....333 can be written as,

0.333... 333 = 3/10 + 3/100 + 3/1000 + ..... (infinity)

--> 3/10[ (1/10)^0 + (1/10)^1 + (1/10)^2 + .... (infinity) ]

--> This is where Geometric progression comes into picture to solve these kind of summations. where a = 3/10 and r = 1/10.

Which implies,

Lt{n->infinity} [ (3/10) ( 1- r^n+1) / (1-r) ]

which can be further simplified as,

--> 3/(10[1-1/10]) [ Lt{n-> infinity} (1) - Lt{n-> infinity} ( r^n+1) ]

--> 3/(10*9/10) [ 1 - (1/10^infinity) ] As Lt{n-> infinity} (1) = 1 & r = 1/10

--> 3/9(1 - 0 ) ( As 1/infinity = 0 and 10^infinity = infinity )

--> 1/3.

0.333...333 = 1/3 Hence Proved.


Now tell me where do you find wrong in above methodology! :

surendar added 5 Minutes and 27 Seconds later...

And in similar methodology like i mentioned for 0.333...333,

0.9999...999 can be written as 1. And there is nothing wrong or harm in it! This is now end of the story

 

[FreddieFan]

One of the proofs on wikiepdia convinced me that 0.999... = 1, which I disliked to start with. I also dislike any proofs based on the fact that 1/3 = 0.3333..., because I'm not convinced it does.

This is the proof that convinced me:

Let x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

As we started off letting x = 0.999..., and finish with x = 1, we must conclude that 0.999... = 1.

This works because the infinite strings of 9s in each number are used to cancel each other out.

 

[surendar]

This is the proof that convinced me:

Let x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

Perhaps, it can also be written in better way,

x = 9/10 + 9/100 + 9/1000 + ....
On Simplifying,

x = 9/10 (1+ 1/10 + 1/100 + 1/1000 + ... ) ---- 1

And then multiplying 1/10 on both sides,

x/10 = 9/10 ( 1/10 + 1/100 + 1/1000 + ... ) ---- 2

2 - 1 => x(1/10 - 1 ) = 9/10 ( 1 )

x(9/10) = 9/10

Hence x = 1. Still, I am satisfied with Geometric progression method which i can see one of the very apt methods for infinite series. No wonder whatsoever the scenario, Mathematics try to prove it.. Its Maths magic

 

[Kshitiz_Indian]

Yeah these proof's are long accepted, but my point is that since infinity is a concept and nothing real, we shouldn't be able to perform mathematical operations on them, because think it in this way, whenever you'll subtrtact you can add more 9's!

 

[surendar]

Yeah these proof's are long accepted, but my point is that since infinity is a concept and nothing real

Those infinity mathematical operations have to considered like every other formula, because it's not defined.

like how a/1 = a ( general formula ) we have,
1/infinity = 0 ( infinity based formula )

Just because infinity is not defined, we cannot consider 1/infinity as such when we come across the same scenario in real applications. We had to substitute with 0.

If 1/infinity is zero, the one which I proved using GP at the top holds good because, it doesn't involve adding or subtraction 9's or 3's or whatever.

If one doesn't agree 1/infinity as zero, then can't help it! Lets leave.. I am more contented with following the standard conventions prescribed for ' mathematics '.

because think it in this way, whenever you'll subtrtact you can add more 9's!

This reminds me Newton's Third law

There are certain things which we can't just sit and brood over it.. Like E = mc^2 where c is the velocity of light. Same Einstein based point, when a spaceship is sent with the velocity of light to space, when it returns back, that spaceship will meet its third generation. ( i am not sure about this wordings as I hardly remember something like I mentioned ).

Wonder is, no one can still make sure that an object can move it with the same speed as velocity of light. These type of things, we cannot be obsessed with them, rather accept those standards and move on. Or, continue the research left by ancestors and waste time. Others will move ahead in the journey of life!

 

[Kshitiz_Indian]

Those infinity mathematical operations have to considered like every other formula, because it's not defined.

like how a/1 = a ( general formula ) we have,
1/infinity = 0 ( infinity based formula )

Just because infinity is not defined, we cannot consider 1/infinity as such when we come across the same scenario in real applications. We had to substitute with 0.

If 1/infinity is zero, the one which I proved using GP at the top holds good because, it doesn't involve adding or subtraction 9's or 3's or whatever.

If one doesn't agree 1/infinity as zero, then can't help it! Lets leave.. I am more contented with following the standard conventions prescribed for ' mathematics '.



This reminds me Newton's Third law

There are certain things which we can't just sit and brood over it.. Like E = mc^2 where c is the velocity of light. Same Einstein based point, when a spaceship is sent with the velocity of light to space, when it returns back, that spaceship will meet its third generation. ( i am not sure about this wordings as I hardly remember something like I mentioned ).

Wonder is, no one can still make sure that an object can move it with the same speed as velocity of light. These type of things, we cannot be obsessed with them, rather accept those standards and move on. Or, continue the research left by ancestors and waste time. Others will move ahead in the journey of life!

There is one thing that quite astonishes me about the current infinity concept, and although it might sound foolish, as you might say you can't multiply infinity, but hey, if you can add or subtract or divide, we should be able to multiply.

1/0 = infinity.

Now taking zero to the other side of the equation, we get 0, because anything into 0 is just 0. So we get 1 = 0. Which is definitely not possible.

Now get those brains thinking.

 

[surendar]

Now get those brains thinking.

There is nothing to think to be honest.. As per the basic mathematics which I learned, " You can't perform 0 * infinity " operation.

Just because 2* 1/2 = 1 this doesn't mean 0 * infinity/0 = infinity and vice versa. : Thats why said, there are some standard conventions in mathematics to be followed. If you deny those conventions, well go ahead and confuse yourself.. At least I am contented in using the standard conventions prescribed for Mathematics.

And this is what I remember learning in my basic mathematics,

0 * infinity - ONP
infinity * 0 - ONP
infinity - infinity -ONP
(+/-) infinity / (+/_) infinity -ONP
[(+/-) infinity]^0 - ONP

[COLOR="Red"][B]ONP = Operation Not Supported![/B][/COLOR]

 

[Harrypotter_fan]

There is one thing that quite astonishes me about the current infinity concept, and although it might sound foolish, as you might say you can't multiply infinity, but hey, if you can add or subtract or divide, we should be able to multiply.

1/0 = infinity.

Now taking zero to the other side of the equation, we get 0, because anything into 0 is just 0. So we get 1 = 0. Which is definitely not possible.

Now get those brains thinking.

You get that one wrong because,
1 / 0 = infinity
(1 / 0) * 0 = infinity * 0
0 / 0 = 0

(Note: Your method implies 0 / 0 = 1) Remember that numbers don't go to the other side or anything, you just +, -, *, / numbers on both the sides.

 

[surendar]

You get that one wrong because,
1 / 0 = infinity
(1 / 0) * 0 = infinity * 0

:shout Infinity * 0 is not defined operation! :shout I don't understand why people misinterpret certain operations which are particularly mentioned as not to be performed or not defined ones in this subject!

 

[Highlander999]

1+1=2

I'm a genius

 

[zimrahil]

1+1=2

I'm a genius

Just wait - someone is going to disprove that soon with some long mathematical theorem

 

[Harrypotter_fan]

If anyone doesn't believe 0.999.... = 1, then give the P/Q form of 0.999.... As it is a rational number (recurring) and rational numbers can be represented by P/Q.

 

[Cricketdude]

Just wait - someone is going to disprove that soon with some long mathematical theorem

Sure, I can do that. Here goes:

Let a = 1 and b = 1.

Therefore a = b, by substitution.

If two numbers are equal, then their squares are equal, too:

a^2 = b^2.

Now subtract b^2 from both sides (if an equation is true, then if
you subtract the same thing from both sides, the result is also
a true equation) so

a^2 - b^2 = 0.

Now the lefthand side of the equation is a form known as "the
difference of two squares" and can be factored into (a-b)*(a+b).
If you don't believe me, then try multiplying it out carefully,
and you will see that it's correct. So:

(a-b)*(a+b) = 0.

Now if you have an equation, you can divide both sides by the same
thing, right? Let's divide by (a-b), so we get:

(a-b)*(a+b) / (a-b) = 0/(a-b).

On the lefthand side, the (a-b)/(a-b) simplifies to 1, right?
and the righthand side simplifies to 0, right? So we get:

1*(a+b) = 0,

and since 1* anything = that same anything, then we have:

(a+b) = 0.

But a = 1 and b = 1, so:

1 + 1 = 0, or 2 = 0.

Now let's divide both sides by 2, and we get:

1 = 0.

Then we add 1 to both sides, and we get what your programming
teacher said, namely:

1 + 1 = 1.

In fact, you can prove that 47 = -3 or anything else you want.
But of course you know that is wrong.

Do you know what I did that was not correct?

Shall I tell you? If you want to work it out for yourself before
viewing my answer, I will space down a few lines so you can hide my
response and work it out for yourself.





hmmm...




not yet...




Okay, here's the bad thing I did. You can divide both sides of an
equation by the same thing ONLY AS LONG AS YOU ARE NOT DIVIDING BY
ZERO. In fact, you cannot ever divide by zero. When I divided by
(a-b), that means a somewhat disguised form of 0, since a = b = 1.
That's where I went wrong. Did you figure that out by yourself, or did
you need the hint?

 

[Kshitiz_Indian]

There is nothing to think to be honest.. As per the basic mathematics which I learned, " You can't perform 0 * infinity " operation.

Just because 2* 1/2 = 1 this doesn't mean 0 * infinity/0 = infinity and vice versa. : Thats why said, there are some standard conventions in mathematics to be followed. If you deny those conventions, well go ahead and confuse yourself.. At least I am contented in using the standard conventions prescribed for Mathematics.

And this is what I remember learning in my basic mathematics,

0 * infinity - ONP
infinity * 0 - ONP
infinity - infinity -ONP
(+/-) infinity / (+/_) infinity -ONP
[(+/-) infinity]^0 - ONP

[COLOR="Red"][B]ONP = Operation Not Supported![/B][/COLOR]

Well exactly, that was my point, which I was trying to explain, if we can't perform operations on infinity, because its just a concept, we shouldn't be able to perform operations on things involved in it either.

Ah well, I wouldn't debate any more, looks like no one is catching my drift, and I don't want to get caught in some serious firepower. I'm not arguing the basis of mathematics, its just that... there is something here that is unbelievable. For an approximation, fine, but definitely not for the equality.

 

[shravi]

Madness, I tell you. MADNESS!

Madness? THIS IS MATHS!

300 reference.

:laugh