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# x = 1 + x

Anyone out there who can explain this statement to poor old uneducated me?

I mean, heck, x = 1 - x means x = 0.5 but x = 1 + x means 0x = 1?

Which means 0 = 1?

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• Wich means it is false and not an equation.  It is an inequality.

x ≠ 1 + x

• But if 0.999... = 1 is an equation, then why not?

As in the entire

1 / 3 = 0.333...

1/3 * 3 = 0.999..

1 = 0.999...

Which also means

1 - 0.000..1 = 1

0.000...1 = 0

Hail DnDOnlinegames!

• I used 2 = 1 rather than your 1 = 0 to prove your x = 1 + x theory:

a = b

a^2 = ab

a^2 + a^2 = a^2 + ab

2(a^2) - 2(ab) = a^2 + ab - 2(ab)

2(a^2) - 2(ab) = a^2 - ab

2(a^2 - ab) = 1(a^2 - ab)

2 = 1

The fallacy lies in the bold section. You are dividing both sides by a^2 - ab. But we've already established a = b, so a^2 - ab = 0, which is a no-no with division.

However, you could also look at x = infinity. Infinity = Infinity + 1.

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Originally posted by serjndestroy

1 - 0.000..1 = 1
0.000...1 = 0

1 - 0 = 0.999.9 => 1 = 1 = 0

• Member UncommonPosts: 2,078

0 or nil has no value. It is only used as a expression for or against a value. Example: 10² = 100 the 0s are not representative to a actual value but are used to express 10². This applies as well to 10 itself. Base numerical values are 1-9 with 0 either adding or subtracting (expression)from the base numerals.  Example  .01 is against the base numeric value as 10 is for the numeral. We see the against or for as - or + while 0 can be neither /or due to it being a expression.

As to .999 9 is the common base of all numeric values. Any character higher than 9 is still a sequence base 9 numeral. A small example is 15.  1+5 is 6 15-6 is 9. Thus the value is true. This is exceptionally helpful when doing division or multiplication. As you can be using very large numbers.

Hope that helps a bit .

• Replacing 0 = 1 with 1 = 2 kinda defies the point, the a = b argument is solved, it relies more deeply on the 0 point here

I also played around with the  Infinity = Infinity + 1 train of thought.. Infinity - 1 = Infinity -1 but Infintity + 1 =  Infinity

Which could mean that 1 = 0.

Hail DnDOnlinegames!

• Member UncommonPosts: 2,078

What part of 0 meaning a "expression" did you not understand?

• Originally posted by serjndestroy

Replacing 0 = 1 with 1 = 2 kinda defies the point, the a = b argument is solved, it relies more deeply on the 0 point here
I also played around with the  Infinity = Infinity + 1 train of thought.. Infinity - 1 = Infinity -1 but Infintity + 1 =  Infinity
Which could mean that 1 = 0.
Well, the topic was "x = x + 1", so I assumed I could substitute whatever I wanted in for x. I thought that's how variables work. :P

But no, just because infinity + 1 = infinity doesn't mean 1 = 0. You can't subtract infinity from both sides, you can't divide it, etc.

I'm positive that there's a fallacy somewhere, but it's been AGES since my last calculus class, and I couldn't even begin to remember the rules for integrals and stuff.

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Member, Newbie CommonPosts: 2,767

Originally posted by serjndestroy

Anyone out there who can explain this statement to poor old uneducated me?
I mean, heck, x = 1 - x means x = 0.5 but x = 1 + x means 0x = 1?
Which means 0 = 1?

Not all equations are true and not all equations have solutions.

1/x = 0 is an example of such (No, infinity is not a real number)

You can't arbitrarily create an equation and expect it to be true and have a solution.

You also can't call .000.... 1 a number, since there can be no number after the ...

Why? Because  ... denotes an infinite sequence of numbers and if it were to terminate with a 1, then it wouldn't be infinite. The very definition of infinite is that it does not terminate finitely: If infinity were to stop at some point, then it would not be infinity.

Also... .3333... is not actually a simple number, but an infinite sequence

When you have an equality like .9999...  = 1, it doesn't mean that .9999... is 1, but that the limit of the sequence .9999... is 1.

This is a sequence of characters intended to produce some profound mental effect, but it has failed.

• Yeah, I knew I was wrong, but still...

By the way, why the restrictions on infinity in equations?

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• Member UncommonPosts: 2,078

Another basic is 1=inf.  So if Inf +inf then it =inf. by using a expression of 0 with inf you still have inf but 0 which is a expression still retains no value. Thus you will still have 1

How hard can it really be.

• Member, Newbie CommonPosts: 2,767

Originally posted by serjndestroy

Replacing 0 = 1 with 1 = 2 kinda defies the point, the a = b argument is solved, it relies more deeply on the 0 point here
I also played around with the  Infinity = Infinity + 1 train of thought.. Infinity - 1 = Infinity -1 but Infintity + 1 =  Infinity
Which could mean that 1 = 0.

No, infinity is not a number, so it doesn't have those properties.

Stop applying simple rules to complex things. You're way out of your league with infinity and the like.

First understand what the NATURAL numbers are, where they came from and the rules of the group ( go google group ).

Second understand what the RATIONAL numbers are, where they come from, and all applicable operations associated with the field.

Even after all of that, you will not be close to having a real grasp on infinity, so buckle up.

This is a sequence of characters intended to produce some profound mental effect, but it has failed.

• Member UncommonPosts: 2,078
Originally posted by //\//\oo

Originally posted by serjndestroy

Replacing 0 = 1 with 1 = 2 kinda defies the point, the a = b argument is solved, it relies more deeply on the 0 point here
I also played around with the  Infinity = Infinity + 1 train of thought.. Infinity - 1 = Infinity -1 but Infintity + 1 =  Infinity
Which could mean that 1 = 0.

No, infinity is not a number, so it doesn't have those properties.

Stop applying simple rules to complex things. You're way out of your league with infinity and the like.

First understand what the NATURAL numbers are, where they came from and the rules of the group ( go google group ).

Second understand what the RATIONAL numbers are, where they come from, and all applicable operations associated with the field.

Even after all of that, you will not be close to having a real grasp on infinity, so buckle up.

Go  //\//\oo!!

• Originally posted by serjndestroy

Yeah, I knew I was wrong, but still...
By the way, why the restrictions on infinity in equations?

Because infinity isn't a value, it's a never-ending sequence of numbers with no limit. Therefore it really has no place in equations like this.

Originally posted by //\//\oo

You can't arbitrarily create an equation and expect it to be true and have a solution.
You also can't call .000.... 1 a number, since there can be no number after the ...
Completely true.

When you have an equality like .9999...  = 1, it doesn't mean that .9999... is 1, but that the limit of the sequence .9999... is 1.
Completely false.
x = .999...
10x = 9.999...
10x - x = 9.999... - .999...
9x = 9
x = 1
Therefore .999... = 1

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Member, Newbie CommonPosts: 2,767

.999... is a sequence that converges to 1.0000...

The = operator and the rest of the operators are different from the same operators with the natural numbers. It makes no sense to say that an infinite sequence IS a number, but that it comes arbitrarily close to a number. That is why there are limits.

The = is notational.

The actual difference is an infinitesimal.

If you don't believe me, then construct the reals using Cantor's binary method (that is sequences of 1's and 0's) where every element is unique, since they can be bijected to the usual construction of the reals.

This is a sequence of characters intended to produce some profound mental effect, but it has failed.

• Apparently you didn't read my proof...

.999... isn't just a sequence of numbers, it can be expressed in real numerical terms. Much like 1/3 = .333...

.999... isn't some equational line diverging to 1... it's a number infinitely close to 1, that in proving, equals 1.

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Yes, I know what natural and rational numbers are, I know that 0 is a placeholder not a value,  that infinity is incomprehensible and not a number, and that is what irritates me...

Hail DnDOnlinegames!

• Originally posted by serjndestroy

infinity is incomprehensible and not a number

The former is false, the latter is true. It's not a number, but it is entirely comprehensible using sigma notation and looking at graphs.

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Member, Newbie CommonPosts: 2,767

Originally posted by ViolentY

Apparently you didn't read my proof...
.999... isn't just a sequence of numbers, it can be expressed in real numerical terms. Much like 1/3 = .333...
.999... isn't some equational line diverging to 1... it's a number infinitely close to 1, that in proving, equals 1.

Divergence? you're the first one to bring that up. Just because you have something that converges to a number does mean that it actually IS that number.

1/3 is just the two natural numbers that generate the sequence through the division operation.

You can come "infinitely close" to something without actually being equal to it. It is just used conveniently, since there is no real number small enough, yet bigger to zero to actually describe the actual difference.

It's just like how lim 1/x as x tends to infinity is equal to 0; the infinity is notational, not a real number.

If you don't believe me, then construct the reals in ZFC making sure that every element is unique and biject them to the reals in the common notation.

.9999... is it's own number. You can't have two elements of a set that are the same, yet unique.

That proof you presented is still using the same convenient notation, which is not a set theoretical equality.

This is a sequence of characters intended to produce some profound mental effect, but it has failed.

• My mistake, I meant convergence. I pulled an all-nighter last night, so everything is still feeling sort of fuzzy. :P

However, 1/x is an expression in terms of x, not a numerical value. As it approaches infinity, it converges to 0. This produces a line graphically.

.999... is a number that exists only between .9 and 1. It is a repeating decimal that repeats infinitely. This produces a point graphically.

I think you are getting your cases of infinity confused. With a number like .999..., the repeating "sequence" (though that term is a little confusing, as a sequence is more along the lines of sigma notation where x -> infinity, n = 0, {1 + 2 + 3 ...}) that just means its infinitely close to 1. If a point is infinitely close to another point, you can say that they are the same point.

Instead of using your 1/x as x approaches infinity, if you put infinity literally in for x, you get 1/(infinity) = 0. This is the basis for integration in calculus...

Once again, check my proof... Or ask any math professor, or do the google work.

::EDITED::

Try using this scenario:

1/3 = .333...

.333... + .333... + .333... = .999... Correct?

1/3      +  1/3     +  1/3     =   1

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Why do you people keep putting equal signs where they shouldn't be?

use: ≈ or ≠

• Member, Newbie CommonPosts: 2,767

Originally posted by ViolentY

My mistake, I meant convergence. I pulled an all-nighter last night, so everything is still feeling sort of fuzzy. :P
However, 1/x is an expression, not a numerical value. As it approaches infinity, it converges to 0. This produces a line graphically.
.999... is a number that exists only between .9 and 1. It is a repeating decimal that repeats infinitely. This produces a point graphically.
I think you are getting your cases of infinity confused. With a number like .999..., the repeating "sequence" (though that term is a little confusing, as a sequence is more along the lines of sigma notation where x -> infinity, n = 0, {1 + 2 + 3 ...}) that just means its infinitely close to 1. If a point is infinitely close to another point, you can say that they are the same point. Once again, check my proof... Or ask any calculus professor, or do the google work.

I'm not confused in the least. I understand what you call a "proof" (although that's not a proof of set theoretical equality) and am sorry to say that it is not applicable when dealing with the reals as individual elements of a set. The = operator in that case does mean that actual elements of the set are the same, but that they are the same in the field of reals.

The problem is that you have not yet taken a class in axiomatic set theory. While the field necessitates that .99... = 1.00..., the two reals are actually not the same when viewed in the real set by itself without the restrictions imposed by the field; the problem is that there doesn't exist a real number to describe the multiplicative inverse of .999.. (that is a, s.t. a*.999 = 1)., so that in order to be consistent in a field, the two numbers have to be the same IN THE FIELD.

In calculus you deal exclusively with the field of real numbers, so I can understand why you think the way you do.

So, the proper thing to say would be: .999.. = 1.00 in the FIELD OF REALS

.999..  and 1.00.. are not the same elements of the reals, but they are EQUAL in the field of reals

If you meant IS as in equality in the field of reals, then that is correct, but you didn't understand that when you typed it. Hopefully now you'll know the difference.

This is a sequence of characters intended to produce some profound mental effect, but it has failed.

•

Originally posted by gpett

Why do you people keep putting equal signs where they shouldn't be?
use: ≈ or ≠
Equal signs are being used appropriately. I think you're being confused by us using ".xxx..." to mean xxx-repeating infinitely. We aren't just trailing off with those periods. :P

Originally posted by //\//\oo

Originally posted by ViolentY

My mistake, I meant convergence. I pulled an all-nighter last night, so everything is still feeling sort of fuzzy. :P
However, 1/x is an expression, not a numerical value. As it approaches infinity, it converges to 0. This produces a line graphically.
.999... is a number that exists only between .9 and 1. It is a repeating decimal that repeats infinitely. This produces a point graphically.
I think you are getting your cases of infinity confused. With a number like .999..., the repeating "sequence" (though that term is a little confusing, as a sequence is more along the lines of sigma notation where x -> infinity, n = 0, {1 + 2 + 3 ...}) that just means its infinitely close to 1. If a point is infinitely close to another point, you can say that they are the same point. Once again, check my proof... Or ask any calculus professor, or do the google work.

I'm not confused in the least. I understand what you call a "proof" (although that's not a proof of set theoretical equality) and am sorry to say that it is not applicable when dealing with the reals as individual elements of a set. The = operator in that case does mean that actual elements of the set are the same, but that they are the same in the field of reals.

The problem is that you have not yet taken a class in axiomatic set theory. While the field necessitates that .99... = 1.00..., the two reals are actually not the same when viewed in the real set by itself without the restrictions imposed by the field; the problem is that there doesn't exist a real number to describe the multiplicative inverse of .999.. (that is a, s.t. a*.999 = 1)., so that in order to be consistent in a field, the two numbers have to be the same IN THE FIELD.

In calculus you deal exclusively with the field of real numbers, so I can understand why you think the way you do.

So, the proper thing to say would be: .999.. = 1.00 in the FIELD OF REALS

.999..  and 1.00.. are not the same elements of the reals, but they are EQUAL in the field of reals

If you meant IS as in equality in the field of reals, then that is correct, but you didn't understand that when you typed it. Hopefully now you'll know the difference.

Well, given the nature of the OP and the topic at hand, I was sure the field of all real numbers was perfect for all intents and purposes. I didn't think we really needed to go through the rigamarole of set theory. While I haven't taken a axiomatic set theory course, I've done work with some set theory stuff in calculus classes. But I couldn't handle math anymore, so I used those credits towards a degree in finance instead. :P

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli

• Member, Newbie CommonPosts: 2,767

Well, I've had a lot of math (B.S.) and just can't let kids go thinking that .99... is the same thing as 1.00.., because they don't have anything else to base equality on other than = in (+,*,R). (by the kid I mean the OP...who thought that 1=0)

I mean.. it's ok when all they're doing is operations in fields, but if they ever decide to go to higher math they're going to have to start from the ground up and it's not a pretty process.

What I don't understand is why they don't let kids split off into two paths, so kids that are more math/science oriented get a stronger theoretical basis while the others get a more applied basis without derivation. I don't have a degree in education, so I guess I shouldn't. Finance? That's a very lucrative field. I'm on my way to getting a degree in that myself... that is an MS in applied math with a concentration in financial applications.

I would have gone with pure math as that was my first academic love, but that doesn't pay as well. This is a sequence of characters intended to produce some profound mental effect, but it has failed.

• Originally posted by //\//\oo

Well, I've had a lot of math (B.S.) and just can't let kids go thinking that .99... is the same thing as 1.00.., because they don't have anything else to base equality on other than = in (+,*,R). (by the kid I mean the OP...who thought that 1=0)
I mean.. it's ok when all they're doing is operations in fields, but if they ever decide to go to higher math they're going to have to start from the ground up and it's not a pretty process.
What I don't understand is why they don't let kids split off into two paths, so kids that are more math/science oriented get a stronger theoretical basis while the others get a more applied basis without derivation. I don't have a degree in education, so I guess I shouldn't. Finance? That's a very lucrative field. I'm on my way to getting a degree in that myself... that is an MS in applied math with a concentration in financial applications.
I would have gone with pure math as that was my first academic love, but that doesn't pay as well. Yeah, I won't sugar-coat it at all; it was the money that drove me there. :P

In reply to the part about having to "start from the ground up," I don't think that would have been an issue had the poor kid thought that .999 = 1. Obviously when he gets into calculus, he'll learn the proofs that they are indeed equal in the set of real numbers, but once he starts learning about set theory, it shouldn't be like starting all over again and trying to catch up.

Sort of like the line in Troy... "When you know how to use it, you won't be taking my orders." :P

_____________________________________
"Io rido, e rider mio non passa dentro;
Io ardo, e l'arsion mia non par di fore."

-Machiavelli 