Zero = Nothing?

[TheDivineFool]

This is from a'book on math I read.

 

1. Solve x + 4 = 4

x + (4 - 4) = 4 - 4

x = 0

 

2. Solve x = x + 1

x - x = x + 1 - x

0 = 1

What's the value of x?

 

I understand that there's no solution to this equation. But if someone were to ask you ''What is the value of x?'' and you must reply in the following format...''The value of x is_____'' what would you write in the blank space?

 

You can't write ''nothing'' because ''nothing'' is zero and zero is definitely NOT a solution to the problem.

 

Or is 'zero' different from 'nothing'?

 

Your valued comments please.

[Strange]

The value of x is undefined

[Sensei]

2. Solve x = x + 1

You can't have such equation.

 

You can have f.e.

 

y=x+1

 

or

 

x'=x+1

 

Equation x=x+1 (aka x++) is used only in software programming, and is incrementing variable.

[Strange]

You can't have such equation.

 

But there it is...

Equation x=x+1 (aka x++) is used only in software programming, and is incrementing variable.

 

That's a statement, not an equation.

[studiot]

Get a better maths book

 

[math]\mathop {\lim }\limits_{x \to \infty } \left( {\frac{x}{{x + 1}}} \right) = 1[/math]

[Greg H.]

 

But there it is...

 

The fact that you can write an equation does not make it a valid equation.

 

And if it's invalid equation, the answer is meaningless to begin with.

[studiot]

 

gregH

 

Where was it specified that the solution has to be a real number?

[derek w]

x = y-1

 

therefore x=y

[Greg H.]

 

Where was it specified that the solution has to be a real number?

My argument wasn't based on the solution, my argument was based on the fact that a value x cannot be equal to itself plus a constant. It literally isn't an equation; those two things cannot be set equal for any value of x.

[studiot]

My argument wasn't based on the solution, my argument was based on the fact that a value x cannot be equal to itself plus a constant. It literally isn't an equation; those two things cannot be set equal for any value of x.

 

Why not?

 

The clue was in my post5, set nearly half an hour before yours.

 

What do you get if you add 1 to infinity?

 

All you need is a suitable number system that includes infinity.

[John]

The OP does provide some context. The second equation is shown to result in 0 = 1. Thus, without redefining 0 or 1, the equation is invalid unless we're working in a structure like the zero ring, in which 0 and 1 coincide.

 

But since the equation is listed next to another one which mentions the element "4," I'm assuming we're not working within such a structure.

As for the OP's first question, I concur with Strange. The value of x is undefined.

For the OP's second question, I'd say that zero can be considered distinct from nothing. For example, the empty set {} is different from the set {0}. Of course, we could also say the first set contains zero elements, but then we're talking about cardinality of the set, rather than about the set itself. The empty set contains nothing, but doesn't contain zero.

 

So in the context of the equations in the OP, in the field of real numbers, the solution set for the first is {0}, whereas the solution set for the second is {}.

[ajb]

Or is 'zero' different from 'nothing'?

Zero is different from 'nothing' if we understand 'nothing' as the empty set.

 

Thus, for your second system of linear equations the solution set is the empty set.

 

This reminds me of a story I was told by my former supervisor about a friend of his teaching in the UK. This guy would have been from Russsia or at least the former Soviet Union. Anyway, in class he asked some question and prompted the class for a reply. Someone answered 'nothing'. He looked a little surprised at this answer and replied '...the answer is zero'.

Edited April 4, 2015 by ajb

[John Cuthber]

 

Where was it specified that the solution has to be a real number?

The OP specifies

"I understand that there's no solution to this equation.".

That rules out any solution that is actually a solution

It's perfectly possible that, in some branch of maths that's not confined to the real numbers, there is a solution- but that's not what we are talking about here.

[Strange]

The fact that you can write an equation does not make it a valid equation.

 

And if it's invalid equation, the answer is meaningless to begin with.

 

Indeed. I was just commenting on the "you can't have such equation" (you know, like people who say "irregardless isn't a word").

[studiot]

 

John Cuthber

The OP specifies

"I understand that there's no solution to this equation.".

That rules out any solution that is actually a solution

 

That is a poor train of reasoning.

 

The OP believes (wrongly) that there is no solution as evinced by the words "I understand that there is no solution."

 

The OP then allows for the possibility that he may be mistaken by asking the question

 

 

I understand that there's no solution to this equation. But if someone were to ask you ''What is the value of x?'' and you must reply in the following format...''The value of x is_____'' what would you write in the blank space?

 

You can't write ''nothing'' because ''nothing'' is zero and zero is definitely NOT a solution to the problem.

 

Or is 'zero' different from 'nothing'?

 

Your valued comments please.

 

 

Are you asserting that infinity is not a correct answer to the question "what still equals itself after one is added?"

[Greg H.]

Infinity is also not a defined value, so the answer undefined is more precise, since it could also be negative infinity, or 2(infinity) or any other "value" that includes an infinity.

[studiot]

Infinity is also not a defined value, so the answer undefined is more precise, since it could also be negative infinity, or 2(infinity) or any other "value" that includes an infinity.

 

So there is more than one solution to the equation.

 

So what, that is not an uncommon situation.

 

In fact some equations have infinitely many solutions.

[TheDivineFool]

Thanks all.

 

1. Some say x = x + 1 is not a "valid" equation. What do you mean by this? Is it something like "x + = % > 3" : a grammatical error of sorts. Or do you mean it is meaningless, undefined? If I remember correctly, there was a time when 3 - 23 had no meaning - negative numbers didn't exist. I was hoping that the equation x = x + 1 is something like this...Is there another type of number which could satisfy this equation?

 

2. Some have suggested infinity as an answer and I can't deny that INFINITY + 1 = INFINITY. But someone remarked that ''infinity" is not really an answer because it too is undefined. I have my doubts about this objection. Infinity is defined. So the objection can't be based on the premise "infinity is undefined". At least that's what I think.

 

3. I also liked the answer from a set perspective. The solution set for x + 4 = 4 is {0}. The solution set for x = x + 1 is { }, the empty set.

This implies that there is a difference between 0 and the "nothingness", if you can call it that, in { }. It is that interests me, in a silly way perhaps.

Consider that before zero was invented/discovered, the solution set for 9 - 9 was { }. That's "nothing" right?

Of course, this "nothingness" is "easier". It is simply an absence of numbers {1, 2, 3,...} later named as "zero". But the "nothingness" of x = x + 1 is the absence of {..., -2, -1, 0, 1, 2, 3,...}. One could say that this "nothingness" is at a different level - the solution is not even nothing.

 

It's quite obvious that the issue is trivial. It doesn't seem to bother mathematicians. But I just was wondering if you all might know how to answer this question as a mathematician. I mean what would a mathematician say? "Don't ask silly questions"...I wonder.

[John]

1. What we mean by that is that in the field of real numbers (that is, the set of all real numbers along with addition and multiplication defined in the usual way), there is no x such that x = x + 1. It's fine syntactically and (I would argue) semantically, but there is no solution to the equation.

2. Infinity isn't undefined, but it also isn't a real number. Thus, as mentioned by studiot earlier, we'd need to move to a structure in which infinity is an element and define addition such that infinity + 1 = infinity. But we still have to be careful about how we operate with infinity.

 

3. I suppose so. Before zero was considered to be a number, one might have said something like 9 - 9 had no solution, though it'd still be clear that taking 9 parts away from 9 parts would leave no parts. However, using modern mathematics, "inventing" some solution for x = x + 1 would either result in the contradictory statement 0 = 1 or involve giving up the (very useful) field structure of the real numbers.

I wouldn't necessarily say the issue is trivial so much as deep enough in the foundations of mathematics that most mathematicians wouldn't worry about it in their day-to-day lives. What a mathematician would say in response to x = x + 1 is that given the standard definitions, there is no solution. If you forced the mathematician to fill in the blank with "The value of x is ______," I would imagine most would say "undefined" as mentioned previously, or perhaps some would say "not a real number."

[studiot]

So, as John said, the OP was not an idle question.

 

One further point is that what we are prepared to accept x to be is very important.

 

We have all agreed it cannot be a Real number.

 

The null set has been proposed as a 'solution' to the equation.

So what do we mean by and equation between sets?

 

Infinity has also been offered and , Fool, you have noted elsewhere that one definition of infinity is "An infinite set is a set that is equivalent to a proper subset of itself.", although you have not come back to that thread for some reason?

 

Of course the null set is a set with no proper subsets - a sort of 'opposite end of the spectrum' to the infinite set.

 

 

 

 

One a further note, the issue of the difference between zero and nothing has emerged for discussion about every four to six months since I have been here.

The site search is very poor in this respect but there are several threads discussing this at great length.

 

One consequence of the null set is the observation that it is possible to build all the numbers from nothing at all.

 

Start with the null set.= {} = 0

Construct a set with one member only, the null set ={{}} = 1

Construct another set with two members, the null set plus one copy of the null set {{},{}} = 2

 

and so on.

[TheDivineFool]

1) x = x + 1

2) x - x = x + 1 - x

3) 0 = 1

 

There are two things going on here.

1. We have mathematical statements

2. We're applying the rules of math to these statements

 

The answer is a contradiction because 0 = 1 AND 0 =/= 1

 

So, something is wrong. The mistake can be either in 1 or 2.

It is not in 2 because this consists of simple algebraic steps and as far as I can see, we've not made any mistakes.

 

Therefore, the error lies in 1. The statement x = x + 1 is FALSE.

There is no real number that can satisfy the equation. In other words x is NOT a real number.

 

Now, we have things called imaginary numbers. The sqrt(-1) is also NOT a number.

 

Is sqrt(-1) is undefined? I think sqrt(-1) is undefined on the set of real numbers. Just like the x in x = x + 1.

 

So, the 'undefined in terms of real numbers' nature of some mathematical entities doesn't mean we can't make sense of it in a different, perhaps useful, way.

[studiot]

 

2. We're applying the rules of math to these statements

 

 

 

Careful.

We are applying some rules of some of mathematics.

 

If what you said in post 21 was completely valid then what about an alternative rearrangement of the equation, such as I offered in post#5, that does not lead to a contradiction ?

 

This view, of course, leads to an alternative view of infinity.

Edited April 4, 2015 by studiot

[John]

Now, we have things called imaginary numbers. The sqrt(-1) is also NOT a number.

 

Is sqrt(-1) is undefined? I think sqrt(-1) is undefined on the set of real numbers. Just like the x in x = x + 1.

 

So, the 'undefined in terms of real numbers' nature of some mathematical entities doesn't mean we can't make sense of it in a different, perhaps useful, way.

 

Well, as mentioned, there are other structures in which we can assign a solution. But the spirit of the OP seems to be that we want to be able to use x with standard operations in standard arithmetical expressions--that is, in some structure similar to or extending the real numbers, and that's a bit tricky.

 

Imaginary numbers are certainly numbers, and the choice of the term "imaginary" is a bit unfortunate. Complex analysis is one of the most useful branches of mathematics, both in mathematics and elsewhere.

 

We can go even further, into systems like the quaternions and even the octonions, where we have multiple square roots of -1. While we do lose the field structure in these more exotic systems, they still find application in various branches of mathematics and physics.

[ajb]

Therefore, the error lies in 1. The statement x = x + 1 is FALSE.

There is no real number that can satisfy the equation. In other words x is NOT a real number.

The usual conclusion here is that there is no real x that satisfies this equation. You could offer more exotic solutions, here plus or minus infinity would work if we are considering the extended real numbers (or some system very similar).

 

Now, we have things called imaginary numbers. The sqrt(-1) is also NOT a number.

It is not a real number, but as we have a field complex numbers are 'numbers', but this is just nomenclature.

 

Is sqrt(-1) is undefined? I think sqrt(-1) is undefined on the set of real numbers. Just like the x in x = x + 1.

x - sqrt(-1) =0 has no real solution, which is what you want to say. In this sense it is undefined.

 

So, the 'undefined in terms of real numbers' nature of some mathematical entities doesn't mean we can't make sense of it in a different, perhaps useful, way.

Indeed, mathematics is not just about real numbers. We can build much more interesting things.

Edited April 5, 2015 by ajb