The two types of Zero

[Balthazzar]

The two types of zero - Could they be relevant in relation to how we perceive the numbers that follow, zero being the starting point for what comes after? One perception being the numbers expand and increase away from the origin and the other perception would be inside the origin itself, expanding and increasing its scope or value while remaining self-contained (as its own universe) as opposed to drifting further away from the source or starting point.

 

[Polymath257]

Negative 3 plus 0 after, equates to negative 30.

Well, that isn't the definition of +, is it? It is specific to the decimal notation for numbers, which is a *representation* of numbers, not the numbers themselves.

But, if you 'add 0' to the end of 3, in your fashion, you would get 30.

If a negative zero were applied, would it still equate to a negative 30 or something else, understanding that there is no negative for the zero?

huh? You also can't put -4 after the 3 in the decimal representation. But you can add -4 to 3 and get (-4)+3=-1.

Anti matter vs matter is where the inquiry came concerning the 0. I never knew it existed until it did. I don't think a negative zero has an existence in mathematics, so I was questioning why?

This still seems confused. The baryon number (for matter/anti-matter) is conserved. if you have a proton (baryon number +1) and an anti-proton (baryon number -1), the combination will leave you with a baryon number of 0. And -0=0 in this situation as well.

The negative of zero does 'have an existence in mathematics'. The negative of 0 is 0: -0=0.

 

[Polymath257]

The two types of zero - Could they be relevant in relation to how we perceive the numbers that follow, zero being the starting point for what comes after? One perception being the numbers expand and increase away from the origin and the other perception would be inside the origin itself, expanding and increasing its scope or value while remaining self-contained (as its own universe) as opposed to drifting further away from the source or starting point.

Well, in standard set theory, the number 0 is defined to be the empty set {}. The number 1 is defined to be {0}={{}}. The number 2 is defined to be {0,1}. The number 3 is defined to be {0,1,2}, etc.

But, at that level no number other than 0 has a negative and -0=0. It is only by some trickery that negative integers can be defined. Getting fractions (rational numbers) and real numbers is also rather tricky when starting at the foundation.

 

[Balthazzar]

Well, in standard set theory, the number 0 is defined to be the empty set {}. The number 1 is defined to be {0}={{}}. The number 2 is defined to be {0,1}. The number 3 is defined to be {0,1,2}, etc.

But, at that level no number other than 0 has a negative and -0=0. It is only by some trickery that negative integers can be defined. Getting fractions (rational numbers) and real numbers is also rather tricky when starting at the foundation.

I've never been very mathematically literate. Question: If (0,1,2) equates to 3, then this suggests that 0 has the value of 1, right? Nice, but very confusing. It implies the opposite of empty.

 

[Polymath257]

I've never been very mathematically literate. Question: If (0,1,2) equates to 3, then this suggests that 0 has the value of 1, right? Nice, but very confusing. It implies the opposite of empty.

Nope. The point is that the set {} has 0 elements, the set {0} has 1 element, the set {0,1} has 2 elements, and the set {0,1,2} has 3 elements.

 

[Balthazzar]

Nope. The point is that the set {} has 0 elements, the set {0} has 1 element, the set {0,1} has 2 elements, and the set {0,1,2} has 3 elements.

I understand.
I only have a basic understanding of the language. I've been wanting to learn a new one. Mathematics would appear to be most useful. Thank you.

 

[Ponder This]

Question: Mathematics have an infinite ability both ways for calculations. My question is why is a negative zero not useful in equations? I'm asking based on the nature of opposites, which zero does not appear to have, due to it being neutral and not a true number, yet it's also most powerful. Is this due to simplicity of calculations or because utilizing a negative counterpart is irrelevant?

You might find it interesting to study the concept of limits from the right and limits from the left, which is a concept useful in the development of Calculus. For example, you have a function like

f(x) = 1 if x>0​

f(x) = 0 if x=0​

f(x) = -1 if x<0​

limit of f(x) as x->0+ is defined as the limit of f(x) as x->0 from the right. This limit happens to be 1.
limit of f(x) as x->0- is defined as the limit of f(x) as x->0 from the left. This limit happens to be -1.

See also Left and Right-Hand Limits.

+0 and -0 might also serve as useful notation for talking about very small nonzero numbers that are nonetheless close in a subjective sense to zero.
For example, 0.000000000000000000000000000000000000000000000001 is close to 0. Talking about +0 could be a way of talking about these numbers which are very close to zero but still positive.

Another way to use the notation of +0 and -0 is in certain variable types in programming that use a bit to denote + or - and the rest of the variable denotes a number (which could be 0). What -0 means in programming is whatever the progammer decides to use it for. Maybe the programmer will use -0 to denote NULL.

The concept of a negative counterpart is not without practical or theoretical utility. I would not say I've given an exhaustive exposition of the potential uses of these concepts.

 

[Balthazzar]

You might find it interesting to study the concept of limits from the right and limits from the left, which is a concept useful in the development of Calculus. For example, you have a function like

f(x) = 1 if x>0​

f(x) = 0 if x=0​

f(x) = -1 if x<0​

limit of f(x) as x->0+ is defined as the limit of f(x) as x->0 from the right. This limit happens to be 1.
limit of f(x) as x->0- is defined as the limit of f(x) as x->0 from the left. This limit happens to be -1.

See also Left and Right-Hand Limits.

+0 and -0 might also serve as useful notation for talking about very small nonzero numbers that are nonetheless close in a subjective sense to zero.
For example, 0.000000000000000000000000000000000000000000000001 is close to 0. Talking about +0 could be a way of talking about these numbers which are very close to zero but still positive.

Another way to use the notation of +0 and -0 is in certain variable types in programming that use a bit to denote + or - and the rest of the variable denotes a number (which could be 0). What -0 means in programming is whatever the progammer decides to use it for. Maybe the programmer will use -0 to denote NULL.

The concept of a negative counterpart is not without practical or theoretical utility. I would not say I've given an exhaustive exposition of the potential uses of these concepts.

Adding more depth to 0 while bridging the divides through decimal separation. It could eventually go back to a 0, then the process would be able to start all over again at another 0 point of separation if structured for that purpose, haha. Multi-directional mathematics or something to that effect with an addition of a "zero-sphere" as whole point - Null could be applied as a single point of origin on the zero-sphere for the resourcing of itself with additions of other numerals ...

 

[Polymath257]

You might find it interesting to study the concept of limits from the right and limits from the left, which is a concept useful in the development of Calculus. For example, you have a function like

f(x) = 1 if x>0​

f(x) = 0 if x=0​

f(x) = -1 if x<0​

limit of f(x) as x->0+ is defined as the limit of f(x) as x->0 from the right. This limit happens to be 1.
limit of f(x) as x->0- is defined as the limit of f(x) as x->0 from the left. This limit happens to be -1.

See also Left and Right-Hand Limits.

+0 and -0 might also serve as useful notation for talking about very small nonzero numbers that are nonetheless close in a subjective sense to zero.
For example, 0.000000000000000000000000000000000000000000000001 is close to 0. Talking about +0 could be a way of talking about these numbers which are very close to zero but still positive.

Another way to use the notation of +0 and -0 is in certain variable types in programming that use a bit to denote + or - and the rest of the variable denotes a number (which could be 0). What -0 means in programming is whatever the progammer decides to use it for. Maybe the programmer will use -0 to denote NULL.

The concept of a negative counterpart is not without practical or theoretical utility. I would not say I've given an exhaustive exposition of the potential uses of these concepts.

Usually, in one-sided limits, the + or - sign is above and to the right of the target value of x, not to the left.

It should also be noted that this does NOT say that 0+ and 0- are actual numbers any more than a limit for x->+infinity implies that +infinity is a number (it isn't).

 

[Ponder This]

Usually, in one-sided limits, the + or - sign is above and to the right of the target value of x, not to the left.

It should also be noted that this does NOT say that 0+ and 0- are actual numbers any more than a limit for x->+infinity implies that +infinity is a number (it isn't).

When we get more precise about what a limit is, we talk about what it means to get close to a number and we can get close to a number from different directions. A limit towards infinity is getting farther away from a number...
But, topologically, the real line is homeomorphic to the open-ended line segment from -1 to 1. We can think of a limit towards +infinity as a limit towards +1 on the line segment from -1 to 1. This means there's something very interesting happening with infinity, the number that is not a number. And we can think of getting close to infinity as if we were getting close to a number by defining what it means to get close to +1 on the open-ended line segment from -1 to +1.

 

[Polymath257]

When we get more precise about what a limit is, we talk about what it means to get close to a number and we can get close to a number from different directions. A limit towards infinity is getting farther away from a number...

Since you mention topology below, it is probably more accurate to say that a limit towards infinity is getting farther away from *every* number.

But, topologically, the real line is homeomorphic to the open-ended line segment from -1 to 1. We can think of a limit towards +infinity as a limit towards +1 on the line segment from -1 to 1. This means there's something very interesting happening with infinity, the number that is not a number. And we can think of getting close to infinity as if we were getting close to a number by defining what it means to get close to +1 on the open-ended line segment from -1 to +1.

There is a standard way of dealing with this for any locally compact hausdorff space, called the one-point compactification. This is a 'smallest' compact hausdorff space in which the original space can be embedded. In the case of the open interval (-1,1), or equivalently, the real line, the one-point compactification is homeomorphic to a circle. In this case a 'limit towards infinity' is simply a limit to the 'one-point' that is added in the compactification.

For the real line specifically, there is also a 'two-point' compactification that essentially gives the closed interval [-1,1]. Limits here correspond to going to +-infinity.

There is also a more general concept called the Stone Cech compactification, which is a 'largest' compact space in which the original is embedded as a dense subspace. These compactifications tend to be very complicated, though.

If you are interested in details, you might want to take this to PM, though.