Are there different types of zero in mathematics?

[Naty1]

With the last year or so I came across a great discussion in these forums about the different types of infinity. It had never occurred to me there would necessarily be different types with different "values".

How about zero? Are there different mathematical concepts regarding it or are all zeros "equal" ? Any different types of zero with different mathematical meanings?? Any difference between approaching from plus, or minus, or "i" ?? Is a 1 x 1 matrix with zero the same as, say, an n x n matrix with all zeros?


[Physically, "zero" for physicsts is not always so clear...like absolute zero; zero energy in a volume of spacetime isn't...it's zero point energy with virtual particles,quantum activity; zero time in cosmology (less than 1 Planck time) etc,etc. ]

Thank you.

PS: for those interested, Wikipedia has a lot of interesting insights on "zero" here:
http://en.wikipedia.org/wiki/Zero

Among them: Zero is an even number; Zero is the smallest non negative integer;there is no calendar year zero but there is a cosmological year zero.

 

[Petek]

You may wish to read https://www.physicsforums.com/showthread.php?t=530207 from the FAQ.

 

[Godswitch]

The nature of zero...great argument

You can have "1 of nothing" and "nothing of 1" both terms have a value of "1" this being a quantitive value.

1 x 0 = 1x [where x=0]

 

[SteveL27]

The nature of zero...great argument

You can have "1 of nothing" and "nothing of 1" both terms have a value of "1" this being a quantitive value.

1 x 0 = 1x [where x=0]

Well ... you can have one zero. Or you can have two zeros. Or a billion zeros. But in each case you have zero.

 

[Naty1]

Well ... you can have one zero. Or you can have two zeros. Or a billion zeros. But in each case you have zero.

really? Are they ALL equal?? All infinities are not equal, so I'm still wondering if there are other subtle aspects of "zero".

Apparently I did not make my question clear:
I posted...

Any difference between approaching from plus, or minus, or "i" ?? Is a 1 x 1 matrix with zero the same as, say, an n x n matrix with all zeros?

It seems to me a 1 x 1 matrix of zeros, say describing some some apect of spacetime in a region, has a different connotation that an n x n matrix of all zeros describing a region of spacetime.

And zero in quantum mechanics just can't be as simple as in classical physics...or maybe it is??

 

[pwsnafu]

It seems to me a 1 x 1 matrix of zeros, say describing some some apect of spacetime in a region, has a different connotation that an n x n matrix of all zeros describing a region of spacetime.

Spacetime? What does that have to do with matrices?
There is precisely one 1 by 1 matrix of zero, namely the zero matrix.
Similarly for any n, there is precisely one n x n matrix with all elements zero, namely the zero matrix.

Heck you wrote "1 x 1 matrix of zeros". How can you have more than one zero in a matrix with one element?

You seem to have muddled up something very fundamental.

And zero in quantum mechanics just can't be as simple as in classical physics...or maybe it is??

Why would it be different? Why should QM affect arithmetic?

 

[Naty1]

Spacetime? What does that have to do with matrices?

You should read up on, say, tensors:

http://en.wikipedia.org/wiki/Tensor
there is a pretty technicolor picture for you to see.


anyway, the zero issue is no big deal...

 

[Hurkyl]

With the last year or so I came across a great discussion in these forums about the different types of infinity. It had never occurred to me there would necessarily be different types with different "values".

And to briefly state the point -- there are lots of different kinds of things one would do that need a notion of infinity.

How about zero? Are there different mathematical concepts regarding it or are all zeros "equal" ?

Zero is somewhat more uniform. It is virtually always used in the following two situations:

• There's an abelian group with operation '+', and 0 is its additive identity. (0+x = x)
• There's a monoid with operation written multiplicatively (e.g. '\times', '\cdot', or just by juxtaposition), and 0 is a nullary element. (0x = 0)

There are lots of different kinds of systems that have a zero -- the zero integer, the zero rational number, the zero real number, the zero complex number, the zero matrix of each size, the zero vector of a vector space, the zero polynomial, the zero rational function, the zero scalar field, the zero vector field, and so forth.

While each of these is technically a different zero, they really do all appear very similar. The only oddity I can think of is in some situations, you might have xy=0, despite neither x=0 nor y=0.


The only substantially different examples I can think of is if you move up a level in algebra, and start talking about things like "the zero group" or "the zero ring" or the "zero vector space". But even those aren't too dissimilar from the usual examples from the perspective of 2-algebra or higher.