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The additive identity element is unique. Does this imply that all zeros are not distinct? Am struggling to explain how a unique additive element does not rule out more than one number zero. I am thinking it only rules out using more than one zero at a time in an arithmetic that includes addition.

Jonathan Cender
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    Considering imaginaries and reals are subsets of the complex number, it's the same zero. (Unless, of course, you don't view reals as a literal subset of the complex numbers, e.g. in constructions of the complex numbers using pairs or matrices - then there's just an isomorphic copy of the reals, technically.) But you're right in your thinking. For example, the $0$ in the integers mod $p$ is not considered the same as the complex zero, although there are certain situations when it's useful to think of them interchangeably... – anon Oct 07 '20 at 00:37
  • @runway44 Is 0 in the integers mod p an additive identity element? I assume it is, but not sure. I appreciate your example that better supports my general point. – Jonathan Cender Oct 07 '20 at 01:48
  • What is "imaginary zero" vs "complex zero"? – rschwieb Oct 07 '20 at 13:53
  • @rschwieb "Imaginary zero" vs "complex zero" is 0i and 0+0i. The imaginaries are numbers in their own right. – Jonathan Cender Oct 07 '20 at 17:06
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    @JonathanCender When you say "imaginaries" do you mean scalar multiple of $i$? – rschwieb Oct 07 '20 at 19:04
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    @rschwieb I mean the numbers on the y-axis of the complex plane, the imaginary part of a complex number. https://en.wikipedia.org/wiki/Imaginary_number – Jonathan Cender Oct 07 '20 at 19:37
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    @JonathanCender I believe that is what I'm saying too. Thanks for clarifying. – rschwieb Oct 07 '20 at 19:38

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"$0$" is just a name given to a particular element of some sets (in this case, an abelian group.)

In two disjoint sets, each one can have a member playing the role of $0$, and since the sets are disjoint the two zeros have to be distinct.

For a subgroup of a group, though, their identities must be the same element. In that case, there is definitely only one element playing the role of the identity in both sets.

Most of the time we think of the integers, rationals, reals and complex numbers as being a chain in which each is a subgroup (a subring even) of the next, and in that picture they all share the same additive identity (and multiplicative identity, for that matter.)

But in principle one could specify two disjoint sets, both isomorphic to $\mathbb Z$, and it would be the case that there are "two zeros," one in each copy.

I could be getting the wrong impression, but the question posed seems to imply that there is some sort of "absolute zero" that must be shared between all additively written abelian groups, which is not the case. "$0$" is just a special notation for a particular element in a set. It's not a universal constant. Apologies if the impression I am laboring under is incorrect.

@rschwieb "Imaginary zero" vs "complex zero" is 0i and 0+0i. The imaginaries are numbers in their own right

This represents a misapprehension about notation, I think. Writing "$0+0i$" is a little ambiguous. When representing complex numbers as pairs of real numbers, each complex number is uniquely represented as $(a,b)$ for two real numbers $a,b\in\mathbb R$.

Now, one can modify the notation by introducing the following conventions: $$i=(0_\mathbb R,1_\mathbb R)$$

$$1_\mathbb C=(1_\mathbb R,0_\mathbb R)$$

$$0_\mathbb C=(0_\mathbb R,0_\mathbb R)$$

, and then one can correctly write that $a1_\mathbb C+bi=a(1_\mathbb R,0_\mathbb R)+b(0_\mathbb R,1_\mathbb R)=(a,b)$.

As you can see, $0_\mathbb R1_\mathbb C+0_\mathbb R i=(0_\mathbb R,0_\mathbb R)=0_\mathbb R i$ are the same element, so there's no point in distinguishing them.

In this scheme, $0_\mathbb R$, viewed only as a coefficient of a linear combination in $\mathbb C$, is not the same thing as $0_\mathbb C$. To view $\mathbb R$ as a subset of this model of $\mathbb C$, one has to make the further identification $a\mapsto a1_\mathbb C=(a,0)$, which of course identifies $1_\mathbb R$ with $1_\mathbb C$.

rschwieb
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  • Thank you. The person I am talking to seems to think there is one zero that is shared by all mathematical structures -- in some sense an "absolute zero" as you say. Since I am introducing a new number zero, an alternative based on a definition different than the empty set that substitutes for 0, dealing with this point about the uniqueness of the additive element precluding anything other than an "absolute zero" is important to me. – Jonathan Cender Oct 07 '20 at 17:21
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    @JonathanCender I've added a bit pertaining to your earlier comment. – rschwieb Oct 07 '20 at 20:39
  • @rschwieb What does (0_R * 1_R) + (0_R * i) mean here (near the "As you can see" paragraph)? The first part of the sum leads to a scalar while the second part is a pair of numbers. – Vi0 Jan 10 '24 at 23:03
  • @Vi0 Sorry i hit the wrong button on my phone, apparently. I think you're right about the R->C update. – rschwieb Jan 11 '24 at 02:27