Group of real numbers under addition I saw this statement in a paper (not a mathematical paper):
"Although the operations of addition and multiplication are defined in the field of real numbers, multiplication is undefined in the group of real numbers under addition."
I would be grateful if someone could explain this in very simple terms. I am absolutely ignorant about this area of mathematics, though I have read the definitions of group and field.
 A: The answer to this question is rather simple. I will present briefly what is going on, without giving the definition of a group. You can go search for that easily, but I think you can follow the answer below without knowing the details, even tho a group is defined quickly, and you do not need much mathematics to understand it. So I recommand you to look it up.
A group is by definition a set $G$ with a binary operation $\circ: G\times G\to G$, which has to fulfill some properties. Here $\circ$ is just a sign. The group operation vary. Sometimes we have a group with $+$. Sometimes we have groups with $\cdot$ as operation. If we look of sets of matrices, we have a group with matrix addition $\oplus$, and so on.
So a group just has one operation, and not two (like the two operations $+$ and $\cdot$ in $\mathbb{R}$).
If we want a group structure on $\mathbb{R}$, we can take the operation $+$.
This fulfills every property there is.
But $\mathbb{R}$ with multiplication $\cdot$ would not give a group, as this does not have every needed property.
It would fail, because one property is that every element needs an inverse, so for every $a\in\mathbb{R}$ there has to be a $b\in\mathbb{R}$ with $a\cdot b=b\cdot a=1$ (1 would here be the so called 'neutral element').
But as you know $0$ has no such inverse. As $a\cdot 0=0$ for every $a\in\mathbb{R}$, and not $1$.
For a more elaborate look on groups you can check the wikipedia page about groups.
However, we could define a group structure on $\mathbb{R}\setminus\{0\}$ denoted of $\mathbb{R}^\ast$, with muliplication as operation.
A: The word "group" in "group ... under addition" means (by the definition of a group) that you are going to use only one operation, and in this case it will be addition. The word "field" means (again by definition) that you are going to be considering two operations, addition and multiplication.
I like analogies, so here's one for this case: When you listen to an MP3 of a song, you have no idea what the color of the lead guitar was, but if you watch a video of that song's performance you probably will. If someone starts telling you how much they liked the cherry red guitar in the MP3 of "The Cradle Will Rock", you're going to say "Wait, you mean in the video, right? You can't see the color of the guitar in the MP3.". When we restrict ourselves to the group $\langle \mathbb R, +\rangle$ we can't see multiplication. If you object that "but multiplication still exists" then perhaps it's better to say that we don't look at it, or don't consider it in the group setting.
BTW, this statement reflects the modern approach to abstract algebra. We try to cut mathematical objects down to as few operations as possible and make as few assumptions about them as possible, and see what we can deduce from them. You can always add more operations, or more assumptions, but we are excruciatingly clear about what we are including, and often use completely different names. So if you said "In the group of the reals, when we multiply ..", my mathematically trained eyebrow would go up, and I would rudely interrupt you with "Multiply? So you mean the field of the reals, not the group?". Like I said, excruciatingly clear.
