# *aa3.2 Let Q be the group of rational numbers under addition

• MHB
• karush
In summary: In $Q$,$\langle\frac{1}{2}\rangle+\langle\frac{1}{4}\rangle=\langle\frac{1}{2+\frac 12}{4}\rangle$In $Q$,$\langle\frac{1}{2}\rangle+\langle\frac{1}{4}\rangle=\langle\frac{1}{2+\frac 12}{4}\rangle$2
karush
Gold Member
MHB
aa3.2

Let Q be the group of rational numbers under addition
and let $Q^∗$ be the group of
nonzero rational numbers under multiplication.
In $Q$, list the elements in $\langle\frac{1}{2} \rangle$,
In ${Q^∗}$ list elements in $\langle\frac{1}{2}\rangle$

ok just had time to post and clueless

Last edited:
Hi karush,

Can you provide some attempt, or at least some initial thoughts please?

I like Serena said:
Hi karush,

Can you provide some attempt, or at least some initial thoughts please?
In Q,
$\langle\frac{1}{2}\rangle =\left\{\cdots,-2,-\frac{3}{2},-1,-\frac{1}{2}, 0,\frac{1}{2},1,\frac{3}{2},2\cdots\right\} =\left\{\frac{n}{2}\vert n \in \Bbb{Z}\right\}$

karush said:
In Q,
$\langle\frac{1}{2}\rangle =\left\{\cdots,-2,-\frac{3}{2},-1,-\frac{1}{2}, 0,\frac{1}{2},1,\frac{3}{2},2\cdots\right\} =\left\{\frac{n}{2}\vert n \in \Bbb{Z}\right\}$

Yep. That is correct. (Nod)

How did you find it?
Can you also find $\langle\frac{1}{2}\rangle$ in $\mathbb Q^*$?

I like Serena said:
Yep. That is correct. (Nod)

How did you find it?
Can you also find $\langle\frac{1}{2}\rangle$ in $\mathbb Q^*$?
In $Q*$
$\langle\frac{1}{2}\rangle =\left\{\cdots, 4,2,1,\frac{1}{2},\frac{1}{4}, \cdots\right\} =\left\{2^n\vert n \in \Bbb{Z}\right\}$
sorta!

$Q$ and $Q^\ast$ as above.
Find the order of each element in $Q$ and $Q^\ast$

ok is this like $4=2^2$

Last edited:
karush said:
In $Q*$
$\langle\frac{1}{2}\rangle =\left\{\cdots, 4,2,1,\frac{1}{2},\frac{1}{4}, \cdots\right\} =\left\{2^n\vert n \in \Bbb{Z}\right\}$
sorta!

$Q$ and $Q^\ast$ as above.
Find the order of each element in $Q$ and $Q^\ast$

ok is this like $4=2^2$

Nope.
The order of an element is the number of times we have to add (respectively multiply) it before we get the identity element.
So in $\mathbb Q$, how many times do we have to add $\frac 12$ before we get $0$?
And how many times do we have to add $0$ before we get $0$?

## 1. What is the group Q under addition?

Q, or the set of rational numbers, is a mathematical group under the operation of addition. This means that any two rational numbers can be added together to get another rational number. The group Q includes all fractions and integers, and is closed under addition, meaning that the sum of any two rational numbers will also be a rational number.

## 2. How is the group Q different from the group of real numbers under addition?

While the group Q includes all fractions and integers, the group of real numbers includes irrational numbers as well. This means that the group Q is a subset of the group of real numbers, as all rational numbers are also real numbers. Additionally, the group of real numbers is not closed under division, while the group Q is closed under both addition and multiplication.

## 3. What is the identity element of the group Q under addition?

The identity element, also known as the identity element of addition, is 0. This means that when 0 is added to any rational number, the result is that same rational number. In other words, 0 is the neutral element in the group Q under addition.

## 4. Is the group Q commutative under addition?

Yes, the group Q is commutative, or abelian, under addition. This means that the order in which the rational numbers are added does not affect the result. For example, 2+3 will always equal 3+2, regardless of the specific rational numbers used.

## 5. How is the group Q related to other mathematical groups?

The group Q is a special case of the group of real numbers under addition, and is also a subgroup of the group of rational numbers under multiplication. It is also isomorphic to the group Z, or the set of integers, under addition. This means that the group Q shares similar properties and behaviors with these other groups, but is unique in its own right.

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