I'm watching YouTube videos to teach myself some abstract algebra, a woman claims that integers under multiplication has a closed operation. But I don't understand why.
For example: $$7x = 3$$
The 3rd element is not an integer.
Please advise
[Math] Integers under multiplication a closed operation
abstract-algebra
Related Solutions
Fun fact to know: in 1992 Ales Drápal proved that if two finite groups agree on 89% on their multiplication tables, the groups must be isomorphic! He conjectured that the same holds true if the tables agree on 75% of their entries. The conjecture has not been proved yet. See also Groups St. Andrews 2001 at Oxford, featuring the paper of Drápal On the distance of 2-groups and 3-groups.
I think all you need has been pointed here. You don't need to write down the associated Cayley table for the presented set and its operation, but for this one it leads you to get the answer graphically:
We can see that the operation is a binary one. Can you find the identity element? What about the inverses ones?
Best Answer
Closure (under multiplication) means: "An integer times an integer is also an integer". It does not mean:
"An integer times something else which results in an integer, means that something else is also an integer."
This is pretty obvious to see:
$2\cdot \frac{1}{2} = 1$.
Here, $2$ is an integer, and the "something else" is $\frac{1}{2}$. Our product is an integer, but it is not the case that we can conclude $\frac{1}{2}$ is an integer; in fact, it is not.
It can be proven that $\Bbb N$ is closed under multiplication (hint: use induction), and using the rules, for $a,b \in \Bbb N$:
$a(-b) = (-a)b = -ab\\ (-a)(-b) = ab$
(If you do not consider $0$ to be a natural number, you have a few more cases to consider, but these are easy)
it is easy to see that if $\Bbb N$ is closed under multiplication, so is $\Bbb Z$.
Perhaps this will be easier to process when you see the differences between rings, and fields. What often throws people off-guard in thinking about this, is that ordinary high-school arithmetic typically takes place in the field of rational numbers, where the non-zero numbers are closed under multiplication, and the equation:
$7x = 3$ has a rational solution, even though it is an integral equation.