Question

Prove that set of rational numbers under usual addition and scalar multiplication is field india

Answer 1

Closure - yes m/n+p/q=(mq+np)/nqm/n+p/q=(mq+np)/nq

Associative - yes m/n+(k/l+p/q)=(m/n+k/l)+p/qm/n+(k/l+p/q)=(m/n+k/l)+p/q

Identity - yes, zero m/n+0/1=m/nm/n+0/1=m/n

Inverse - yes m/n+(−m)/n=0/n=0/1m/n+(−m)/n=0/n=0/1

And, as Jim pointed out, the rational numbers in fact firm a field under addition and multiplication.

Answer 2

Answer:

The set of rational numbers is closed when added to, however, it is not closed when multiplied by a scalar.

Explanation:

In any field containing integers, rational numbers, along with addition and multiplication, produce a field that also contains the integers. In other words, a field only has feature zero if and only if it contains the field of rational numbers as a subfield. The field of rational numbers is a prime field. If you can add any two digits in a set and end up with a number in the set, the set is said to be closed under addition. If you can multiply any two items in a set using (scalar) multiplication and the result remains a number in the set, the set is said to be closed.

As a result, we can observe that the outcome of addition, subtraction, and multiplication is always a rational integer. The closure of rational numbers under addition, subtraction, and multiplication are indicated by this.

So, here we proved that rational numbers are closed under addition and not closed under scalar multiplication.