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Are rational numbers are closed under addition subtraction multiplication and division?
They are closed under all except that division by zero is not defined.
What is a counterexample to show that the repeating decimals are closed under subtraction false?
In fact, the statement is true. Consequently, there is not a proper counterexample. The fallacy is in asserting that a terminating decimal is not a repeating decimal. First, there is the trivial argument that any terminating decimal can be written with a repeating string of trailing zeros. But, Cantor or Dedekind (I can't remember which) proved that any terminating decimal can also be expressed as a repeating decimal. For example, 2.35 can be written as 2.3499... Or 150,000 as 149,999.99... Thus, a terminating decimal becomes a recurring decimal. As a consequence, all real numbers can be expressed as infinite decimals. And that proves closure under addition.
Are rational numbers closed under division?
No.
If a non-terminating decimal that is not a fraction is called irrational what do you call a non-terminating decimal that is a ratio of rational numbers such as 1 over 7?
It is called a rational number. The rational numbers are closed under the operations of addition, subtraction, multiplication, and division (not dividing by 0). The fact that it is not terminating is not important at all. In fact, if we use other bases besides base 10, we will see that the set of numbers that are rational or irrational doesn't change. However, if we use another base, for example base 3, then the number 1/3 in base 3 can be represented with a terminating "decimal" (technically not decimal). The set of rational numbers that have terminating "decimals" depends on the base.
What set of numbers is closed under division?
Integers are closed under division I think o.o. It's either counting numbers, integers or whole numbers . I cant remember :/
