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Numbers that cannot be expressed as terminating or repeating decimals?
irrational numbers
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.
A repeating decimal is a real number?
Yes repeating decimals are real numbers. They can fall under the category of rational numbers under real numbers since their repeating decimal patterns allows them to be converted into a fraction. Nonreal numbers are imaginary numbers which are expressed with i, or sqrt(-1).
What is an example of a terminating decimal?
A terminating decimal is a decimal number whose digits don't go on forever, like 3.45. A non-terminating decimal is a decimal number that goes on forever, like 1/3 = 0.3333333... since the 3's go on forever. So any repeating decimal is non-terminating. Also, numbers like pi go on forever: 3.1415926535897932384626.......
What are terminating and non terminating numbers in rational numbers?
Terminating numbers are decimal representations of rational numbers. Nonterminating numbers may or may not be rational numbers.
