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Whichs et of number sis always rational?

intergers
positive numbers
negative numbers
decimals

3 answers

The set of integers is always rational.

Rational numbers are defined as numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Since integers can be written as fractions (e.g., \(5\) can be expressed as \(\frac{5}{1}\)), they are always rational.

To clarify the other sets:

  • Positive numbers: This set includes both rational (like \(1\) or \(0.5\)) and irrational numbers (like \(\sqrt{2}\)) and thus is not always rational.

  • Negative numbers: Similarly, this set can include both rational (like \(-1\) or \(-0.5\)) and irrational numbers (like \(-\sqrt{2}\)) and is not restricted to rational numbers.

  • Decimals: This set includes both terminating decimals (which are rational, like \(0.5\)) and non-terminating, non-repeating decimals (which are irrational, like \( \pi\)) and is thus not always rational.

So, the set of integers is the only one in your list that is always rational.

are you sure id jsut like to make sure.

Yes, I am sure. The set of integers is indeed the only one among the options provided that is always rational. To recap:

  1. Integers: Each integer can be expressed as a fraction (e.g., \(5 = \frac{5}{1}\), \(-3 = \frac{-3}{1}\)), making every integer a rational number.

  2. Positive numbers: This set includes positive rational numbers (like \(0.5\)) and can also include irrational numbers (like \(\sqrt{2}\)). Therefore, this set is not exclusively rational.

  3. Negative numbers: Similar to positive numbers, this set can include negative rational numbers (like \(-1\)) and negative irrational numbers (like \(-\sqrt{2}\)), so it is not exclusively rational either.

  4. Decimals: This group includes both terminating decimals (which are rational) and non-terminating decimals (like \(\pi\)) that are irrational.

So, throughout our examination, the integers stand out as the only set that is definitively composed of rational numbers. If you have further questions or need clarification on any point, feel free to ask!