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Question Video: The Properties of Real Numbers Mathematics • 8th Grade

Every integer is also ๏ผฟ.

02:33

Video Transcript

Every integer is also a what. Option (A) a natural number, option (B) a real number.

In this question, we need to determine which of two different types of numbers is applicable to every integer. We can start by recalling that the integers means all of the whole numbers. That is the set of all the positive whole numbers such as one, two, three, and so on, all of the negative whole numbers such as negative one, negative two, negative three, and so on, and also zero. Similarly, we can recall that the natural numbers are the set of nonnegative whole numbers, so they are the numbers zero, one, two, three, and so on.

It is worth noting that some texts do not include zero as a natural number; however, this will not affect the question. We can see that negative one is a whole number, so it is an integer. However, it is not a positive whole number or zero, so negative one is not a natural number. Since we have found an integer that is not a natural number, we can see that option (A) is incorrect since every integer is not necessarily a natural number.

We can follow a similar process for option (B). We start by recalling that the set of real numbers is the union of the set of rational numbers and irrational numbers. We can also recall that the set of rational numbers is the set of all quotients of integers, that is, any number of the form ๐‘Ž over ๐‘, where ๐‘Ž and ๐‘ are integers and ๐‘ is nonzero.

We can use this definition to show that all integers are rational numbers. We know that dividing any integer by one will not affect its value, so all integers can be written as the quotient of two integers with a denominator of one. Thus, all integers are rational numbers.

Since the set of real numbers includes all rational numbers, we can say that the set of integers is a subset of the set of rational numbers, which in turn is a subset of the set of real numbers. This means that all integers are rational numbers and that they are also all real numbers, which is option (B).

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