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Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.
The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in [link] .
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?
Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as [link] .
The set of natural numbers includes the numbers used for counting: $\text{\hspace{0.17em}}\left\{1,2,3,\mathrm{...}\right\}.$
The set of whole numbers is the set of natural numbers plus zero: $\text{\hspace{0.17em}}\left\{0,1,2,3,\mathrm{...}\right\}.$
The set of integers adds the negative natural numbers to the set of whole numbers: $\text{\hspace{0.17em}}\left\{\mathrm{...},\mathrm{-3},\mathrm{-2},\mathrm{-1},0,1,2,3,\mathrm{...}\right\}.$
The set of rational numbers includes fractions written as $\text{\hspace{0.17em}}\left\{\frac{m}{n}\text{\hspace{0.17em}}|m\text{and}n\text{areintegersand}n\ne 0\right\}.$
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: $\text{\hspace{0.17em}}\left\{h|h\text{isnotarationalnumber}\right\}.$
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