Classify numbers within the real number system

The real number system organizes numbers into nested and related subsets: natural (counting) numbers, whole numbers, integers, rational numbers, and irrational numbers, with every rational and irrational number belonging to the real numbers. Rational numbers can be expressed as a ratio of integers and have terminating or repeating decimal representations, while irrational numbers, such as √2, have nonterminating, nonrepeating decimals; both correspond to points on the number line. This classification supports work with radicals and excludes complex numbers and more advanced generalizations beyond the real number system.

Detailed Explanation: Classify numbers within the real number system

To classify a number, check which number sets it belongs to. The sets are nested:

\text{natural} \subset \text{whole} \subset \text{integers} \subset \text{rational} \subset \text{real}

Natural numbers: $1,2,3,\ldots$
Whole numbers: $0,1,2,3,\ldots$
Integers: $\ldots,-2,-1,0,1,2,\ldots$
Rational numbers: numbers that can be written as $\frac{a}{b}$ , where $a$ and $b$ are integers and $b\ne 0$ . Their decimals terminate or repeat.
Irrational numbers: real numbers that cannot be written as a ratio of integers. Their decimals do not terminate or repeat.

Every rational and irrational number is a real number.

Worked example: Classify $\boldsymbol{-\sqrt{18}}$

Step 1: Simplify the radical.

Since $18=9\cdot 2$ ,

-\sqrt{18}=-\sqrt{9\cdot 2}=-3\sqrt{2}.

Step 2: Decide whether it is rational or irrational.

The number $2$ is not a perfect square, so $\sqrt{2}$ is irrational. Multiplying it by $-3$ does not make it rational. Therefore,

-3\sqrt{2}\text{ is irrational}.

Step 3: Classify it in the real number system.

Irrational numbers are real numbers, so $-\sqrt{18}$ is real.

It is not:

natural, because it is negative;
whole, because it is negative and not a whole number;
an integer, because it is not a counting number with no decimal part;
rational, because it cannot be written as a ratio of integers.

Therefore, the complete classification is:

\boxed{-\sqrt{18}\text{ is an irrational real number}.}

Learn by doing: Classify numbers within the real number system

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Number Types (Real) - Number to Description - Whole, Natural, Integer, Rational, Irrational Numbers

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