Classify numbers as natural, whole, integer, rational, irrational, or real

Numbers are classified as natural numbers (the positive counting numbers), whole numbers (natural numbers together with 0), integers (whole numbers and their opposites), rational numbers (numbers expressible as a ratio of integers with a nonzero denominator), irrational numbers (real numbers that are not rational), or real numbers (all numbers located on the number line). These categories form the containment chain natural ⊂ whole ⊂ integer ⊂ rational ⊂ real, while irrational numbers are real but not rational; terminating or repeating decimals are rational, whereas nonterminating, nonrepeating decimals are irrational. Complex numbers and formal constructions of the real number system are beyond this classification.

Detailed Explanation: Classify numbers as natural, whole, integer, rational, irrational, or real

To classify a number, first identify what kind of number it is, then use the containment chain:

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

Natural numbers: $1,2,3,\ldots$
Whole numbers: $0,1,2,3,\ldots$
Integers: positive and negative whole numbers, including $0$
Rational numbers: numbers that can be written as $\frac{a}{b}$ , where $a$ and $b$ are integers and $b\ne 0$
Irrational numbers: real numbers that cannot be written as a ratio of integers
Real numbers: all numbers on the number line

Worked example: Classify $-\frac{6}{3}$

Step 1: Simplify the number.

-\frac{6}{3}=-2

Step 2: Check whether $-2$ is natural or whole.

It is not natural because natural numbers are positive. It is not whole because whole numbers are $0$ and positive numbers.

Step 3: Check whether it is an integer.

Yes. $-2$ is a negative whole number, so it is an integer.

Step 4: Check whether it is rational.

Yes, because it can be written as a ratio of integers:

-2=\frac{-2}{1}

So it is rational.

Step 5: Check whether it is real.

Every integer and rational number is real, so $-2$ is real.

Therefore, $-\frac{6}{3}$ is an:

\boxed{\text{integer, rational, and real number}}

It is not natural, whole, or irrational.

Learn by doing: Classify numbers as natural, whole, integer, rational, irrational, or real

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

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