Distinguish rational and irrational numbers

Rational numbers are real numbers expressible as a quotient $a/b$ of integers with $b\ne0$ ; their decimal representations terminate or repeat, while irrational numbers cannot be written in that form and have nonterminating, nonrepeating decimals. Classification includes integers, fractions, terminating or repeating decimals, and familiar radicals such as $\sqrt2$ , recognizing that the real number system is partitioned into rational and irrational numbers; advanced results about transcendental numbers or generalized irrationality are not included.

Detailed Explanation: Distinguish rational and irrational numbers

A rational number can be written as

\frac{a}{b},

where $a$ and $b$ are integers and $b\ne 0$ . Rational numbers include:

integers, such as $-3$ and $8$ ;
fractions, such as $\frac{2}{5}$ ;
terminating decimals, such as $0.75$ ;
repeating decimals, such as $0.\overline{3}$ .

An irrational number cannot be written as a fraction of two integers. Its decimal goes on forever without repeating a pattern. Familiar examples include $\sqrt{2}$ and $\pi$ .

Worked example

Classify each number as rational or irrational:

-4,\qquad \frac{7}{8},\qquad 0.121212\ldots,\qquad \sqrt{2},\qquad \sqrt{49}

Step 1: Classify $-4$ .

Every integer can be written as a fraction:

-4=\frac{-4}{1}.

Therefore, $-4$ is rational.

Step 2: Classify $\frac{7}{8}$ .

It is already written as a quotient of two integers, with a nonzero denominator. Therefore, $\frac{7}{8}$ is rational.

Step 3: Classify $0.121212\ldots$ .

The digits $12$ repeat forever. A repeating decimal is rational, so $0.121212\ldots$ is rational.

Step 4: Classify $\sqrt{2}$ .

$2$ is not a perfect square, so $\sqrt{2}$ cannot be written as a fraction of integers. Its decimal is nonterminating and nonrepeating. Therefore, $\sqrt{2}$ is irrational.

Step 5: Classify $\sqrt{49}$ .

Since $49$ is a perfect square,

\sqrt{49}=7.

Because $7$ is an integer, $\sqrt{49}$ is rational.

Thus, the classifications are:

\boxed{-4\text{ rational},\quad \frac78\text{ rational},\quad 0.121212\ldots\text{ rational},\quad \sqrt2\text{ irrational},\quad \sqrt{49}\text{ rational}}

Learn by doing: Distinguish rational and irrational numbers

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

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