Question

classify each number below as a rational number or an irrational number.

	rational	irrational
-√26	○	○
21.\overline{18}	○	○
-√25	○	○
28.16	○	○
19π	○	○

Explanation:

Step1: Recall definitions

A rational number can be expressed as \(\frac{p}{q}\) where \(p,q\) are integers and \(q
eq0\), or has a terminating/repeating decimal. An irrational number is non - repeating, non - terminating and cannot be expressed as \(\frac{p}{q}\).

Step2: Classify \(-\sqrt{26}\)

\(26\) is not a perfect square (\(5^2 = 25\), \(6^2=36\)), so \(\sqrt{26}\) is irrational. Multiplying by \(- 1\) (a rational number) still gives an irrational number. So \(-\sqrt{26}\) is irrational.

Step3: Classify \(21.\overline{18}\)

\(21.\overline{18}\) is a repeating decimal (the "18" repeats). Repeating decimals are rational because they can be expressed as a fraction. Let \(x = 21.\overline{18}=21.181818\cdots\). Then \(100x=2118.1818\cdots\). Subtract \(x\) from \(100x\): \(100x - x=2118.1818\cdots-21.1818\cdots\), \(99x = 2097\), \(x=\frac{2097}{99}\), so it is rational.

Step4: Classify \(-\sqrt{25}\)

\(\sqrt{25}=5\), so \(-\sqrt{25}=- 5\). \(-5\) can be written as \(\frac{-5}{1}\), so it is rational.

Step5: Classify \(28.16\)

\(28.16\) is a terminating decimal. Terminating decimals are rational (e.g., \(28.16=\frac{2816}{100}\)).

Step6: Classify \(19\pi\)

\(\pi\) is irrational, and multiplying an irrational number (\(\pi\)) by a non - zero rational number (\(19\)) gives an irrational number. So \(19\pi\) is irrational.

Answer:

• \(-\sqrt{26}\): irrational
• \(21.\overline{18}\): rational
• \(-\sqrt{25}\): rational
• \(28.16\): rational
• \(19\pi\): irrational

To fill the table:

Number	rational	irrational
\(21.\overline{18}\)	\(\circ\)
\(-\sqrt{25}\)	\(\circ\)
\(28.16\)	\(\circ\)
\(19\pi\)		\(\circ\)