Question

classify each number below as a rational number or an irrat\
\

	rational	irrational	\
---	---	---	\
$-2\sqrt{6}$	$\circ$	$\circ$	\
$-9\pi$	$\circ$	$\circ$	\
$-\dfrac{17}{15}$	$\circ$	$\circ$	\
$48.\overline{46}$	$\circ$	$\circ$	\
$\sqrt{1}$	$\circ$	$\circ$

Explanation:

To solve this, we recall the definitions: A rational number can be expressed as a fraction of two integers (\( \frac{p}{q}, q
eq 0 \)) or has a terminating/repeating decimal. An irrational number cannot be expressed as such and has a non - terminating, non - repeating decimal.

For \(-2\sqrt{6}\)

• Step 1: Analyze \(\sqrt{6}\). Since 6 is not a perfect square, \(\sqrt{6}\) is an irrational number (its decimal expansion is non - terminating and non - repeating).
• Step 2: When we multiply an irrational number (\(\sqrt{6}\)) by a non - zero rational number (\(-2\)), the result \(-2\sqrt{6}\) is still irrational. So, \(-2\sqrt{6}\) is irrational.

For \(-9\pi\)

• Step 1: Recall that \(\pi\) is an irrational number (its decimal expansion is non - terminating and non - repeating: \(3.1415926535\cdots\)).
• Step 2: When we multiply an irrational number (\(\pi\)) by a non - zero rational number (\(-9\)), the result \(-9\pi\) is irrational. So, \(-9\pi\) is irrational.

For \(-\frac{17}{15}\)

• Step 1: By the definition of a rational number, a number in the form of \(\frac{p}{q}\) where \(p=- 17\) and \(q = 15\) (both are integers and \(q

eq0\)) is a rational number. So, \(-\frac{17}{15}\) is rational.

For \(48.\overline{46}\)

• Step 1: The notation \(\overline{46}\) means that the digits 46 repeat indefinitely. A repeating decimal can be expressed as a fraction of two integers. Let \(x = 48.\overline{46}\). Then \(100x=4846.\overline{46}\), and \(100x - x=4846.\overline{46}-48.\overline{46}\), \(99x = 4846 - 48=4798\), \(x=\frac{4798}{99}\). So, \(48.\overline{46}\) is rational.

For \(\sqrt{1}\)

• Step 1: Calculate \(\sqrt{1}\). We know that \(1\times1 = 1\), so \(\sqrt{1}=1\).
• Step 2: The number 1 can be written as \(\frac{1}{1}\), which is in the form of a fraction of two integers. So, \(\sqrt{1}\) is rational.

Final Classification Table

Number	Rational	Irrational
\(-9\pi\)		\(\circ\)
\(-\frac{17}{15}\)	\(\circ\)
\(48.\overline{46}\)	\(\circ\)
\(\sqrt{1}\)	\(\circ\)

Answer:

To solve this, we recall the definitions: A rational number can be expressed as a fraction of two integers (\( \frac{p}{q}, q
eq 0 \)) or has a terminating/repeating decimal. An irrational number cannot be expressed as such and has a non - terminating, non - repeating decimal.

For \(-2\sqrt{6}\)

• Step 1: Analyze \(\sqrt{6}\). Since 6 is not a perfect square, \(\sqrt{6}\) is an irrational number (its decimal expansion is non - terminating and non - repeating).
• Step 2: When we multiply an irrational number (\(\sqrt{6}\)) by a non - zero rational number (\(-2\)), the result \(-2\sqrt{6}\) is still irrational. So, \(-2\sqrt{6}\) is irrational.

For \(-9\pi\)

• Step 1: Recall that \(\pi\) is an irrational number (its decimal expansion is non - terminating and non - repeating: \(3.1415926535\cdots\)).
• Step 2: When we multiply an irrational number (\(\pi\)) by a non - zero rational number (\(-9\)), the result \(-9\pi\) is irrational. So, \(-9\pi\) is irrational.

For \(-\frac{17}{15}\)

• Step 1: By the definition of a rational number, a number in the form of \(\frac{p}{q}\) where \(p=- 17\) and \(q = 15\) (both are integers and \(q

eq0\)) is a rational number. So, \(-\frac{17}{15}\) is rational.

For \(48.\overline{46}\)

• Step 1: The notation \(\overline{46}\) means that the digits 46 repeat indefinitely. A repeating decimal can be expressed as a fraction of two integers. Let \(x = 48.\overline{46}\). Then \(100x=4846.\overline{46}\), and \(100x - x=4846.\overline{46}-48.\overline{46}\), \(99x = 4846 - 48=4798\), \(x=\frac{4798}{99}\). So, \(48.\overline{46}\) is rational.

For \(\sqrt{1}\)

• Step 1: Calculate \(\sqrt{1}\). We know that \(1\times1 = 1\), so \(\sqrt{1}=1\).
• Step 2: The number 1 can be written as \(\frac{1}{1}\), which is in the form of a fraction of two integers. So, \(\sqrt{1}\) is rational.

Final Classification Table

Number	Rational	Irrational
\(-9\pi\)		\(\circ\)
\(-\frac{17}{15}\)	\(\circ\)
\(48.\overline{46}\)	\(\circ\)
\(\sqrt{1}\)	\(\circ\)