To determine which of the given numbers is irrational, we need to analyze each option step-by-step.
1. Option a: [tex]\(\sqrt{12}\)[/tex]
The square root of 12 is [tex]\(\sqrt{12}\)[/tex]. We need to check if this number can be expressed as a ratio of two integers.
[tex]\(\sqrt{12}\)[/tex] simplifies as follows:
[tex]\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\][/tex]
[tex]\(\sqrt{3}\)[/tex] is known to be an irrational number (it cannot be expressed as a ratio of two integers). Therefore, [tex]\(2\sqrt{3}\)[/tex] is also irrational.
2. Option b: [tex]\(\sqrt{9}\)[/tex]
The square root of 9 is [tex]\(\sqrt{9}\)[/tex]. We can simplify this directly.
[tex]\[
\sqrt{9} = 3
\][/tex]
Since 3 is an integer, it is a rational number.
3. Option c: [tex]\(\frac{7}{3}\)[/tex]
The number [tex]\(\frac{7}{3}\)[/tex] is a fraction where both the numerator (7) and the denominator (3) are integers.
A number in the form of a fraction [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers with [tex]\(q \neq 0\)[/tex] is by definition a rational number. Thus, [tex]\(\frac{7}{3}\)[/tex] is a rational number.
4. Option d: [tex]\(\frac{0}{11}\)[/tex]
The number [tex]\(\frac{0}{11}\)[/tex] is a fraction where the numerator is 0 and the denominator is 11 (an integer).
Any number with 0 in the numerator is equal to 0, provided the denominator is a non-zero integer.
[tex]\[
\frac{0}{11} = 0
\][/tex]
Since 0 is an integer, it is a rational number.
After analyzing all the options:
- Option a ([tex]\(\sqrt{12}\)[/tex]) is irrational.
- Option b ([tex]\(\sqrt{9}\)[/tex]) is rational.
- Option c ([tex]\(\frac{7}{3}\)[/tex]) is rational.
- Option d ([tex]\(\frac{0}{11}\)[/tex]) is rational.
Hence, the irrational number among the given choices is:
[tex]\[
\boxed{1}
\][/tex]
