Question

which one doesnt belong? consider the numbers shown.
-23.\overline{2} -23\frac{1}{5} -\sqrt{23} -23.2
which number does not belong with the others and why? choose the best response.
a) -23.\overline{2}; it is the only number that is an irrational number.
b) -23\frac{1}{5}; it is the only number that is a rational number.
c) -\sqrt{23}; it is the only number that is an irrational number.
d) -23.2; it is the only number that is a rational number.

Explanation:

Step1: Analyze each number's type

• \( -23.\overline{2} \): Repeating decimals are rational (can be expressed as a fraction).
• \( -23\frac{1}{5} \): Fractions are rational (equals \( -23.2 \), a terminating decimal).
• \( -\sqrt{23} \): \( \sqrt{23} \) is irrational (23 is not a perfect square), so \( -\sqrt{23} \) is irrational.
• \( -23.2 \): Terminating decimals are rational.

Step2: Evaluate options

• A: \( -23.\overline{2} \) is rational, so A is wrong.
• B: All others (except \( -\sqrt{23} \)) are rational, so B is wrong.
• C: \( -\sqrt{23} \) is irrational, while the rest are rational. This matches.
• D: \( -23.2 \) is rational, but others (except \( -\sqrt{23} \)) are also rational, so D is wrong.

Answer:

C) \( -\sqrt{23} \); It is the only number that is an irrational number.