A rational number is any number that can be written as a fraction, where the numerator and denominator are both integers, and the denominator is not zero. In other words, a rational number can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.

Let's examine each option:

A. $\frac{5}{6}$

• This is a fraction of two integers, so it is a rational number.

B. $\sqrt{11}$

• $\sqrt{11}$ is an irrational number because it cannot be expressed as a fraction of integers. The square root of 11 is a non-repeating, non-terminating decimal.

C. $6.565656\ldots$

• This is a repeating decimal. It can be written as a fraction (e.g., $\frac{65}{99}$), so it is a rational number.

D. $0.23$

• This is a terminating decimal, and it can be written as $\frac{23}{100}$, so it is a rational number.

E. $0.32416\ldots$

• This is a non-repeating, non-terminating decimal, and it cannot be expressed as a fraction, so it is an irrational number.

F. $-538$

• This is an integer, and all integers are rational numbers because they can be expressed as a fraction with a denominator of 1 (e.g., $\frac{-538}{1}$).

Rational Numbers:

• A. $\frac{5}{6}$
• C. $6.565656\ldots$
• D. $0.23$
• F. $-538$

Irrational Numbers:

• B. $\sqrt{11}$
• E. $0.32416\ldots$

Would you like more details on any of these numbers or their classifications?

Here are 5 related questions:

1. What makes a number irrational?
2. How do you convert a repeating decimal into a fraction?
3. What are the characteristics of a terminating decimal?
4. Why are integers always rational numbers?
5. How do you identify a rational number from its decimal expansion?

Tip: When you're unsure if a decimal is rational, check if it repeats or terminates—those are the key signs!