Let's determine whether each number is rational or irrational based on the image.

1. √11:

   • This is an irrational number because the square root of 11 cannot be expressed as a fraction of two integers, and its decimal form is non-terminating and non-repeating.

2. 1/11:

   • This is a rational number because it is a fraction of two integers (1 and 11).

3. 0.1̅ (repeating decimal):

   • This is a rational number because repeating decimals can always be expressed as a fraction of two integers.

4. √111:

   • This is an irrational number because the square root of 111 cannot be expressed as a fraction, and its decimal form is non-terminating and non-repeating.

Summary:

• √11: Irrational
• 1/11: Rational
• 0.1̅: Rational
• √111: Irrational

Would you like a more detailed explanation for any of these numbers?

Here are 5 related questions:

1. Why are square roots of non-perfect squares irrational?
2. Can a repeating decimal ever be irrational?
3. What is the difference between terminating and repeating decimals?
4. How can we prove a number is rational?
5. Are all fractions rational numbers?

Tip: If a decimal terminates or repeats, it's a rational number!