Let's go through each part of the Algebra assignment focusing on rational and irrational numbers.

Part A: Identifying Rational and Irrational Numbers

1. Circle the rational numbers in the list below:

   • $\frac{1}{2}$: Rational (can be expressed as a fraction)
   • $\sqrt{3}$: Irrational (cannot be expressed as a simple fraction; its decimal expansion is non-repeating and infinite)
   • 0.25: Rational (equivalent to $\frac{1}{4}$)
   • $\pi$: Irrational (its decimal expansion is non-terminating and non-repeating)
   • $\frac{5}{3}$: Rational (can be expressed as a fraction)
   • *1.414...* (assuming it refers to the square root of 2): Irrational (square root of 2 is known to be irrational)
   • $\sqrt{4}$: Rational (equal to 2, which can be expressed as $\frac{2}{1}$)

   Rational numbers circled: $\frac{1}{2}$, 0.25, $\frac{5}{3}$, $\sqrt{4}$.

2. Write down whether the following numbers are rational or irrational:

   • $\frac{7}{8}$: Rational
   • $\sqrt{5}$: Irrational
   • 2.75: Rational (equivalent to $\frac{11}{4}$)
   • $\frac{9}{2}$: Rational
   • $\pi$: Irrational

Part B: Basic Understanding

3. In your own words, define a rational number:

   • Definition: A rational number is any number that can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$. This includes all integers, finite decimals, and repeating decimals.

4. In your own words, define an irrational number:

   • Definition: An irrational number is a number that cannot be expressed as a simple fraction. Its decimal expansion is non-terminating and does not repeat. Examples include $\pi$ and $\sqrt{2}$.

If you need more details or have any questions, feel free to ask!

Questions to Consider:

1. Why is every integer considered a rational number?
2. How can you determine if a square root is rational or irrational without a calculator?
3. What is the difference between non-repeating decimals and repeating decimals?
4. Can an irrational number become rational by any mathematical operations?
5. Why is the number $\pi$ so significant in mathematics beyond its classification as irrational?

Tip:

When examining whether a square root is rational or irrational, consider the properties of the number under the square root. If it's not a perfect square, the root is typically irrational.