📚 Part 1: Classifying Real Numbers

1. Which of the following is a rational number?

π

√2

0.75

√3

2. Which statement about irrational numbers is true?

They can be written as fractions

They have terminating decimal expansions

They have non-terminating, non-repeating decimal expansions

They are always negative

3. Select all the rational numbers from the list below:

-3

2/5

√9

√7

0.333...

π

4. The decimal 0.1666... (where 6 repeats forever) is:

Rational, because it has a repeating pattern

Irrational, because it goes on forever

Neither rational nor irrational

Not a real number

✏️ Part 2: Understanding and Explaining

5. Explain in your own words what makes a number rational. Give two examples.

6. Why is √2 an irrational number? Explain your reasoning.

7. Complete the following statements:

a) All integers are _____________ numbers.

b) The decimal expansion of an irrational number is _____________ and _____________.

c) The set of real numbers includes both _____________ and _____________ numbers.

8. Place the following numbers on the number line below by writing them in the appropriate positions:

Numbers: -2, -0.5, √2, 3.5, π

←—————|—————|—————|—————|—————|—————→
-3 -2 -1 0 1 2 3 4

🌟 Part 3: Real-World Applications

9. In New Zealand, the exact measurement of a circular garden's circumference involves π. If the garden has a radius of 2 metres, explain why the exact circumference cannot be written as a simple fraction.

10. A carpenter needs to cut a piece of wood that is exactly √8 metres long. Explain whether this measurement is rational or irrational, and describe how the carpenter might work with this measurement practically.