SECTIONA: QUESTION 1 1.1. From the list below, classify the the set of numbers: $$ \frac{7}{4} ; 3 \pi ; 0,41 ; \sqrt[3]{18} ; \sqrt{9} ; \frac{0}{8} ; 5 ;-10 $$ 1.1.1. $$ Q= $$ $$ \qquad $$ 1.1.2. $$ Q^{\prime}= $$ $$ \qquad $$ 1.1.3. $$ z= $$ $$ \qquad $$ 1.4. $$ N= $$ $$ \qquad $$ List the prime numbers up to 30 . $$ \qquad $$

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We are given the following list of numbers:

7/4  ,  3π  ,  0,41  ,  ∛18  ,  √9  ,  0/8  ,  5  ,  –10

Since in many texts the notations mean:

• Q (or ℚ): the set of all rational numbers.
  • Q′ (or ℚ′): the set of irrational numbers.
  • ℤ: the set of all integers.
  • ℕ: the set of natural numbers (here taken as the positive integers).

Let’s analyze each number:

1. 7/4  
  – This is expressed as a fraction, so it is rational.

2. 3π  
  – Since π is irrational, any nonzero multiple (like 3π) is also irrational.

3. 0,41  
  – Written with a comma to denote the decimal point (common in many countries), this is 0.41. Because it has a decimal expansion that terminates (0.41 = 41/100), it is rational.

4. ∛18 (cube root of 18)  
  – 18 is not a perfect cube; therefore, its cube root cannot be expressed as a fraction of integers. It is irrational.

5. √9  
  – Since 9 is a perfect square, √9 = 3, which is rational.

6. 0/8  
  – This is 0, and 0 is a rational number.

7. 5  
  – An integer, hence rational.

8. –10  
  – An integer (negative), thus rational.

Now we classify them into the given sets.

1. 1.1.1. Q = {the rational numbers}  
  • From our list, these are: 7/4, 0,41, √9 (which equals 3), 0/8 (0), 5, and –10.

2. 1.1.2. Q′ = {the irrational numbers}  
  • From our list, these are: 3π and ∛18.

3. 1.1.3. ℤ = {the integers}  
  • The integers among our list are: √9 ( = 3), 0/8 ( = 0), 5, and –10.

4. 1.4. ℕ = {the natural numbers}  
  • Typically, the natural numbers are defined as the positive integers. Hence, from our list, we have: √9 ( = 3) and 5.
   (Notice that although some definitions of ℕ include 0, in many contexts ℕ is taken as {1, 2, 3, …}, and here we assume 0 is not included.)

Finally, the question asks to list the prime numbers up to 30. The prime numbers (natural numbers greater than 1 that have no positive divisors other than 1 and themselves) up to 30 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Summary of Answers:

1.1.1. Q = {7/4, 0,41, √9, 0/8, 5, –10}  
  1.1.2. Q′ = {3π, ∛18}  
  1.1.3. ℤ = {√9, 0/8, 5, –10}  
  1.4. ℕ = {√9, 5}  (with √9 = 3)

Primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
**Classification of Numbers:** 1. **Rational Numbers (Q):** 7/4, 0.41, √9 (which is 3), 0/8 (which is 0), 5, –10 2. **Irrational Numbers (Q′):** 3π, ∛18 3. **Integers (ℤ):** √9 (3), 0/8 (0), 5, –10 4. **Natural Numbers (ℕ):** √9 (3), 5 **Prime Numbers up to 30:** 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

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