Unit Numbers can $$ \mathbb{N}=\{1 ; 2 ; 3 ; 4 ; \ldots\} $$ $$ \mathbb{N}_{0}=\{0 ; 1 ; 2 ; 3 ; 4 ; \ldots\} $$ $$ \mathbb{Z}=\{\ldots-4 ;-3 ;-2 ;-1 ; 0 ; 1 ; 2 $$, 3 the When one integer is divided For example, $$ 1 \div 4=\frac{1}{4}=0,25 $$, by a nother, the answer is not always an integer. To include $$ a \div b $$, where $$ a

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Sure! Let's work through **Exercise 1.1** step by step.

---

## **Exercise 1.1**

### **1. Write down the following numbers in the form $$ \frac{a}{b} $$:**

#### **1.1.1. $$ 1.17 $$**

To convert the decimal $$ 1.17 $$ to a fraction:

$$
1.17 = \frac{117}{100}
$$

Since 117 and 100 have no common divisors other than 1, the fraction is already in its simplest form.

$$
1.17 = \frac{117}{100}
$$

#### **1.1.2. $$ 1.2 $$**

To convert the decimal $$ 1.2 $$ to a fraction:

$$
1.2 = \frac{12}{10} = \frac{6}{5}
$$

(Simplified by dividing numerator and denominator by 2.)

$$
1.2 = \frac{6}{5}
$$

#### **1.1.3. $$ 2 \frac{3}{4} $$**

This is a mixed number. To convert it to an improper fraction:

$$
2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}
$$

$$
2 \frac{3}{4} = \frac{11}{4}
$$

#### **1.1.4. $$ 1.40,5 $$**

Assuming this is a typographical error and should be two separate numbers: $$ 1.40 $$ and $$ 0.5 $$.

- **For $$ 1.40 $$:**

$$
1.40 = \frac{140}{100} = \frac{7}{5}
$$

(Simplified by dividing numerator and denominator by 20.)

$$
1.40 = \frac{7}{5}
$$

- **For $$ 0.5 $$:**

$$
0.5 = \frac{5}{10} = \frac{1}{2}
$$

(Simplified by dividing numerator and denominator by 5.)

$$
0.5 = \frac{1}{2}
$$

#### **1.1.5. $$ 6 \frac{1}{2} $$**

This is a mixed number. To convert it to an improper fraction:

$$
6 \frac{1}{2} = \frac{6 \times 2 + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}
$$

$$
6 \frac{1}{2} = \frac{13}{2}
$$

---

### **2. Classify the following numbers fully as natural numbers, whole numbers, integers and/or rational numbers:**

#### **2.1. $$ 1 \frac{1}{4} $$**

- **Mixed Number:** $$ 1 \frac{1}{4} $$
- **Improper Fraction:** $$ \frac{5}{4} $$
  
**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{5}{4} $$)
- **Natural Number:** No (not a whole number)
- **Whole Number:** No
- **Integer:** No

#### **2.2. $$ -198 $$**

**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{-198}{1} $$)
- **Integer:** Yes
- **Whole Number:** No
- **Natural Number:** No

#### **2.3. $$ 3456 $$**

**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{3456}{1} $$)
- **Integer:** Yes
- **Whole Number:** Yes
- **Natural Number:** Yes

#### **2.4. $$ 1.26 $$**

**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{126}{100} $$ or simplified to $$ \frac{63}{50} $$)
- **Integer:** No
- **Whole Number:** No
- **Natural Number:** No

#### **2.5. $$ 0 $$**

**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{0}{1} $$)
- **Integer:** Yes
- **Whole Number:** Yes
- **Natural Number:** No (natural numbers typically start from 1)

#### **2.6. $$ 2 \frac{5}{8} $$**

- **Mixed Number:** $$ 2 \frac{5}{8} $$
- **Improper Fraction:** $$ \frac{21}{8} $$
  
**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{21}{8} $$)
- **Integer:** No
- **Whole Number:** No
- **Natural Number:** No

#### **2.7. $$ \sqrt{25} $$**

First, compute $$ \sqrt{25} $$:

$$
\sqrt{25} = 5
$$

**Classification of 5:**
- **Rational Number:** Yes (can be expressed as $$ \frac{5}{1} $$)
- **Integer:** Yes
- **Whole Number:** Yes
- **Natural Number:** Yes

#### **2.8. $$ 5 \frac{7}{8} $$**

- **Mixed Number:** $$ 5 \frac{7}{8} $$
- **Improper Fraction:** $$ \frac{47}{8} $$
  
**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{47}{8} $$)
- **Integer:** No
- **Whole Number:** No
- **Natural Number:** No

#### **2.9. $$ -3.589 $$**

**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{-3589}{1000} $$)
- **Integer:** No
- **Whole Number:** No
- **Natural Number:** No

#### **2.10. $$ 0.\dot{3} $$**

The notation $$ 0.\dot{3} $$ represents a repeating decimal:

$$
0.\overline{3} = 0.333\ldots
$$

To convert to a fraction:

$$
x = 0.\overline{3} \
10x = 3.\overline{3} \
$$
Subtract the first equation from the second:

$$
10x - x = 3.\overline{3} - 0.\overline{3} \
9x = 3 \
x = \frac{3}{9} = \frac{1}{3}
$$

$$
0.\overline{3} = \frac{1}{3}
$$

**Classification:**
- **Rational Number:** Yes (can be expressed as $$ \frac{1}{3} $$)
- **Integer:** No
- **Whole Number:** No
- **Natural Number:** No

---

## **Summary of Classifications**

| Number          | Natural Number | Whole Number | Integer | Rational Number |
|-----------------|----------------|--------------|---------|------------------|
| $$ 1 \frac{1}{4} $$ | No             | No           | No      | Yes              |
| $$ -198 $$           | No             | No           | Yes     | Yes              |
| $$ 3456 $$           | Yes            | Yes          | Yes     | Yes              |
| $$ 1.26 $$           | No             | No           | No      | Yes              |
| $$ 0 $$              | No             | Yes          | Yes     | Yes              |
| $$ 2 \frac{5}{8} $$ | No             | No           | No      | Yes              |
| $$ \sqrt{25} $$ -> 5 | Yes            | Yes          | Yes     | Yes              |
| $$ 5 \frac{7}{8} $$ | No             | No           | No      | Yes              |
| $$ -3.589 $$         | No             | No           | No      | Yes              |
| $$ 0.\dot{3} $$ -> $$ \frac{1}{3} $$ | No             | No           | No      | Yes              |

---

Feel free to ask if you need further explanations or assistance with other exercises!
**Exercise 1.1 Solutions** 1. **Convert the following decimals and mixed numbers to fractions:** - $$ 1.17 = \frac{117}{100} $$ - $$ 1.2 = \frac{6}{5} $$ - $$ 2 \frac{3}{4} = \frac{11}{4} $$ - $$ 1.40 = \frac{7}{5} $$ and $$ 0.5 = \frac{1}{2} $$ - $$ 6 \frac{1}{2} = \frac{13}{2} $$ 2. **Classify the numbers:** - $$ 1 \frac{1}{4} $$: Rational - $$ -198 $$: Integer, Rational - $$ 3456 $$: Natural, Whole, Integer, Rational - $$ 1.26 $$: Rational - $$ 0 $$: Whole, Integer, Rational - $$ 2 \frac{5}{8} $$: Rational - $$ \sqrt{25} = 5 $$: Natural, Whole, Integer, Rational - $$ 5 \frac{7}{8} $$: Rational - $$ -3.589 $$: Rational - $$ 0.\dot{3} = \frac{1}{3} $$: Rational **Summary:** - **Natural Numbers:** 3456, 5 - **Whole Numbers:** 0, 3456, 5 - **Integers:** -198, 0, 3456, 5 - **Rational Numbers:** All numbers listed If you have any more questions or need further assistance, feel free to ask!

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