1.1) Indicate which of the following numbers are rational or irrational $$ \pi: 2 \frac{1}{6} ; 0,8 ; -\sqrt{3} $$ 1.2) Convert $$ 0,2 \ddot{4} $$; to a common fraction in its simplest form (show all necessary working). (3) 1.3) Convert 5,36 to an improper fraction without using a calculator. (3) 1.4) Determine between which 2 integers the following surds lie: 14.1) $$ \sqrt{30} $$ 14.2) $$ -\sqrt[3]{20} $$ 1.5) Write each number in the form $$ \frac{a}{b} $$, where $$ a $$ and $$ b \in \mathbb{Z} $$ and $$ a $$ and $$ b $$ have no common factors, ie. Simplify (without a calculator) 15.1) $$ -1 \frac{3}{4} \times \frac{8}{9} $$ 15.2) $$ \frac{1}{2}+\frac{2}{3}-\frac{3}{4} $$

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**1.1) Determining Rationality**

- $$\pi$$ is known to be irrational.
- $$2\frac{1}{6}=\frac{13}{6}$$ is a rational number (it can be written as a fraction of two integers).
- $$0,8$$ (interpreted as $$0.8$$) is rational since $$0.8=\frac{4}{5}$$.
- $$-\sqrt{3}$$ is irrational because $$\sqrt{3}$$ is irrational (changing the sign does not affect irrationality).

---

**1.2) Converting $$0,2\ddot{4}$$ to a Fraction**

Let 
$$
x=0.2\overline{4}
$$
Then multiplying both sides by 10 (since the repeating block "4" has 1 digit):
$$
10x=2.\overline{4}
$$
Subtract the original equation from this:
$$
10x - x = 2.\overline{4} - 0.2\overline{4}
$$
Since the repeating parts cancel:
$$
9x=2.2
$$
Express $$2.2$$ as a fraction:
$$
2.2=\frac{22}{10}
$$
Thus:
$$
9x=\frac{22}{10} \quad \Longrightarrow \quad x=\frac{22}{90}=\frac{11}{45}
$$

---

**1.3) Converting 5,36 to an Improper Fraction**

Interpret $$5,36$$ as $$5.36$$. Write it as a sum of its integer and decimal parts:
$$
5.36=5+\frac{36}{100}
$$
Simplify the decimal fraction:
$$
\frac{36}{100}=\frac{9}{25}
$$
Express the integer part with denominator 25:
$$
5=\frac{125}{25}
$$
Add the fractions:
$$
5.36=\frac{125}{25}+\frac{9}{25}=\frac{134}{25}
$$

---

**1.4) Determining Integer Bounds for the Surds**

*1.4.1)* For $$\sqrt{30}$$ note:
$$
5^2=25 \quad \text{and} \quad 6^2=36
$$
Thus:
$$
5 < \sqrt{30} < 6
$$

*1.4.2)* For $$-\sqrt[3]{20}$$ note:
$$
2^3=8 \quad \text{and} \quad 3^3=27
$$
Thus:
$$
2 < \sqrt[3]{20} < 3
$$
Taking the negative, the order reverses:
$$
-3 < -\sqrt[3]{20} < -2
$$

---

**1.5) Expressing in the Form $$\frac{a}{b}$$ in Simplest Form**

*1.5.1)* Compute $$-1 \frac{3}{4} \times \frac{8}{9}$$:

First, convert the mixed number:
$$
-1\frac{3}{4}=-\frac{7}{4}
$$
Multiply by $$\frac{8}{9}$$:
$$
-\frac{7}{4}\times\frac{8}{9}=-\frac{7\times8}{4\times9}=-\frac{56}{36}
$$
Simplify by dividing numerator and denominator by 2:
$$
-\frac{56}{36}=-\frac{28}{18}=-\frac{14}{9}
$$

*1.5.2)* Compute $$\frac{1}{2}+\frac{2}{3}-\frac{3}{4}$$:

Find a common denominator, which is 12:
$$
\frac{1}{2}=\frac{6}{12},\quad \frac{2}{3}=\frac{8}{12},\quad \frac{3}{4}=\frac{9}{12}
$$
Then:
$$
\frac{6}{12}+\frac{8}{12}-\frac{9}{12}=\frac{6+8-9}{12}=\frac{5}{12}
$$
**1.1) Rational and Irrational Numbers** - $$\pi$$ is irrational. - $$2\frac{1}{6} = \frac{13}{6}$$ is rational. - $$0.8 = \frac{4}{5}$$ is rational. - $$-\sqrt{3}$$ is irrational. --- **1.2) Converting $$0,2\ddot{4}$$ to a Fraction** $$0,2\ddot{4} = \frac{11}{45}$$. --- **1.3) Converting 5,36 to an Improper Fraction** $$5,36 = \frac{134}{25}$$. --- **1.4) Integer Bounds for Surds** - $$\sqrt{30}$$ lies between 5 and 6. - $$-\sqrt[3]{20}$$ lies between -3 and -2. --- **1.5) Simplifying Expressions** - $$-1\frac{3}{4} \times \frac{8}{9} = -\frac{14}{9}$$. - $$\frac{1}{2} + \frac{2}{3} - \frac{3}{4} = \frac{5}{12}$$.

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