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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Sure! Let's dive into some fun facts about rational and irrational numbers, and how they pop up in the world around us. Can you believe that irrational numbers like \(\pi\) and \(\sqrt{3}\) have fascinated mathematicians for centuries? \(\pi\) represents the ratio of a circle's circumference to its diameter, and it’s not just a bunch of random digits; it starts with 3.14159 and stretches infinitely without repeating! This little wonder is crucial in various fields, from engineering to quantum physics, proving that sometimes, the most mind-boggling numbers just love to roll in circles! Now, onto our pal \(0.2\underline{4}\)—it’s a repeating decimal, making it quite a magical creature! To convert \(0.2\underline{4}\) into a fraction, let’s say \(x = 0.2\underline{4}\). Multiply both sides by 100 to get \(100x = 24.24\underline{4}\), and then we also have \(10x = 2.4\underline{4}\). By subtracting these, you'll capture the essence of \(x\), leading you to the fraction \(\frac{22}{90}\), which simplifies nicely to \(\frac{11}{45}\). See? Easy as pie!