Question

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) \( \frac{1}{2}+\frac{2}{3}-\frac{3}{4} \)

Ask by Frank Mcguire. in South Africa
Mar 11,2025

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Answer

**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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Bonus Knowledge

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!

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