Rational Approximations of Irrational Numbers
Math ⇒ Number and Operations
Rational Approximations of Irrational Numbers starts at 8 and continues till grade 12.
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Describe how you would use a calculator to find a rational approximation of an irrational number.
Describe one method to find a rational approximation of an irrational number.
Explain why \pi is considered an irrational number, but \frac{22}{7} is often used as its rational approximation.
Explain why 1.414 is a rational approximation of \sqrt{2} , but not its exact value.
Explain why rational approximations are important when working with irrational numbers in real-world applications.
Given the context: A carpenter needs to cut a piece of wood to a length of \sqrt{8} meters. What rational approximation should he use to the nearest tenth?
Given the context: A circle has a diameter of 10 cm. Using \frac{22}{7} as a rational approximation for \pi , calculate the circumference.
Given the context: An engineer needs to use \sqrt{13} in a calculation. What rational approximation should she use to two decimal places?
Given the irrational number \sqrt{5} , provide a rational approximation to two decimal places.
If \sqrt{7} \approx 2.645751 , what is its rational approximation to three decimal places?
Is the number 0.333... (repeating) a rational or irrational number?
A student claims that the decimal 2.718 is a rational approximation of Euler's number e . Explain whether this claim is correct and justify your answer.
Explain how continued fractions can be used to find increasingly accurate rational approximations of \pi .
Given that \sqrt{17} \approx 4.1231 , find a rational approximation of \sqrt{17} as a fraction accurate to three decimal places.
Given the context: An architect needs to use \sqrt{12} in a calculation for a building design. Provide a rational approximation of \sqrt{12} as a fraction in simplest form, accurate to two decimal places.
Prove that for any irrational number, there exist infinitely many rational numbers that are closer to it than any given positive distance.
