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 to use a calculator to find a rational approximation of an irrational number to two decimal places.
Describe the difference between a rational approximation and an exact value for an irrational number.
Describe the process of finding a rational approximation for an irrational number using continued fractions.
Explain how the error in a rational approximation of an irrational number can be reduced.
Explain why 22/7 is considered a rational approximation of π.
Explain why rational approximations are important in scientific calculations involving irrational numbers.
Explain why the decimal 1.4142 is not an exact value for √2.
Given the irrational number π, provide a rational approximation with a denominator less than 100.
If a rational number r approximates an irrational number x such that |x - r| < 0.001, what does this inequality represent?
A student claims that 19/6 is a better rational approximation for π than 22/7. Is this claim correct? Justify your answer with calculations.
Determine a rational approximation for √13 with a denominator less than 20, and calculate its percentage error compared to the actual value.
Explain how the method of continued fractions can be used to generate increasingly accurate rational approximations of π. Provide the first two convergents and their decimal values.
Given that √5 is an irrational number, find a rational approximation of √5 with a denominator less than 50, and calculate the absolute error of your approximation.
Given the context: An engineer needs to use the value of e (Euler's number) in a calculation but can only use fractions with denominators less than 100. Suggest a suitable rational approximation and justify your choice.
