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.
Given the context: A student claims that 1.618 is an exact value for the golden ratio. Is this claim correct?
Given the context: A student uses 1.732 as an approximation for √3 in a calculation. Is this a rational approximation?
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.
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.
Which of the following fractions is closest to the value of √3?
(1) 1.732
(2) 1.700
(3) 1.750
(4) 1.700
Which of the following is a better rational approximation for π?
(1) 22/7
(2) 355/113
(3) 3.14
(4) 3.1
Which of the following is a rational approximation of e (Euler's number)?
(1) 2.718
(2) 3.142
(3) 1.618
(4) 1.414
Which of the following is NOT a rational approximation of π?
(1) 22/7
(2) 355/113
(3) 3.141592653589793...
(4) 333/106
Fill in the blank: The decimal 3.1416 is a rational approximation of ________.
Fill in the blank: The decimal expansion of an irrational number is ________ and non-repeating.
Fill in the blank: The fraction ________ is a commonly used rational approximation for the golden ratio (φ).
Fill in the blank: The fraction 17/12 is a rational approximation for ________.
Given the context: A student claims that 1.618 is an exact value for the golden ratio. Is this claim correct?
Given the context: A student uses 1.732 as an approximation for √3 in a calculation. Is this a rational approximation?
True or False: Every irrational number can be approximated arbitrarily closely by rational numbers.
True or False: The decimal 1.618 is a rational approximation of the golden ratio.
