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.
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 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.
Which of the following fractions is closest to \( \sqrt{3} \)?
(1) \( \frac{5}{3} \)
(2) \( \frac{7}{4} \)
(3) \( \frac{17}{10} \)
(4) \( \frac{13}{8} \)
Which of the following is a rational approximation of \( \sqrt{6} \) to two decimal places?
(1) 2.44
(2) 2.45
(3) 2.46
(4) 2.47
Which of the following is a rational approximation of \( e \) (Euler's number)?
(1) 2.71
(2) 3.14
(3) 1.41
(4) 1.73
Which of the following is NOT a rational approximation of \( \sqrt{2} \)?
(1) 1.414
(2) \( \frac{99}{70} \)
(3) 1.5
(4) \( \sqrt{2} \)
Fill in the blank: The decimal 1.732 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 \( \frac{19}{6} \) is a rational approximation of ________.
Fill in the blank: The fraction \( \frac{355}{113} \) is a rational approximation of ________.
True or False: Every irrational number can be approximated by rational numbers as closely as desired.
True or False: The decimal 2.236 is a rational approximation of \( \sqrt{5} \).
True or False: The decimal 3.1415926535 is a rational approximation of \( \pi \).
True or False: The decimal expansion of a rational number always terminates or repeats.
