Questions and Answers
What type of decimal expansion does the rational number 3.3333... have?
What is the significance of the bar above the digits in the decimal expansion of a rational number?
The bar above the digits indicates the block of digits that repeats in the decimal expansion.
The decimal expansion of a rational number can be non-terminating non-recurring.
False
The decimal expansion of a rational number is either ______________________ or non-terminating recurring.
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Match the following decimal expansions with their types:
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What can be concluded about a number like 1.272727...?
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The terminating decimal expansion of a rational number can always be expressed in the form p/q, where p and q are integers and q ≠ 0.
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Express 0.3333... in the form p/q, where p and q are integers and q ≠ 0.
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What is the maximum number of digits in the repeating block of digits in the decimal expansion of 17/7?
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Irrational numbers do not satisfy the commutative law for addition.
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What happens when you add, subtract, multiply, or divide (except by zero) two rational numbers?
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Rational numbers are 'closed' with respect to addition, subtraction, multiplication, and ____________________.
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Match the following numbers with their classification as rational or irrational:
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Which of the following numbers has a non-terminating non-recurring decimal expansion?
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The decimal expansion of an irrational number is always terminating.
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What can you say about the decimal expansion of a rational number?
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What is the approximate value of π often taken?
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Archimedes was the first to compute 5 digits in the decimal expansion of π.
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Who is credited with finding the value of π correct to four decimal places?
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The Sulbasutras is a mathematical treatise of the __________ period.
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Match the following ancient mathematicians with their contributions:
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What is the value of √2 according to the Sulbasutras?
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π has been computed to over 1.24 billion decimal places.
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What is the value of the irrational number between 1/7 and 2/7?
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What is a characteristic of every whole number?
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Every integer is a rational number.
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Write one way to find a rational number between two given numbers, r and s.
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The fraction _______ is equivalent to 1/2.
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Match the following types of numbers with their descriptions:
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Why is 3/5 not an integer?
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Every rational number is an integer.
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Find one rational number between 1 and 2 using the method of adding and dividing by 2.
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What can be said about the sum, difference, quotient, and product of irrational numbers?
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The decimal expansion of an irrational number is always terminating.
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What is the result of adding a rational number to an irrational number?
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The decimal expansion of _______________ numbers is non-terminating and non-recurring.
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The product of a rational number and an irrational number is always rational.
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What can be said about the numbers 2 + 21 and π – 2?
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Match the expressions with their simplified forms:
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What is the result of multiplying an irrational number by a rational number?
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Study Notes
Equivalent Fractions
- On the number line, there are infinitely many fractions equivalent to 1/2, and we choose 1/2 to represent all of them.
Types of Numbers
- Every whole number is not a natural number (e.g., 0 is a whole number but not a natural number).
- Every integer is a rational number (e.g., every integer m can be expressed as m/1).
- Not every rational number is an integer (e.g., 3/5 is a rational number but not an integer).
Finding Rational Numbers
- To find a rational number between two numbers, add the two numbers and divide by 2.
- Five rational numbers between 1 and 2 are 3/2, 5/4, 7/4, 11/8, and 13/8.
Decimal Expansions of Rational Numbers
- The decimal expansion of a rational number is either terminating or non-terminating recurring.
- Examples of non-terminating recurring decimal expansions include 3.333... and 0.142857...
- The decimal expansion of an irrational number can be terminating or non-terminating recurring.
Irrational Numbers
- An irrational number is a number that cannot be expressed as a finite decimal or fraction.
- Examples of irrational numbers include π, 2, and the square root of 2.
- Irrational numbers can be expressed as non-terminating non-recurring decimals.
Operations on Real Numbers
- Rational numbers satisfy the commutative, associative, and distributive laws for addition and multiplication.
- Irrational numbers also satisfy these laws, but the results of operations may not always be irrational.
- Examples of irrational numbers resulting from operations include 2 + √3, 2√3, and π - 2.
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Solve examples about different types of numbers, including whole numbers, natural numbers, integers, and rational numbers. Decide whether given statements are true or false and provide reasons.