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Questions and Answers
What type of decimal expansion does the rational number 3.3333... have?
What type of decimal expansion does the rational number 3.3333... have?
- Non-terminating non-recurring
- Neither terminating nor recurring
- Non-terminating recurring (correct)
- Terminating
What is the significance of the bar above the digits in the decimal expansion of a rational number?
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.
The decimal expansion of a rational number can be non-terminating non-recurring.
False (B)
The decimal expansion of a rational number is either ______________________ or non-terminating recurring.
The decimal expansion of a rational number is either ______________________ or non-terminating recurring.
Match the following decimal expansions with their types:
Match the following decimal expansions with their types:
What can be concluded about a number like 1.272727...?
What can be concluded about a number like 1.272727...?
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.
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.
Express 0.3333... in the form p/q, where p and q are integers and q ≠0.
Express 0.3333... in the form p/q, where p and q are integers and q ≠0.
What is the maximum number of digits in the repeating block of digits in the decimal expansion of 17/7?
What is the maximum number of digits in the repeating block of digits in the decimal expansion of 17/7?
Irrational numbers do not satisfy the commutative law for addition.
Irrational numbers do not satisfy the commutative law for addition.
What happens when you add, subtract, multiply, or divide (except by zero) two rational numbers?
What happens when you add, subtract, multiply, or divide (except by zero) two rational numbers?
Rational numbers are 'closed' with respect to addition, subtraction, multiplication, and ____________________.
Rational numbers are 'closed' with respect to addition, subtraction, multiplication, and ____________________.
Match the following numbers with their classification as rational or irrational:
Match the following numbers with their classification as rational or irrational:
Which of the following numbers has a non-terminating non-recurring decimal expansion?
Which of the following numbers has a non-terminating non-recurring decimal expansion?
The decimal expansion of an irrational number is always terminating.
The decimal expansion of an irrational number is always terminating.
What can you say about the decimal expansion of a rational number?
What can you say about the decimal expansion of a rational number?
What is the approximate value of π often taken?
What is the approximate value of π often taken?
Archimedes was the first to compute 5 digits in the decimal expansion of π.
Archimedes was the first to compute 5 digits in the decimal expansion of π.
Who is credited with finding the value of π correct to four decimal places?
Who is credited with finding the value of π correct to four decimal places?
The Sulbasutras is a mathematical treatise of the __________ period.
The Sulbasutras is a mathematical treatise of the __________ period.
Match the following ancient mathematicians with their contributions:
Match the following ancient mathematicians with their contributions:
What is the value of √2 according to the Sulbasutras?
What is the value of √2 according to the Sulbasutras?
Ï€ has been computed to over 1.24 billion decimal places.
Ï€ has been computed to over 1.24 billion decimal places.
What is the value of the irrational number between 1/7 and 2/7?
What is the value of the irrational number between 1/7 and 2/7?
What is a characteristic of every whole number?
What is a characteristic of every whole number?
Every integer is a rational number.
Every integer is a rational number.
Write one way to find a rational number between two given numbers, r and s.
Write one way to find a rational number between two given numbers, r and s.
The fraction _______ is equivalent to 1/2.
The fraction _______ is equivalent to 1/2.
Match the following types of numbers with their descriptions:
Match the following types of numbers with their descriptions:
Why is 3/5 not an integer?
Why is 3/5 not an integer?
Every rational number is an integer.
Every rational number is an integer.
Find one rational number between 1 and 2 using the method of adding and dividing by 2.
Find one rational number between 1 and 2 using the method of adding and dividing by 2.
What can be said about the sum, difference, quotient, and product of irrational numbers?
What can be said about the sum, difference, quotient, and product of irrational numbers?
The decimal expansion of an irrational number is always terminating.
The decimal expansion of an irrational number is always terminating.
What is the result of adding a rational number to an irrational number?
What is the result of adding a rational number to an irrational number?
The decimal expansion of _______________ numbers is non-terminating and non-recurring.
The decimal expansion of _______________ numbers is non-terminating and non-recurring.
The product of a rational number and an irrational number is always rational.
The product of a rational number and an irrational number is always rational.
What can be said about the numbers 2 + 21 and π – 2?
What can be said about the numbers 2 + 21 and π – 2?
Match the expressions with their simplified forms:
Match the expressions with their simplified forms:
What is the result of multiplying an irrational number by a rational number?
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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