Classifying Numbers: Whole, Natural, Integer, Rational

Questions and Answers

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?

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.

terminating Signup and view all the answers

Match the following decimal expansions with their types:

3.3333... = Non-terminating recurring 0.142857142857... = Non-terminating recurring 3.142678 = Terminating 1.27 = Non-terminating recurring Signup and view all the answers

What can be concluded about a number like 1.272727...?

It is a rational number Signup and view all the answers

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.

True Signup and view all the answers

Express 0.3333... in the form p/q, where p and q are integers and q ≠ 0.

1/3 Signup and view all the answers

What is the maximum number of digits in the repeating block of digits in the decimal expansion of 17/7?

It depends on the divisor Signup and view all the answers

Irrational numbers do not satisfy the commutative law for addition.

False Signup and view all the answers

What happens when you add, subtract, multiply, or divide (except by zero) two rational numbers?

You get a rational number Signup and view all the answers

Rational numbers are 'closed' with respect to addition, subtraction, multiplication, and ____________________.

division Signup and view all the answers

Match the following numbers with their classification as rational or irrational:

23/25 = Rational 0.3796 = Rational 7.478478... = Irrational 1.101001000100001... = Irrational Signup and view all the answers

Which of the following numbers has a non-terminating non-recurring decimal expansion?

None of the above Signup and view all the answers

The decimal expansion of an irrational number is always terminating.

False Signup and view all the answers

What can you say about the decimal expansion of a rational number?

It is either terminating or non-terminating recurring Signup and view all the answers

What is the approximate value of π often taken?

22/7 Signup and view all the answers

Archimedes was the first to compute 5 digits in the decimal expansion of π.

False Signup and view all the answers

Who is credited with finding the value of π correct to four decimal places?

Aryabhatta Signup and view all the answers

The Sulbasutras is a mathematical treatise of the __________ period.

Vedic Signup and view all the answers

Match the following ancient mathematicians with their contributions:

Archimedes = Computed π to 2 digits. Aryabhatta = Computed π to 4 digits. Ancient Indians = Developed Sulbasutras. Greeks = Used division method for finding digits of 2. Signup and view all the answers

What is the value of √2 according to the Sulbasutras?

1.4142156 Signup and view all the answers

π has been computed to over 1.24 billion decimal places.

False Signup and view all the answers

What is the value of the irrational number between 1/7 and 2/7?

0.285714 Signup and view all the answers

What is a characteristic of every whole number?

It is an integer Signup and view all the answers

Every integer is a rational number.

True Signup and view all the answers

Write one way to find a rational number between two given numbers, r and s.

Add r and s, and divide the sum by 2 Signup and view all the answers

The fraction _______ is equivalent to 1/2.

2/4 Signup and view all the answers

Match the following types of numbers with their descriptions:

Natural Numbers = Start from 1 and include all positive integers Whole Numbers = Include zero and all positive integers Integers = Include all positive and negative integers, and zero Rational Numbers = Can be expressed in the form m/n, where m and n are integers Signup and view all the answers

Why is 3/5 not an integer?

Because the denominator is not 1 Signup and view all the answers

Every rational number is an integer.

False Signup and view all the answers

Find one rational number between 1 and 2 using the method of adding and dividing by 2.

3/2 Signup and view all the answers

What can be said about the sum, difference, quotient, and product of irrational numbers?

They are not always irrational Signup and view all the answers

The decimal expansion of an irrational number is always terminating.

False Signup and view all the answers

What is the result of adding a rational number to an irrational number?

an irrational number Signup and view all the answers

The decimal expansion of _______________ numbers is non-terminating and non-recurring.

irrational Signup and view all the answers

The product of a rational number and an irrational number is always rational.

False Signup and view all the answers

What can be said about the numbers 2 + 21 and π – 2?

They are both irrational Signup and view all the answers

Match the expressions with their simplified forms:

2² + 5³ = 3² + 2³ 2 – 3³ = 3² + 2³ (2 + 1)² + (5 – 3)³ = 3² + 2³ Signup and view all the answers

What is the result of multiplying an irrational number by a rational number?

an irrational number Signup and view all the answers

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.

Classifying Numbers: Whole, Natural, Integer, Rational

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 decimal expansion of a rational number can be non-terminating non-recurring.

The decimal expansion of a rational number is either ______________________ or non-terminating recurring.

Match the following decimal expansions with their types:

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.

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?

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?

Rational numbers are 'closed' with respect to addition, subtraction, multiplication, and ____________________.

Match the following numbers with their classification as rational or irrational:

Which of the following numbers has a non-terminating non-recurring decimal expansion?

The decimal expansion of an irrational number is always terminating.

What can you say about the decimal expansion of a rational number?

What is the approximate value of π often taken?

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?

The Sulbasutras is a mathematical treatise of the __________ period.

Match the following ancient mathematicians with their contributions:

What is the value of √2 according to the Sulbasutras?

π 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 a characteristic of every whole number?

Every integer is a rational number.

Write one way to find a rational number between two given numbers, r and s.

The fraction _______ is equivalent to 1/2.

Match the following types of numbers with their descriptions:

Why is 3/5 not 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.

What can be said about the sum, difference, quotient, and product of irrational numbers?

The decimal expansion of an irrational number is always terminating.

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 product of a rational number and an irrational number is always rational.

What can be said about the numbers 2 + 21 and π – 2?

Match the expressions with their simplified forms:

What is the result of multiplying an irrational number by a rational number?

Study Notes

Equivalent Fractions

Types of Numbers

Finding Rational Numbers

Decimal Expansions of Rational Numbers

Irrational Numbers

Operations on Real Numbers

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