Podcast
Questions and Answers
Is zero a rational number?
Is zero a rational number?
Yes, zero can be written in the form p/q where p = 0 and q = 1.
The collection of natural numbers is denoted by the symbol ___
The collection of natural numbers is denoted by the symbol ___
N
The whole numbers are denoted by the symbol ___
The whole numbers are denoted by the symbol ___
W
The collection of integers is denoted by the symbol ___
The collection of integers is denoted by the symbol ___
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The collection of rational numbers is denoted by the symbol ___
The collection of rational numbers is denoted by the symbol ___
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Every whole number is a natural number.
Every whole number is a natural number.
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Every integer is a rational number.
Every integer is a rational number.
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Every rational number is an integer.
Every rational number is an integer.
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What are the numbers on the number line that cannot be written in the form p/q called?
What are the numbers on the number line that cannot be written in the form p/q called?
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Which of the following statements is true?
Which of the following statements is true?
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What form must a number r take to be classified as a rational number?
What form must a number r take to be classified as a rational number?
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What defines the decimal expansion of irrational numbers?
What defines the decimal expansion of irrational numbers?
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How can you locate the square root of 2 on the number line?
How can you locate the square root of 2 on the number line?
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Which of the following is an example of a rational number?
Which of the following is an example of a rational number?
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Corresponding to every real number, there is a point on the ___ number line.
Corresponding to every real number, there is a point on the ___ number line.
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Which of the following forms a rational number?
Which of the following forms a rational number?
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What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$?
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$?
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All rational numbers have a non-terminating decimal expansion.
All rational numbers have a non-terminating decimal expansion.
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Express $0.6$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
Express $0.6$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
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Express $0.47$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
Express $0.47$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
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Express $0.001$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
Express $0.001$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
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What is the decimal expansion of $\frac{1}{7}$?
What is the decimal expansion of $\frac{1}{7}$?
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The sum of a rational number and an irrational number is always irrational.
The sum of a rational number and an irrational number is always irrational.
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Classify the number $23$. Is it rational or irrational?
Classify the number $23$. Is it rational or irrational?
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Classify the number $0.3796$. Is it rational or irrational?
Classify the number $0.3796$. Is it rational or irrational?
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Classify the number $7.478478...$. Is it rational or irrational?
Classify the number $7.478478...$. Is it rational or irrational?
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Classify the number $1.101001000100001...$. Is it rational or irrational?
Classify the number $1.101001000100001...$. Is it rational or irrational?
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Study Notes
Here are the study notes for the text in detailed bullet points, focusing on key facts with context:
Introduction to Number Systems
- We have learned about the number line and how to represent various types of numbers on it.
- The number line is infinite in both directions, and we can collect different types of numbers and store them in a bag.
Natural Numbers, Whole Numbers, and Integers
- Natural numbers (N) are positive integers, starting from 1 (1, 2, 3, ...).
- Whole numbers (W) include zero and all natural numbers (0, 1, 2, 3, ...).
- Integers (Z) include all whole numbers, both positive and negative (...,-3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers
- A rational number (Q) is a number that can be written in the form p/q, where p and q are integers, and q ≠ 0.
- Rational numbers include all natural numbers, whole numbers, and integers.
- Rational numbers can be represented on the number line.
- There are infinitely many rational numbers between any two given rational numbers.
Irrational Numbers
- An irrational number is a number that cannot be written in the form p/q, where p and q are integers, and q ≠ 0.
- Irrational numbers cannot be represented as a finite decimal or fraction.
- Examples of irrational numbers include √2, √3, and π.
- There are infinitely many irrational numbers.
Real Numbers
- Real numbers (R) are either rational or irrational numbers.
- Every real number can be represented by a unique point on the number line.
- Every point on the number line represents a unique real number.
Decimal Expansions of Real Numbers
- The decimal expansion of a rational number is either terminating or non-terminating recurring.
- A number with a non-terminating non-recurring decimal expansion is irrational.
- Examples of decimal expansions include 0.5 (terminating), 0.142857 (non-terminating recurring), and π (non-terminating non-recurring).
Key Results
- The decimal expansion of a rational number is either terminating or non-terminating recurring.
- A number with a non-terminating non-recurring decimal expansion is irrational.
- Every number with a non-terminating non-recurring decimal expansion is irrational.
Examples and Exercises
- Examples of locating irrational numbers on the number line, such as √2 and √3.
- Exercises to practice identifying and working with rational and irrational numbers, and their decimal expansions.Here are the study notes:
Rational and Irrational Numbers
- A rational number is a number that can be written in the form
p/qwherepandqare integers andq ≠ 0. - An irrational number is a number that cannot be written in the form
p/qwherepandqare integers andq ≠ 0.
Decimal Expansions
- The decimal expansion of a rational number is either terminating or non-terminating recurring.
- A number whose decimal expansion is terminating or non-terminating recurring is rational.
- The decimal expansion of an irrational number is non-terminating non-recurring.
- A number whose decimal expansion is non-terminating non-recurring is irrational.
Operations on Real Numbers
- When you add, subtract, multiply, or divide two rational numbers, the result is always a rational number.
- When you add, subtract, multiply, or divide a rational number and an irrational number, the result may be rational or irrational.
- The sum, difference, product, or quotient of two irrational numbers may be rational or irrational.
Square Roots and Radical Signs
- A square root of a positive real number
ais a numberbsuch thatb² = aandb > 0. - The symbol
√is called the radical sign. -
acan be extended toa^(1/n)whereais a real number andnis a positive integer.
Rationalizing Denominators
- To rationalize the denominator of an expression, you multiply it by a suitable form of
1to make the denominator a rational number. - This is useful when working with expressions that involve square roots.
Laws of Exponents
- The laws of exponents can be extended to rational exponents.
- Let
a > 0be a real number andpandqbe rational numbers. Then:-
a^p × a^q = a^(p+q) -
(a^p)^q = a^(pq) -
a^p / a^q = a^(p-q) -
a^p × b^p = (ab)^p
-
I hope these notes are helpful! Let me know if you have any questions.
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Description
This quiz covers the basics of number systems, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
