Questions and Answers
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 ___
N
The whole numbers are denoted by the symbol ___
W
The collection of integers is denoted by the symbol ___
Signup and view all the answers
The collection of rational numbers is denoted by the symbol ___
Signup and view all the answers
Every whole number is a natural number.
Signup and view all the answers
Every integer is a rational number.
Signup and view all the answers
Every rational number is an integer.
Signup and view all the answers
What are the numbers on the number line that cannot be written in the form p/q called?
Signup and view all the answers
Which of the following statements is true?
Signup and view all the answers
What form must a number r take to be classified as a rational number?
Signup and view all the answers
What defines the decimal expansion of irrational numbers?
Signup and view all the answers
How can you locate the square root of 2 on the number line?
Signup and view all the answers
Which of the following is an example of a rational number?
Signup and view all the answers
Corresponding to every real number, there is a point on the ___ number line.
Signup and view all the answers
Which of the following forms a rational number?
Signup and view all the answers
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$?
Signup and view all the answers
All rational numbers have a non-terminating decimal expansion.
Signup and view all the answers
Express $0.6$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
Signup and view all the answers
Express $0.47$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
Signup and view all the answers
Express $0.001$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q ≠ 0$.
Signup and view all the answers
What is the decimal expansion of $\frac{1}{7}$?
Signup and view all the answers
The sum of a rational number and an irrational number is always irrational.
Signup and view all the answers
Classify the number $23$. Is it rational or irrational?
Signup and view all the answers
Classify the number $0.3796$. Is it rational or irrational?
Signup and view all the answers
Classify the number $7.478478...$. Is it rational or irrational?
Signup and view all the answers
Classify the number $1.101001000100001...$. Is it rational or irrational?
Signup and view all the answers
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers the basics of number systems, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
