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Questions and Answers
What is a fundamental concept stated by the Fundamental Theorem of Arithmetic?
What is a fundamental concept stated by the Fundamental Theorem of Arithmetic?
What is a characteristic of irrational numbers?
What is a characteristic of irrational numbers?
Which of the following is an example of an irrational number?
Which of the following is an example of an irrational number?
What is the Commutative Property of Addition?
What is the Commutative Property of Addition?
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What is the Associative Property of Multiplication?
What is the Associative Property of Multiplication?
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What is the Distributive Property?
What is the Distributive Property?
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Which property states that the order in which real numbers are added does not change the result?
Which property states that the order in which real numbers are added does not change the result?
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What is a consequence of the Fundamental Theorem of Arithmetic?
What is a consequence of the Fundamental Theorem of Arithmetic?
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What is the prime factorization of the number 24?
What is the prime factorization of the number 24?
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What is the characteristic that π and e have in common?
What is the characteristic that π and e have in common?
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What is the result of multiplying any real number by 1?
What is the result of multiplying any real number by 1?
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What is the result of adding 0 to any real number?
What is the result of adding 0 to any real number?
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What is the expression that illustrates the Distributive Property of real numbers?
What is the expression that illustrates the Distributive Property of real numbers?
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What is the characteristic of the decimal representation of irrational numbers?
What is the characteristic of the decimal representation of irrational numbers?
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What is the relationship between the numbers a and -a in real number arithmetic?
What is the relationship between the numbers a and -a in real number arithmetic?
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What is the result of rearranging the factors in a multiplication problem involving real numbers?
What is the result of rearranging the factors in a multiplication problem involving real numbers?
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Study Notes
Fundamental Theorem of Arithmetic
- Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
- This theorem provides a way to write a positive integer as a product of prime numbers, known as the prime factorization of the integer.
- The prime factorization of a positive integer is unique, up to the order of the prime numbers.
Irrational Numbers
- An irrational number is a real number that cannot be expressed as a finite decimal or fraction.
- Irrational numbers have an infinite number of digits that never repeat in a predictable pattern.
- Examples of irrational numbers include:
- Pi (π)
- Euler's number (e)
- The square root of 2 (√2)
- The square root of 3 (√3)
Properties of Real Numbers
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Commutative Property of Addition: The order in which real numbers are added does not change the result.
- a + b = b + a
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Commutative Property of Multiplication: The order in which real numbers are multiplied does not change the result.
- a × b = b × a
-
Associative Property of Addition: The order in which real numbers are added does not change the result, even when more than two numbers are involved.
- (a + b) + c = a + (b + c)
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Associative Property of Multiplication: The order in which real numbers are multiplied does not change the result, even when more than two numbers are involved.
- (a × b) × c = a × (b × c)
-
Distributive Property: The product of a real number and the sum of two real numbers is equal to the sum of the products of the real number and each of the two real numbers.
- a × (b + c) = a × b + a × c
Fundamental Theorem of Arithmetic
- Every positive integer has a unique prime factorization, excluding the order of prime numbers.
- The prime factorization of a positive integer is unique, up to the order of the prime numbers.
Irrational Numbers
- An irrational number is a real number that cannot be expressed as a finite decimal or fraction.
- Irrational numbers have an infinite number of digits that never repeat in a predictable pattern.
- Examples of irrational numbers include Pi (π), Euler's number (e), the square root of 2 (√2), and the square root of 3 (√3).
Properties of Real Numbers
- The commutative property of addition states that the order of real numbers being added does not change the result: a + b = b + a.
- The commutative property of multiplication states that the order of real numbers being multiplied does not change the result: a × b = b × a.
- The associative property of addition states that the order of real numbers being added does not change the result, even when more than two numbers are involved: (a + b) + c = a + (b + c).
- The associative property of multiplication states that the order of real numbers being multiplied does not change the result, even when more than two numbers are involved: (a × b) × c = a × (b × c).
- The distributive property states that the product of a real number and the sum of two real numbers is equal to the sum of the products of the real number and each of the two real numbers: a × (b + c) = a × b + a × c.
Fundamental Theorem of Arithmetic
- Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which they are listed.
- Example: 12 = 2 × 2 × 3 (unique prime factorization)
Irrational Numbers
- A real number that cannot be expressed as a finite decimal or fraction (ratio of integers).
- Examples: π (pi), e (Euler's number), √2 (square root of 2)
- Irrational numbers are non-terminating and non-repeating decimals.
- Irrational numbers cannot be expressed as a simple fraction (numerator and denominator are integers).
Properties of Real Numbers
Closure Properties
- The sum of two real numbers is always a real number.
- The product of two real numbers is always a real number.
Commutative Properties
- a + b = b + a (addition)
- a × b = b × a (multiplication)
Associative Properties
- (a + b) + c = a + (b + c) (addition)
- (a × b) × c = a × (b × c) (multiplication)
Distributive Property
- a × (b + c) = a × b + a × c
Existence of Identities
- Additive identity: 0 (a + 0 = a)
- Multiplicative identity: 1 (a × 1 = a)
Existence of Additive Inverses
- For each real number a, there exists a number -a such that a + (-a) = 0
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Description
Learn about the fundamental theorem of arithmetic and irrational numbers, including their definitions and properties.
