Questions and Answers
What is a fundamental property of real numbers that allows for the rearrangement of numbers in an equation, such as a + b = b + a, and a × b = b × a?
Commutative Property
What is the term for a real number that can be expressed as the ratio of two integers, such as 3/4 or 22/7?
Rational number
What is the term for a real number that cannot be expressed as the ratio of two integers, such as π or √2?
Irrational number
What is the property of real numbers that states that the order in which numbers are added or multiplied does not change the result, such as (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)?
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What is the term for the number that, when added to a real number, results in zero, such as -a?
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What is the term for the number that, when multiplied by a real number, results in one, such as 1/a?
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What is the property of real numbers that states that multiplication distributes over addition, such as a × (b + c) = a × b + a × c?
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What is the value that, when used as an exponent, does not change the value of a real number, such as a^1 = a?
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Study Notes
Definition of Real Numbers
- A real number is a value that can be represented on the number line
- It is a number that can be expressed as a finite or infinite decimal
- Real numbers can be either rational or irrational
Rational Numbers
- A rational number is a real number that can be expressed as the ratio of two integers, e.g. 3/4 or 22/7
- Rational numbers can be expressed as a finite decimal or a repeating decimal
- Examples of rational numbers: 0.5, 0.25, 1.333...
Irrational Numbers
- An irrational number is a real number that cannot be expressed as the ratio of two integers
- Irrational numbers have an infinite number of digits that never repeat in a predictable pattern
- Examples of irrational numbers: π, e, √2
Properties of Real Numbers
- Commutative Property: a + b = b + a and a × b = b × a
- Associative Property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = a × b + a × c
- Existence of Additive and Multiplicative Identities: 0 and 1, respectively
- Existence of Additive and Multiplicative Inverses: -a and 1/a, respectively
Operations on Real Numbers
- Addition and Subtraction: standard rules apply
- Multiplication and Division: standard rules apply, except for division by zero which is undefined
- Exponentiation: standard rules apply, with special cases for negative exponents and fractional exponents
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