Questions and Answers
What is the set of numbers that includes all positive integers?
Which property states that a + b = b + a?
What is the set of numbers that includes all real and imaginary numbers?
What is the number that, when added to any number, does not change its value?
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Which property states that a × (b + c) = a × b + a × c?
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What is the set of numbers that includes fractions and can be expressed as a finite decimal?
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Study Notes
Classification of Numbers
- Natural Numbers: 1, 2, 3, ... (positive integers)
- Whole Numbers: 0, 1, 2, 3, ... (non-negative integers)
- Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (positive and negative whole numbers)
- Rational Numbers: fractions, e.g. 3/4, 22/7, can be expressed as a finite decimal
- Irrational Numbers: cannot be expressed as a finite decimal, e.g. π, e, sqrt(2)
- Real Numbers: includes rational and irrational numbers
- Complex Numbers: includes real and imaginary numbers, e.g. 3 + 4i
Properties of Numbers
- Commutative Property: a + b = b + a, a × b = b × a
- Associative Property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c)
- Distributive Property: a × (b + c) = a × b + a × c
- Additive Identity: 0, when added to any number, does not change its value
- Multiplicative Identity: 1, when multiplied to any number, does not change its value
- Additive Inverse: for each number a, there exists -a, such that a + (-a) = 0
- Multiplicative Inverse: for each number a, there exists 1/a, such that a × (1/a) = 1
Classification of Numbers
- Natural numbers are positive integers, starting from 1 and going upwards (1, 2, 3,...).
- Whole numbers include natural numbers and 0, representing non-negative integers (0, 1, 2, 3,...).
- Integers encompass both positive and negative whole numbers (...,-3, -2, -1, 0, 1, 2, 3,...).
- Rational numbers consist of fractions that can be expressed as finite decimals (e.g., 3/4, 22/7).
- Irrational numbers cannot be expressed as finite decimals and include numbers like π, e, and sqrt(2).
- Real numbers comprise both rational and irrational numbers.
- Complex numbers combine real and imaginary numbers, such as 3 + 4i.
Properties of Numbers
- The commutative property allows for swapping numbers during addition and multiplication: a + b = b + a, a × b = b × a.
- The associative property enables rearranging numbers during addition and multiplication: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
- The distributive property facilitates expanding a single operation across multiple terms: a × (b + c) = a × b + a × c.
- The additive identity, 0, maintains a number's value when added to it.
- The multiplicative identity, 1, preserves a number's value when multiplied by it.
- Each number has an additive inverse (-a) that results in 0 when added together: a + (-a) = 0.
- Each number has a multiplicative inverse (1/a) that equals 1 when multiplied together: a × (1/a) = 1.
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Description
Quiz about different types of numbers, including natural, whole, integers, rational, irrational, real, and complex numbers
