Questions and Answers
What are natural numbers?
Numbers used for counting.
What are whole numbers?
All natural numbers and 0.
What are integers?
The natural numbers, 0, and the negative numbers.
What are rational numbers?
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What are irrational numbers?
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What is the additive inverse?
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What is the multiplicative inverse?
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What does the commutative property of addition state?
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What does the commutative property of multiplication state?
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What is the closure property of addition?
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What is the closure property of multiplication?
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What does the associative property of addition state?
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What does the associative property of multiplication state?
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What is the identity of addition?
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What is the identity of multiplication?
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What is the inverse of multiplication?
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What is the distributive property?
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What is absolute value?
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What is a variable?
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What is an algebraic expression?
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What does it mean to evaluate an expression?
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What is a term?
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What is a coefficient?
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What is a solution of the equation?
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What is the reflexive property?
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What is the symmetric property?
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What is the transitive property?
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What is the addition property?
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What is the subtraction property?
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What is the multiplication property?
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What is the division property?
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What is the substitution property?
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What is a compound inequality?
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What is an extraneous solution?
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Study Notes
Number Sets
- Natural Numbers: Used for counting; includes positive integers (1, 2, 3, …).
- Whole Numbers: Comprise all natural numbers plus zero (0, 1, 2, 3, …).
- Integers: Include whole numbers and their negative counterparts (…, -2, -1, 0, 1, 2, …).
- Rational Numbers: Can be expressed as fractions or quotients of integers with a non-zero denominator; represented as either terminating or repeating decimals.
- Irrational Numbers: Cannot be expressed as fractions; their decimal forms neither terminate nor repeat.
Inverses and Properties
- Additive Inverse: For any number 'a', its inverse is '-a' such that a + (-a) = 0.
- Multiplicative Inverse: For any non-zero number 'a', its inverse is 1/a; multiplying a number by its reciprocal yields 1.
Commutative and Associative Properties
- Commutative Property of Addition: Order of addition does not affect the sum (a + b = b + a).
- Commutative Property of Multiplication: Order of multiplication does not affect the product (ab = ba).
- Associative Property of Addition: Grouping in addition does not change the sum ((a + b) + c = a + (b + c)).
- Associative Property of Multiplication: Grouping in multiplication does not change the product ((ab)c = a(bc)).
Identity and Inverse Properties
- Identity Property of Addition: Adding zero does not change a number (a + 0 = a).
- Identity Property of Multiplication: Multiplying by one does not change a number (a × 1 = a).
- Inverse Property of Multiplication: A number multiplied by its reciprocal equals one (a × (1/a) = 1, where a ≠ 0).
Other Important Properties
- Distributive Property: Multiplying a number by a sum equals the sum of the products (a(b + c) = ab + ac).
- Absolute Value: Represented as |x|, it denotes the distance from zero on the number line, always non-negative.
Algebraic Concepts
- Variable: A symbol representing an unknown value, typically a letter (e.g., x, y).
- Algebraic Expression: Consists of numbers, variables, and operations; does not have an equals sign.
- Term: A single mathematical expression such as a number, variable, or the product of numbers and variables.
- Coefficient: The numerical factor in a term (e.g., in 3x, 3 is the coefficient).
- Evaluate: The process of substituting values for variables in an expression and following order of operations.
Equation and Properties of Equality
- Solution of the Equation: A number that satisfies the equation, making it true.
- Reflexive Property: Any quantity is equal to itself (a = a).
- Symmetric Property: If one quantity equals another, then vice versa (if a = b, then b = a).
- Transitive Property: If two quantities are equal to the same quantity, they are equal to each other (if a = b and b = c, then a = c).
Additional Properties
- Addition Property: If two quantities are equal, adding the same value to both maintains equality (if a = b, then a + c = b + c).
- Subtraction Property: If a = b, then subtracting the same value from both maintains equality (if a = b, then a - c = b - c).
- Multiplication Property: If a = b, then multiplying both by the same value maintains equality (if a = b, then ac = bc).
- Division Property: If a = b and c ≠ 0, dividing both by c maintains equality (if a = b, then a/c = b/c).
- Substitution Property: If a = b, b can replace a in any expression to yield an equivalent expression.
Complex Concepts
- Compound Inequality: Two inequalities connected by "and" or "or".
- Extraneous Solution: A solution obtained from an original equation that does not satisfy the original equation.
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