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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?

<p>Numbers that can be written as quotients of integers.</p> Signup and view all the answers

What are irrational numbers?

<p>Numbers that cannot be written as quotients of integers.</p> Signup and view all the answers

What is the additive inverse?

<p>a + (-a) = 0</p> Signup and view all the answers

What is the multiplicative inverse?

<p>1/a</p> Signup and view all the answers

What does the commutative property of addition state?

<p>a + b = b + a</p> Signup and view all the answers

What does the commutative property of multiplication state?

<p>ab = ba</p> Signup and view all the answers

What is the closure property of addition?

<p>a + b is a real number.</p> Signup and view all the answers

What is the closure property of multiplication?

<p>ab is a real number.</p> Signup and view all the answers

What does the associative property of addition state?

<p>(a + b) + c = a + (b + c)</p> Signup and view all the answers

What does the associative property of multiplication state?

<p>(ab)c = a(bc)</p> Signup and view all the answers

What is the identity of addition?

<p>a + 0 = a; 0 + a = a</p> Signup and view all the answers

What is the identity of multiplication?

<p>a(1) = a; 1(a) = a</p> Signup and view all the answers

What is the inverse of multiplication?

<p>a(1/2) = 1; a cannot = 0</p> Signup and view all the answers

What is the distributive property?

<p>a(b + c) = ab + ac</p> Signup and view all the answers

What is absolute value?

<p>The distance from zero on the number line.</p> Signup and view all the answers

What is a variable?

<p>A symbol, usually a letter.</p> Signup and view all the answers

What is an algebraic expression?

<p>An expression that contains one or more variables.</p> Signup and view all the answers

What does it mean to evaluate an expression?

<p>Substituting numbers for the variables and following the order of operations.</p> Signup and view all the answers

What is a term?

<p>A number, variable, or the product of a number and one or more variables.</p> Signup and view all the answers

What is a coefficient?

<p>The numerical factor in a term.</p> Signup and view all the answers

What is a solution of the equation?

<p>A number that makes the equation true.</p> Signup and view all the answers

What is the reflexive property?

<p>a = a</p> Signup and view all the answers

What is the symmetric property?

<p>If a = b then b = a</p> Signup and view all the answers

What is the transitive property?

<p>If a = b and b = c, then a = c.</p> Signup and view all the answers

What is the addition property?

<p>If a = b, then a + c = b + c.</p> Signup and view all the answers

What is the subtraction property?

<p>If a = b, then a - c = b - c.</p> Signup and view all the answers

What is the multiplication property?

<p>If a = b, then ac = bc.</p> Signup and view all the answers

What is the division property?

<p>If a = b and c does not = 0, then a/c = b/c.</p> Signup and view all the answers

What is the substitution property?

<p>If a = b, then b may be substituted for a in any expression.</p> Signup and view all the answers

What is a compound inequality?

<p>A pair of inequalities joined by and or or.</p> Signup and view all the answers

What is an extraneous solution?

<p>A solution derived from an original equation that is not a solution of the original equation.</p> Signup and view all the answers

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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