Algebra Vocabulary

Types of Numbers

You should be able to

  1. Recognize a number, tell what “types” it is.
  2. Give a few examples of any type, including one that is not in an earlier type.
  3. Be able to follow instructions when told to factor a number.
  4. Be able to follow instructions when told to completely factor, or to give the prime factorization.
  5. Realize that “break it down” is insufficient as a definition for “factor.”

 

Counting Numbers                 {1, 2, 3, …}

Whole Numbers                     Counting numbers and zero.

Integers                                  Counting numbers, opposites of counting numbers, and zero

Rational Numbers                  Numbers that can be written as , where a and b are integers, and b ¹ 0,

                                                or, equivalently,

                                                Numbers that can be written as repeating or terminating decimals.

Irrational Numbers                Numbers that can be written as nonterminating, nonrepeating decimals.

Real Numbers                        All of the above.

Prime Numbers                      Counting numbers greater than one, that have only trivial factors.

Composite Numbers              Counting numbers greater than one, that have nontrivial factors.

Factor of a number                  A whole number that divides into the number with no remainder.

Factor (verb)                           To write a number as the product of two or more numbers.

Operations and Their Properties

You should be able to

  1. Recognize operations in expressions.
  2. Quickly give examples of operations.
  3. Follow instructions if told to distribute, commute, or change associations,
  4. Recognize a simple change in an expression as a valid change by naming the property that guarantees it.
  5. Realize that there is more than one Commutative property, more than one Associative Property, etc., so that we need to specify which Commutative property we are talking about.

 

Commutative Property of Addition                                a + b = b + a

Associative Property of Addition                                   (a + b) + c = a + (b + c)

Identity Property of Addition                                         a + 0 = a

Inverse Property of Addition                                         Each number has an additive inverse:

                                                                                    a + –a = 0

Commutative Property of Multiplication             ab = ba

Associative Property of Multiplication               (ab)c = a(bc)

Identity Property of Multiplication                                 1a = a

Inverse Property of Multiplication                                  Each number has a multiplicative inverse:

 

Distributive Property of Multiplication over Addition      a(b + c) = ab + ac

Expressions

You should

  1. Be able to recognize an expression, distinguish between expression and equation.
  2. Be able to follow instructions when told to evaluate or simplify an expression.
  3. Realize that the value of a variable is required to evaluate an expression.
  4. Know the importance of equivalence in expressions.
  5. Be able to recognize an expression as a sum, difference, product, or fraction.
  6. Be able to recognize parts of an expression as terms or factors.
  7. Know why “break it down” is insufficient as a definition for “simplify.”
  8. Know why “make it simpler” is insufficient as a definition for “simplify.”

 

Expression                  A set of instructions that tells you how to calculate a result

Constant                     A number that will be the same each time anyone follows the instructions.

Variable                      A number that may vary when different people follow the instructions.

Value (variable)         The particular number used for a variable when following the instructions.

Value (expression)     The number that results from following the instructions.

Evaluate                     Find the value. Follow the instructions and get the result.

Equivalent                  Expressions are equivalent when they always have equal values.

Simplify                       Replace the expression with an equivalent expression that will work better.

    Alt. definition:         Make it simpler, but keep it equivalent.

Sum                             An expression in which the last operation is addition.

Difference      `           An expression in which the last operation is subtraction.

Product                       An expression in which the last operation is multiplication.

Quotient (fraction)     An expression in which the last operation is division

Terms                         Parts of an expression being added together.

Factors                        Parts of an expression being multiplied together.

Equations

You should

  1. Be able to recognize an equation, distinguish between expression and equation.
  2. Be able to decide whether or not a number is a solution to an equation.
  3. Be able to follow instructions when told to solve an equation.
  4. Know why “solving” an expression doesn’t make sense.
  5. Know why “break it down” is insufficient as a definition for “solve.”

 

Equation                     A sentence that says that two expressions have the same value.

(The equation may be saying that the expressions are always equal, or it may be saying that they are sometimes equal.)

Solution                       A value of the variable that makes the expressions have the same value.

    Alt. definitions:       A number that makes the equation true.

                                    A number that makes the equation work.

                                    A number that works in the equation.

The main definition is more precise, but the alternates are generally good enough.

Solution Set                The set of all solutions to an equation.

Equivalent                  Equations are equivalent when they have the same solution set.

Solve                           Obtain the solution set to an equation.

    Alt. definitions:       Give all the numbers that make the equation true.

                                    Give all the numbers that make the equation work.

Solve for x                  Write an equivalent equation that has x alone on one side, and no x occurring

                                    on the other side.

Inequality                   A sentence that compares the values of two expressions, but does not say that they are strictly equal.

Solution, solution set, equivalent, solve, and solve for x mean the same thing for inequalities that they do for equations.