(/) Search CliffsNotes Literature Notes (/literature) Test Prep (/test-prep) Study Guides (/study-guides) Log In (/users/log-in) Sign Up (/users/sign-up) Algebra I Home (/) » Study Guides (/Study-Guides) » Algebra I (/Study-Guides/Algebra/Algebra-I) » Properties Of Basic Mathematical Operations (/Study-Guides/Algebra/Algebra-I/Preliminaries-And-BasicOperations/Properties-Of-Basic-Mathematical-Operations) Properties of Basic Mathematical Operations Preliminaries and Basic Operations () Mathematical Operations (/studyra-i/preliminaries-and-basics-of-basic-mathematical-operations) Signed Numbers Fractions and Percents () rs (Positive Numbers and Negative Numbers) ra/algebra-i/signed-numbers-fractions-andasic Mathematical Operations (/study-numbers-positive-numbers-and-negativera-i/preliminaries-and-basic-operations/quizTerminology Sets and Expressions () udy-guides/algebra/algebra-i/terminology-setsmathematical-operations) z-set-theory) ding des/algebra/algebra-i/signed-numbersUsing Zero (/study-guides/algebra/algebraEquations Ratios and Proportions () portions (/study-guides/algebra/algebrants/fractions) basic-operations/multiplying-and-dividing- All Subjec Back to Top d-proportions/quiz-ratios-and-proportions) aic Expressions (/study-guides/algebra/algebrady-guides/algebra/algebra-i/signed-numbersnd-expressions/variables-and-algebraicEquations with Two Variables () (/studys of Equations (Simultaneous Equations) nts/quiz-fractions) dides/algebra/algebra-i/equations-ratios-andDividing Using Zero (/studyra-i/equations-with-two-variables/quiz-solvingns) ra-i/preliminaries-and-basic-operations/quizs-simultaneous-equations) and Complex Fractions (/studyAlgebraic ding-using-zero) Expressions (/study- and Factoring () Monomials Polynomials guides/algebra/algebra-i/monomialsra-i/signed-numbers-fractions-anddy-guides/algebra/algebra-i/equations-ratios-andra-i/terminology-sets-and-expressions/quizquations (Simultaneous Equations) (/studytoring/polynomials) -fractions-and-complex-fractions) uations) aic-expressions) ra-i/equations-with-two-variables/solvingAlgebraic Fractions () (/studyebraic Fractions (/study-guides/algebra/algebratudy-guides/algebra/algebra-i/monomialss-simultaneous-equations) ctions and Complex Fractions ns (/study-guides/algebra/algebra-i/equations(/study-guides/algebra/algebraoperations-with-algebraic-fractions) toring/quiz-polynomials) ra-i/signed-numbers-fractions-andns/ratios-and-proportions) nd-expressions/evaluating-expressions) Graphing and Absolute Value () hdes/algebra/algebra-i/monomials-polynomialsAlgebraic Fractions (/studyer LineInequalities (/study-guides/algebra/algebraressions (/study-guides/algebra/algebrang-and-absolute-value/graphing-on-a-numberra-i/algebraic-fractions/quiz-operations-withng) Coordinate Geometry () nd dy-guides/algebra/algebra-i/monomialsHalf-Planes (/study-guides/algebra/algebraFractions? (/study-guides/algebra/algebrary/linear-inequalities-and-half-planes) Number toring/quiz-factoring) Line (/study-guides/algebra/algebrawhat-are-algebraic-fractions) ng-and-absolute-value/quiz-graphing-on-aFunctions and (/studyVariations () ties dy-guides/algebra/algebra-i/functions-andand Half-Planes uides/algebra/algebra-i/monomialstoring/monomials) tions) ra-i/coordinate-geometry/quiz-linear-inequalitiesdy-guides/algebra/algebra-i/inequalitiesRoots and Radicals () udy-guides/algebra/algebra-i/monomialsare des/algebra/algebra-i/functions-andRoots (/study-guides/algebra/algebra-i/rootste-value/absolute-value) mplifying-square-roots) /study-guides/algebra/algebra-i/coordinatetoring/quiz-monomials) e-graphs) (/study-guides/algebra/algebra-i/inequalitiesQuadratic Equations () are dy-guides/algebra/algebra-i/functions-andtic Equations Roots (/study-guides/algebra/algebra-i/roots(/study-guides/algebra/algebrate-value/quiz-absolute-value) aphs ons-with-square-roots) s/quiz-solving-quadratic-equations) ions)(/study-guides/algebra/algebra-i/coordinatentaining inate-graphs) Absolute Value (/studyWord Problems () h uations ides/algebra/algebra-i/functions-andSquare (/study-guides/algebra/algebraRoots (/study-guides/algebra/algebrases (/study-guides/algebra/algebra-i/wordra-i/inequalities-graphing-and-absolutequiz-operations-with-square-roots) s/solving-quadratic-equations) ) -and-phrases) ons-containing-absolute-value) Quiz (/study-guides/algebra/algebra-i/algebra-i-quizzes/algebra-i-quizzes) s and Radicals (/study-guides/algebra/algebrady-guides/algebra/algebra-i/wordintroduction-to-roots-and-radicals) erest) oots (/study-guides/algebra/algebra-i/roots-and/study-guides/algebra/algebra-i/wordsquare-roots) -interest) Properties of Basic Mathematical Operations (/study-guides/algebra/algebra-i/wordproportion) Some mathematical operations have properties that can make them easier to work with and can actually es/algebra/algebra-i/word-problems/percent) save you time. dy-guides/algebra/algebra-i/wordange)Some properties (axioms) of addition You should know the definition of each of the following properties of addition and how each can be used. Closure is when all answers fall into the original set. If you add two even numbers, the answer is still an even number (2 + 4 = 6); therefore, the set of even numbers is closed under addition (has closure). If you add two odd numbers, the answer is not an odd number (3 + 5 = 8); therefore, the set of odd numbers is not closed under addition (no closure). Commutative means that the order does not make any difference in the result. Back to Top Note: Commutative does not hold for subtraction. Associative means that the grouping does not make any difference in the result. The grouping has changed (parentheses moved), but the sides are still equal. Note: Associative does not hold for subtraction. The identity element for addition is 0. Any number added to 0 gives the original number. The additive inverse is the opposite (negative) of the number. Any number plus its additive inverse equals 0 (the identity). Some properties (axioms) of multiplication You should know the definition of each of the following properties of multiplication and how each can be used. Back to Top Closure is when all answers fall into the original set. If you multiply two even numbers, the answer is still an even number (2 × 4 = 8); therefore, the set of even numbers is closed under multiplication (has closure). If you multiply two odd numbers, the answer is an odd number (3 × 5 = 15); therefore, the set of odd numbers is closed under multiplication (has closure). Commutative means the order does not make any difference. Note: Commutative does not hold for division. Associative means that the grouping does not make any difference. The grouping has changed (parentheses moved) but the sides are still equal. Note: Associative does not hold for division. The identity element for multiplication is 1. Any number multiplied by 1 gives the original number. Back to Top The multiplicative inverse is the reciprocal of the number. Any nonzero number multiplied by its reciprocal equals 1. ; therefore, 2 and are multiplicative inverses. ; therefore, a and are multiplicative inverses (provided a 0). A property of two operations The distributive property is the process of passing the number value outside of the parentheses, using multiplication, to the numbers being added or subtracted inside the parentheses. In order to apply the distributive property, it must be multiplication outside the parentheses and either addition or subtraction inside the parentheses. Note: You cannot use the distributive property with only one operation. 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