SAMPLE College Math
Real numbers are used extensively in mathematicsandarerepresentedonacontinuousnumberline. Thenumberlineincludesthesetofallthewholenumbers{0,1,2,3,4…},naturalnumbers{1,2,3,4…}, integers{…-3,-2,-1,0,1,2,3,…},fractions(numberswrittenintheformof ,suchthattheratioisnot equaltozero)andirrationalnumbers(likesquarerootof2,givenby ).Onthenumberline,negative 2 values are on the left side of zero while positive values are on the right side of zero, as shown below: The properties of real numbers deal with four major operations: addition (+), subtraction ( ), − multiplication ( ) and division ( ). Addition, subtraction, and multiplication of real numbers will ×, ∙ ÷, / result inarealnumber.Divisionofrealnumbersisalsopossible,providedthatthedenominatorofthe fraction is not equal to zero. If the denominator is equal to zero, then the result is undefined. The Properties of Real Number are listed below (assume , , and are real numbers): 1. Cumulative law states that when we add or multiply two real numbers, their order does not matter. That is, and + = + × = × 2. Associative law states that while adding or multiplying more than two real numbers thenthe effect of parentheses does not matter. That is, and ( + ) + = + ( + ) ( × )× = ×( × ) 3. Distributive law applies to the caseswhereonemakesuseofbothadditionandmultiplication operations. and ×( + ) = × + × ( + )× = × + × 4. The additive identity is known as 0 and 1 is a multiplicative identity. That is, and + 0 = ×1 = 5. Additiveinverseisdenotedbyanegativesign.Thatis,theadditiveinverseof wouldbe( ). − Addition of a real number with its additive inverse would be 0. So, + (− ) = 0 6. Multiplicative inverse of would be -1 or . The multiplication of a real number with its −1 1 multiplicative inverse would be equal to 1, Therefore, × −1 = 1 7. Cancellation law for addition: if , then . + = + = 8. Cancellation law for multiplication: if , then , such that . × = × = ≠0 9. Cancellation law for division: , provided and . = ≠0 ≠0 Absolute values, or the magnitude of an integer, are another vital concept to know in algebra. It is definedasthedistancebetweentheintegerandthezerovalueonthenumberline.Absolutevaluesare denoted by the modulus symbol where can be any integer, negative or positive. If we have to | |
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