Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }


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1 ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line Negtive integers Positive integers RATIONAL NUMBERS All numers tht cn e written in the form >, where nd re integers nd Z 0 IRRATIONAL NUMBERS Rel numers tht cnnot e written s the quotient of two integers ut cn e represented on the numer line REAL NUMBERS Include ll numers tht cn e represented on the numer line, tht is, ll rtionl nd irrtionl numers Irrtionl Numers 5VN 3, VN, p, etc Rel Numers Zero 4_ Rtionl Numers 3, 4, 1, 0, 06, 1, etc Integers p 3,, 1, 0, 1,, 3, p Whole Numers 0, 1,, 3, p 5 Importnt Properties PROPERTIES OF ADDITION Identity Property of Zero: Inverse Property: Commuttive Property: Associtive Property: PROPERTIES OF MULTIPLICATION Property of Zero: Identity Property of One: # 1 =, when Z 0 Inverse Property: # 1, when = 1 Commuttive Property: Associtive Property: PROPERTIES OF DIVISION 0 Property of Zero:, when Z 0 = 0 Property of One:, when Z 0 = 1 Identity Property of One: Asolute Vlue # 0 = = + 1 = 0 + = c = c # = # Z 0 # 1 # c = 1 # # c 1 = # 1 The solute vlue of numer is lwys 0 If 7 0, ƒ ƒ = If 6 0,  ƒ = For exmple, ƒ 5 ƒ = 5 nd ƒ 5 ƒ = 5 In ech cse, the nswer is positive Nturl Numers 1,, 3, p PRIME NUMBERS A prime numer is numer greter thn 1 tht hs only itself nd 1 s fctors Some exmples:, 3, nd 7 re prime numers COMPOSITE NUMBERS A composite numer is numer tht is not prime For exmple, 8 is composite numer since 8 = # # = 3 ISBN13: ISBN10:
2 Key Words nd Symols The following words nd symols re used for the opertions listed Addition Sum, totl, increse, plus ddend ddend = sum Sutrction Difference, decrese, minus minuend sutrhend = difference Multipliction Product, of, times *, #, 11, fctor fctor = product Division Quotient, per, divided y Order of Opertions dividend divisor = quotient 1 st : Prentheses Simplify ny expressions inside prentheses nd : Exponents Work out ny exponents 3 rd : Multipliction nd Division Solve ll multipliction nd division, working from left to right 4 th : Addition nd Sutrction These re done lst, from left to right For exmple, Integers > 15  # , 3 = 15  # 3 + 7, 9 = = 1 ADDING AND SUBTRACTING WITH NEGATIVES   = = = + Some exmples: = = = 419 = 15 Integers MULTIPLYING AND DIVIDING WITH NEGATIVES Some exmples: 3 # 5 = = 4 14>18 = or Frctions  # =   #  =   = , =  LEAST COMMON MULTIPLE The LCM of set of numers is the smllest numer tht is multiple of ll the given numers For exmple, the LCM of 5 nd 6 is 30, since 5 nd 6 hve no fctors in common GREATEST COMMON FACTOR The GCF of set of numers is the lrgest numer tht cn e evenly divided into ech of the given numers For exmple, the GCF of 4 nd 7 is 3, since oth 4 nd 7 re divisile y 3, ut they re not oth divisile y ny numers lrger thn 3 FRACTIONS Frctions re nother wy to express division The top numer of frction is clled the numertor, nd the ottom numer is clled the denomintor ADDING AND SUBTRACTING FRACTIONS Frctions must hve the sme denomintor efore they cn e dded or sutrcted, with d Z 0 d + d = + d, with d Z 0 d  d =  d If the frctions hve different denomintors, rewrite them s equivlent frctions with common denomintor Then dd or sutrct the numertors, keeping the denomintors the sme For exmple, = = 11 1
3 Frctions Equivlent frctions re found y multiplying the numertor nd denomintor of the frction y the sme numer In the previous exmple, 1 nd 4 = 1 # 3 4 # = MULTIPLYING AND DIVIDING FRACTIONS When multiplying nd dividing frctions, common denomintor is not needed To multiply, tke the product of the numertors nd the product of the denomintors: To divide frctions, invert the second frction nd then multiply the numertors nd denomintors:, c d = # d c = d c Some exmples: 3 = # 4 3 # 4 = 8 1 # c d = # c # = c d d 3 5 # 7 = , 1 = 5 1 # 1 = 10 1 = 5 6 REDUCING FRACTIONS To reduce frction, divide oth the numertor nd denomintor y common fctors In the lst exmple, 10 1 = 10, 1, = 5 6 MIXED NUMBERS A mixed numer hs two prts: whole numer prt nd frctionl prt An exmple of mixed numer is This relly represents 5 + 3, 8 which cn e written s = 43 8 Similrly, n improper frction cn e written s mixed numer For exmple, 0 cn e written s 6 3, 3 since 0 divided y 3 equls 6 with reminder of Rtes, Rtios, Proportions, nd Percents RATES AND RATIOS A rte is comprison of two quntities with different units For exmple, cr tht trvels 110 miles in hours is moving t rte of 110 miles/ hours or 55 mph A rtio is comprison of two quntities with the sme units For exmple, clss with 3 students hs 3 student techer rtio of 3:1 or PROPORTIONS A proportion is sttement in which two rtios or rtes re equl An exmple of proportion is the following sttement: 30 dollrs is to 5 hours s 60 dollrs is to 10 hours This is written $30 5 hr = $60 10 hr A typicl proportion prolem will hve one unknown quntity, such s 1 mile 0 min = x miles 60 min We cn solve this eqution y cross multiplying s shown: 0x = 60 # 1 x = 60 0 = 3 So, it tkes 60 minutes to wlk 3 miles PERCENTS A percent is the numer of prts out of 100 To write percent s frction, divide y 100 nd drop the percent sign For exmple, 57% = To write frction s percent, first check to see if the denomintor is 100 If it is not, write the frction s n equivlent frction with 100 in the denomintor Then the numertor ecomes the percent For exmple, 4 5 = = 80% To find percent of quntity, multiply the percent y the quntity For exmple, 30% of 5 is 30 # 5 = = 3 1 3
4 Bsic Mth Review Deciml Numers The numers fter the deciml point represent frctions with denomintors tht re powers of 10 The deciml point seprtes the whole numer prt from the frctionl prt 9 For exmple, 09 represents Plce Vlue Chrt illions ADDING AND SUBTRACTING DECIMAL NUMBERS To dd or sutrct deciml numers, line up the numers so tht the deciml points re ligned Then dd or sutrct s usul, keeping the deciml point in the sme plce For exmple, = 300 MULTIPLYING AND DIVIDING DECIMAL NUMBERS To multiply deciml numers, multiply them s though they were whole numers The numer of deciml plces in the product is the sum of the numer of deciml plces in the fctors For exmple, 37 * 45 is hundred thousnds hundred millions ten millions millions ten thousnds thousnds hundreds tens ones tenths Whole numers hundredths thousndths ten thousndths Decimls deciml plces 1 deciml plce 3 deciml plces millionths hundred thousndths To divide deciml numers, first mke sure the divisor is whole numer If it is not, move the deciml plce to the right (multiply y 10, 100, nd so on) to mke it whole numer Then move the deciml point the sme numer of plces in the dividend For exmple, 04, 1 = 4, The deciml point in the nswer is plced directly ove the new deciml point in the dividend Percents to Decimls nd Decimls to Percents To chnge numer from percent to deciml, divide y 100 nd drop the percent sign: 58% = 58/100 = 058 To chnge numer from deciml to percent, multiply y 100 nd dd the percent sign: 073 = 73 * 100 = 73% Simple Interest Given the principl (mount of money to e orrowed or invested), interest rte, nd length of time, the mount of interest cn e found using the formul where I = interest 1dollr mount p = principl r = percentge rte of interest t = time period For exmple, find the mount of simple interest on $3800 lon t n nnul rte of 55% for 5 yers: p = $3800 r = 55% = 0055 t = 5 yers I = = 1045 The mount of interest is $1045 Scientific Nottion I = p # r # t Scientific nottion is convenient wy to express very lrge or very smll numers A numer in this form is written s * 10 n, where 1 ƒ ƒ 6 10 nd n is n integer For exmple, 36 * 10 5 nd 1 * 104 re expressed in scientific nottion To chnge numer from scientific nottion to numer without exponents, look t the power of ten If tht numer is positive, move the deciml point to the right If it is negtive, move the deciml point to the left The numer tells you how mny plces to move the deciml point For exmple, 397 * 10 3 = 3970 To chnge numer to scientific nottion, move the deciml point so it is to the right of the first nonzero digit If the deciml point is moved n plces to the left nd this mkes the numer smller, n is positive; otherwise, n is negtive If the deciml point is not moved, n is 0 For exmple, = 16 *
5 Scientific Nottion MULTIPLYING AND DIVIDING IN SCIENTIFIC NOTATION To multiply or divide numers in scientific nottion, we cn chnge the order nd grouping, so tht we multiply or divide first the deciml prts nd then the powers of 10 For exmple, Sttistics There re severl wys to study list of dt Men, or verge, is the sum of ll the dt vlues divided y the numer of vlues Medin is the numer tht seprtes the list of dt into two equl prts To find the medin, list the dt in order from smllest to lrgest If the numer of dt is odd, the medin is the middle numer If the numer of dt is even, the medin is the verge of the two middle numers Mode is the numer in the list tht occurs the most frequently There cn e more thn one mode For exmple, consider the following list of test scores: {87, 56, 69, 87, 93, 8} To find the men, first dd: = 474 Then divide y 6: = 79 The men score is 79 To find the medin, first list the dt in order: 56, 69, 8, 87, 87, 93 Since there is n even numer of dt, we tke the verge of 8 nd 87: = 169 = 845 The medin score is 845 The mode is 87, since this numer ppers twice nd ech of the other numers ppers only once Distnce Formul 137 * 103 # 15 * 10 8 = 137 * 5 # * 10 8 = 95 * 10 5 Given the rte t which you re trveling nd the length of time you will e trveling, the distnce cn e found y using the formul d = r # t where d = distnce r = rte t = time US Mesurement Units in = inch ft = foot min = minute sec = second gl = gllon yd = yrd pt = pint Mesurements Metric Units mm = millimeter cm = centimeter km = kilometer m = meter ml = milliliter cl = centiliter L = liter kl = kiloliter mg = milligrm cg = centigrm g = grm kg = kilogrm US AND METRIC CONVERSIONS US oz = ounce c = cup mi = mile hr = hour l = pound qt = qurt T = ton 1 in = 1 ft 3 ft = 1 yd 1760 yd = 1 mi 580 ft = 1 mi c = 1 pt 1 c = 8 oz 4 qt = 1 gl pt = 1 qt 000 l = 1 T 16 oz = 1 l Metric 1000 mm = 1 m 100 cm = 1 m 1000 m = 1 km 100 cl = 1 L 1000 ml = 1 L 100 cg = 1 g 1000 mg = 1 g 1000 g = 1 kg 0001 m = 1 mm 001 m = 1 cm 0001 g = 1 mg 001 g = 1 cg 0001 L = 1 ml 001 L = 1 cl 5
6 Geometry The perimeter of geometric figure is the distnce round it or the sum of the lengths of its sides The perimeter of rectngle is times the length plus times the width: W P = L + W The perimeter of squre is 4 times the length of side: P = 4s Are is lwys expressed in squre units, since it is twodimensionl The formul for re of rectngle is A = L # W The formul for re of squre is A = s # s or A = s s The re of tringle is onehlf the product of the height nd se: L s Geometry PYTHAGOREAN THEOREM In ny right tringle, if nd re the lengths of the legs nd c is the length of the hypotenuse, then + = c CIRCLES Are: A = p # r Circumference: C = p # d = # p # r where d is the dimeter, r is the rdius, or hlf the dimeter, nd p is pproximtely 314 or A circle hs n ngle of 360 degrees A stright line hs n ngle of 180 degrees c d 7 r The sum of ll three ngles in ny tringle lwys equls 180 degrees x A = 1 # h y x + y + z = 180 A right tringle is tringle with 90 (right) ngle The hypotenuse of right tringle is the side opposite the right ngle h z hypotenuse Algeric Terms Vrile: A vrile is letter tht represents numer ecuse the numer is unknown or ecuse it cn chnge For exmple, the numer of dys until your vction chnges every dy, so it could e represented y vrile, x Constnt: A constnt is term tht does not chnge For exmple, the numer of dys in the week, 7, does not chnge, so it is constnt Expression: An lgeric expression consists of constnts, vriles, numerls nd t lest one opertion For exmple, x + 7 is n expression Eqution: An eqution is siclly mthemticl sentence indicting tht two expressions re equl For exmple, x + 7 = 18 is n eqution Solution: A numer tht mkes n eqution true is solution to tht eqution For exmple, in using the ove eqution, x + 7 = 18, we know tht the sttement is true if x =
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