Defining the Rational Numbers


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1 MATH10 College Mthemtis  Slide Set 2 1. Rtionl Numers 1. Define the rtionl numers. 2. Redue rtionl numers.. Convert etween mixed numers nd improper frtions. 4. Express rtionl numers s deimls.. Express deimls in the form /. 6. Multiply nd divide rtionl numers.. Add nd sutrt rtionl numers. 8. Use the order of opertions greement with rtionl numers. 9. Solve prolems involving rtionl numers. Copyright 201 R. Lurie 1 Defining the Rtionl Numers The set of rtionl numers is the set of ll numers whih n e expressed in the form where nd re integers nd is not equl to 0. The integer is lled the numertor The integer is lled the denomintor Exmples of rtionl numers: ¼, ½, ¾,, 0 Equivlent Rtionl Numers To redue rtionl numer to its lowest terms divide numertor nd denomintor y their gretest ommon divisor = Copyright 201 R. Lurie 2 Reduing Rtionl Numer 10 4 Redue to lowest terms. Solution: Begin y finding the Gretest Common Ftor of 10 nd 4. Thus, 10 = 2 1, nd 4 = 1 Divide the numertor nd the denomintor of the given rtionl numer y 1 or = = 2 or 10 4 = = 2 There re no ommon divisors of 2 nd other thn 1. 2 Thus, the rtionl numer is in its lowest terms. Copyright 201 R. Lurie Mixed Numers, Improper Frtions, nd Deiml A mixed numer onsists of the sum of n integer nd rtionl numer, expressed without the use of n ddition sign. Exmple: An improper frtion is rtionl numer whose numertor is 19 greter thn denomintor Any rtionl numer n e expressed s deiml numer y dividing the denomintor into the numertor.8 19 is lrger thn Copyright 201 R. Lurie 4 Copyright 201 R. Lurie Pge 1 of
2 MATH10 College Mthemtis  Slide Set 2 Convert Mixed Numer to Improper Frtion 1. Multiply the denomintor of the rtionl numer y the integer nd dd the numertor to this produt. 2. Ple the sum in step 1 over the denomintor of the mixed numer. Convert Improper Frtion to Mixed Numer 1. Divide the denomintor into the numertor. Reord the quotient nd the reminder. 2. Write the mixed numer using the following form: 42 Convert to mixed numer. Solution: Step 1 Divide denomintor into the numertor Step 2 Copyright 201 R. Lurie Copyright 201 R. Lurie 6 Expressing Rtionl Numers s Deimls Express eh rtionl numer s deiml Notie the digits 6 repet over nd over 20 indefinitely. This is lled 16 repeting deiml Notie the deiml stops with reminder = 0. This is terminting deiml. 0 Copyright 201 R. Lurie M Expressing Deimls s Frtion Express terminting deiml s quotient of integers: Solution:. 0. = euse the is in the tenths position = 100 euse the digit on the right, 9, is in the hundredths position = 1000 = = 6 12 euse the digit on the right, 8, is thousndths position nd n e redued to lowest terms Copyright 201 R. Lurie 8 Copyright 201 R. Lurie Pge 2 of
3 MATH10 College Mthemtis  Slide Set 2 Multiplying Rtionl Numers The produt of two rtionl numers is the produt of their numertors divided y the produt of their denomintors. If nd re multiplied, then =. d d d ( )( 2 94 ) =( 2)( 9) = = = 2 or Multiply ross. Presimplify Exmple 1 ( 2 = or 1 )( 94) Simplify to lowest terms. Copyright 201 R. Lurie 9 Dividing Rtionl Numers The quotient of two rtionl numers is produt of the first numer nd the reiprol of the seond numer Flip lst numer nd multiply y first numer If nd re rtionl d numers, then d = d = d = = = Chnge to multiplition y using the reiprol. Multiply ross. Copyright 201 R. Lurie 10 Add nd Sutrt Rtionl Numers The sum or differene of two rtionl numers with identil denomintors is the sum or differene of numertors over ommon denomintor. If nd re rtionl numers, then + = + Exmples: + 2 =+2 = nd = = 12 = 6 12 = = = = + = = = or Copyright 201 R. Lurie Add nd Sutrt Rtionl Numers The sum or differene of two rtionl numers with different denomintors, we use the Lest Common Multiple of their denomintors to rewrite the rtionl numers. The Lest Common Multiple of their denomintors is lled the Lest Common Denomintor or LCD = = = 12 We multiply the first rtionl numer y / nd the seond one y 2/2 to otin 12 in the denomintor for eh numer. Notie, we hve 12 in the denomintor for eh numer. Add numertors nd put this sum over the lest ommon denomintor. Copyright 201 R. Lurie 12 Copyright 201 R. Lurie Pge of
4 MATH10 College Mthemtis  Slide Set 2 Exerise: Simplify using PEMDAS 2 ( ( 1 ) )2( 18)= 4 = Pro = Pro Copyright 201 R. Lurie The Irrtionl Numers 1. Define the irrtionl numers. 2. Simplify squre roots.. Perform opertions with squre roots. 4. Rtionlize the denomintor. The set of irrtionl numers is the set of numers whose deiml representtions re neither terminting nor repeting. π Copyright 201 R. Lurie 14 Squre Roots The prinipl squre root of nonnegtive numer n, written n, is the positive numer tht when multiplied y itself gives n. For exmple, 6=6 euse 6 6 = 6. Notie tht 6 is rtionl numer euse 6 is terminting deiml. Not ll squre roots re irrtionl. For exmple, here re few perfet squres: 0 = = = = 2 0=0 1=1 4=2 9= The squre root of perfet squre is rtionl numer Copyright 201 R. Lurie 1 The Produt Rule For Squre Roots If nd represent nonnegtive numers = nd = The squre root of produt is the produt of the squre roots. Simplify, if possile: = 2 = 2 = 00= 100 = 100 = 10 Copyright 201 R. Lurie 16 Copyright 201 R. Lurie Pge 4 of
5 MATH10 College Mthemtis  Slide Set 2 Adding nd Sutrting Squre Roots The numer tht multiplies squre root is lled the squre root s oeffiient. Squre roots with the sme rdind n e dded or sutrted y dding or sutrting their oeffiients: 2+ 2=(+ ) 2= =(2 6) = 4 Copyright 201 R. Lurie 1 Produt Rule Exerises = nd = 2 = 2 = 10 = 49= 6 12= 6 12= 2= 6 2= 6 2=6 2 (2 + 1) =0+4 Pro =1 Pro Copyright 201 R. Lurie 18 Dividing Squre Roots = Quotient Rule If nd represent nonnegtive rel numers nd 0, then The quotient of two squre roots is the squre root of the quotient = 90 2 = = = 2= nd = = 4= 9 = 9 = Copyright 201 R. Lurie 19 Time is Reltive Plnet of the Apes (1968) Einstein s Speil Reltivity Eqution R = Reltive Age Astronut R f = Reltive Age Friend on Erth v = Veloity = Speed of light R =R f 1 ( v ) 2 Copyright 201 R. Lurie 20 Copyright 201 R. Lurie Pge of
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