VII - განათლების ხარისხის...
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gamomcemloba inteleqti
Transcript of VII - განათლების ხარისხის...
gamomcemloba
inteleqti
VII
guram gogiSvili, Teimuraz vefxvaZe,
ia mebonia, lamara qurCiSvili
maTematika
VII klasis saxelmZRvanelo
gamomcemloba inteleqtiTbilisi 2012
redaqtori Teimuraz vefxvaZe
grifi mieniWa 2012 wels ssip ganaTlebis xarisxis ganviTarebis
erovnuli centris (brZaneba # 375, 18. 05. 2012) mier
guram gogiSvili, Teimuraz vefxvaZe,
ia mebonia, lamara qurCiSvili
m a T e m a t i k a
VII klasis saxelmZRvanelo
gamomcemloba inteleqti
Tbilisi 2012
Guram Gogishvili, Teimuraz Vepkhvadze,
Ia Mebonia, Lamara Kurchishvili
MaTHS VII
InTeLeKTI Publishers
Tbilisi 2012
ISBN 978-9941-439-16-2
© gamomcemloba `inteleqti~, 2012.
gamomcemloba inteleqtiTbilisi, ilia WavWavaZDis gamz. 5. tel.: 225 05 22
www.intelekti.ge [email protected] [email protected]
intelekti Publishers5 ilia Chavchavadze Ave., Tbilisi, Georgia. Tel.: (995 32) 225 05 22
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Cveno keTilo megobaro!
mogesalmebiT da gilocavT axali saswavlo wlis dadgomas!imedia, Cveni saxelmZRvanelo karg megzurobas gagiwevT maTema-
tikis didebul samyaroSi mogzaurobisas. SevecadeT, praqtikuli amocanebis ganxilviT dagarwmunoT maTematikis Seswavlis aucile-blobaSi. gaxsovdeT, aq miRebuli codna da Cvevebi sxva sagnebis ukeT aTvisebaSic dagexmarebaT.
Tu pirveli gacnobisas romelime sakiTxi kargad ver gaiazreT, ar daRondeT, gaagrZeleT muSaoba _ amave sakiTxs saxelmZRvaneloSi kvlav araerTxel SexvdebiT da aucileblad daZlevT mas.
warmatebiT gevloT maTematikis Seswavlis gzaze.
avtorTa madloba
axali saxelmZRvaneloebis Seqmna da maTi skolaSi danergva mimdinare ga-naTlebis reformis mniSvnelovani nawilia. Cven Tavidanve CaverTeT am saqmeSi _ VII klasis pirveli saxelmZRvanelo 2005-2006 wels SemuSavebuli erovnuli saswavlo gegmis mixedviT Seiqmna. man warmatebiT gaiara pilotireba, erovnu-li saswavlo gegmebis da Sefasebis centris eqspertebisa da maswavleblebis rCevebis gaTvaliswinebiT daixvewa. es TanamSromloba Semdegi konkursebis drosac gagrZelda.
madlobas movaxsenebT maT nayofieri TanamSromlobisTvis. maTi araerTi mosazreba asaxulia am axal saxelmZRvaneloSic, romelic 2011-2016 wlebis erovnuli saswavlo gegmis mixedviTaa Sedgenili. gaTvaliswinebulia yvela is cvlileba, rac erovnul saswavlo gegmaSi ganxorcielda.
specialuri niSnakebiTaa gamoyofili is sakiTxebi, romlebic mxolod maTematikis gaZlierebuli swavlebis programas Seesabameba.
gaviTvaliswineT maswavleblebisa da moswavleebis Txovna da wigni or nawilad davyaviT.
madlobas movaxsenebT yovel pedagogsa da moswavles, romlebic TavianT arCevans am wignze SeaCereben. vimedovnebT, es wigni gaamarTlebs maT imeds.
guram gogiSvili, Teimuraz vefxvaZe,ia mebonia, lamara qurCiSvili
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rCevebi wigniT sargeblobisTvis
gameoreba. naturaluri ricxvebi. geometriuli figurebi.1
1.1. naturaluri ricxvebi. naturaluri ricxvebis Caweris sxvadasxva forma1.2. naturaluri ricxvebis gayofa. gayofadobis niSnebi1.3. martivi da Sedgenili ricxvebi. udidesi saerTo gamyofi da umciresi saerTo jeradi1.4. simravle. qvesimravle1.5. simravleTa gaerTianeba da TanakveTa. venis diagramebi1.6. geometriuli figurebi1.7. kuTxe. kuTxeTa urTierTganlageba1.8. monakveTisa da kuTxis gazomva1.9. kuTxeTa klasifikacia. or wrfes Soris kuTxe
racionaluri ricxvebi. monacemTa analizi da statistika. zo gierTi geometriuli figura2
2.1. mTeli ricxvebi2.2. mTeli ricxvis moduli (absoluturi mniSvneloba). mTeli ricxvebis Sedareba2.3. mTeli ricxvebis Sekreba da gamokleba2.4. mTeli ricxvebis gamravleba da gayofa2.5. dadebiTi racionaluri ricxvebi2.6. racionaluri ricxvebi2.7. racionaluri ricxvis naturalurmaGCvenebliani xarisxi2.8. monacemTa analizi da statistika2.9. texili. mravalkuTxedi2.10. oTxkuTxedebi
cvladiani gamosaxuleba. wrfivi erTucnobiani gantoleba3
3.1. cvladiani gamosaxuleba. cvladiani gamosaxulebis mniSvnelobis povna3.2. naturalurmaCvenebliani xarisxis Tvisebebi3.3. erTwevri. mravalwevri. moqmedebebi mravalwevrebze3.4. Semoklebuli gamravlebis formulebi3.5. mravalwevris mamravlebad daSla3.6. gantoleba. gantolebis fesvi
sarCevi
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3.7. wrfivi erTucnobiani gantoleba amocanebi maTematikuri SejibrebebisTvis
koordinatebi. geometriuli gardaqmna4
4.1. koordinatebi wrfeze. ricxviTi Sualedebi4.2. wertilis koordinatebi sibrtyeze4.3. sakoordinato meoTxedebi4.4. amocanebis amoxsna gantolebis saSualebiT4.5. amocanebis amoxsnis sxvadasxva xerxi4.6. RerZuli simetria4.7. paraleluri gadatana
sidideebs Soris damokidebulebebi.samkuTxedebis tolobis niSnebi5
5.1. Sefardeba5.2. ricxvis dayofa proporciul nawilebad5.3. proporcia5.4. pirdapirproporciuli sidideebi5.5. samkuTxedi. samkuTxedis saxeebi kuTxeebis mixedviT5.6. samkuTxedis elementebi5.7. mimdevrobebi5.8. samkuTxedebis tolobis niSnebi5.9. procenti5.10. vagrZelebT procentze amocanebis ganxilvas amocanebi maTematikuri SejibrebisTvis
statistikis elementebi. amocanebi agebaze6
6.1. monacemTa Semajamebeli ricxviTi maxasiaTeblebi6.2. samkuTxedis saxeebi gverdebis mixedviT. tolferda samkuTxedi6.3. paraleluri wrfeebi6.4. samkuTxedis kuTxeebis jami6.5. marTkuTxa samkuTxedi6.6. monakveTis SuamarTobis Tviseba, wrewiris qorda da mxebi6.7. amocanebi agebaze amocanebi maTematikuri SejibrebebisTvis
maTematikuri terminebis leqsikoni sagnobrivi saZiebeli maTematikuri niSnebi berZnuli anbani laTinuri anbani pasuxebi
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am niSnakiT drodadro SegaxsenebT, rom yvela amocana unda gadaita noT rveulSi da DSemdeg SeasruloT. wignSi Canawerebi ar unda gakeTdes!
rv
eul
Si!
gaufrTxildiT wigns!
masSi Tqvenma Canawerebma wigni SeiZleba sxvebisTvis gamousadegari gaxados!
rogor visargebloT wigniT
wigniT sargebloba rom gagiadvildeT, ramdenime rCevas mogcemT.
saxelmZRvanelos TiToeuli Tavi masSi mocemuli ZiriTadi sakiTxebis CamonaTvaliT iwyeba.
TiToeuli Tavi paragrafebisgan Sedgeba. paragrafebi, rogorc wesi, ornawiliania. pirveli nawili Teoriul masalas eTmoba. misi gadmocema praqtikuli magaliTebis ganxilviT mimdinareobs. am nawils Seswavlili masalis SejamebiT vasrulebT.
es niSnaki migvaniSnebs paragrafis meore nawilis dawyebas _ `sxva-da sxva~. aq istoriuli sakiTxebi da sxva damxmare masalaa mo-cemuli.
s
es niSnaki amocanebis CamonaTvals win uZRvis.a
SemogTavazebT jgufuri muSaobis proeqtebs _ es TanaklaselebT-an erToblivad muSaobis gamocdilebas Seg ma tebT. aq SeiZleba dag-WirdeT informaciis damo u ki de be li moZiebac da misi damuSavebac.
j
zogierTi paragrafis Teoriul nawilsa da amocanebs mosdevs sa-varjiSoebi, romliTac Seswavlil (maT Soris _ gasul wlebSic) sakiTxebs gaimeorebT. am nawilis dawyebas gvauwyebs es sagangebo niSnaki.
g
teqstis ukeT gagebis mizniT saWirod CavTvaleT zo gi er Ti si-tyvis warmomavlobis, misi msgavsi mniSv ne lo be bi sa an sinonimebis miTiTeba. aRniSnuli miTiTebebi Cvens leq si konSia mocemuli. am saTauriT gamoyofil Ca na werebs Tqven xSi rad SexvdebiT. igi Tqvens sityvaTa ma rag sac gaamdidrebs da ufro gasagebs gaxdis wignis Si na arss.
Cveni
leqsikoni
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gameoreba. naturaluri ricxvebi. geometriuli figurebi.
1.1 naturaluri ricxvebi.naturaluri ricxvebis Caweris sxvadasxva forma
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am TavSi gaiRrmavebT Tqvens codnas naturaluri ricxvebisa da geometri uli figurebis Sesaxeb. ZiriTadi sakiTxebia:
• gayofadobis niSnebi• naSTiani gayofa• martivi da Sedgenili ricxvebi• simravle. qvesimravle. venis diagramebi• geometriuli figurebi
naturalur ricxvebs Tqven ukve icnobT, xSirad iyenebT maT. magaliTad, Tvlis dros. am wignis gverdebi naturaluri ricxvebiTaa gadanomrili.
naturaluri ricxvebis Casawerad aT cifrs viyenebT: 0; 1; 2; 3; 4; 5; 6; 7; 8; 9. es cifrebi sakmarisia nebismieri naturaluri ricxvis aTobiT sistemaSi Casawerad.
Tu naturaluri ricxvi erTi cifriTaa Cawerili, mas erTniSna ricxvi vuwodeT. 49292 xuTniSna ricxvia, igi xuTi cifriTaa Cawerili. am CanawerSi pirveli Tanrigis (erTeulis Tanrigis) cifria 2; aseulis Tanrigis cifric 2-ia, igi warmoadgens or aseuls _ 200 erTeuls. erTi da igive cifri CanawerSi sxvadasxva mniSvnelobiT gamoiyeneba _ es mniSvneloba cifris adgilze (poziciaze) aris damokidebuli. amitom ricxvebis Caweris am sistemas aTobiTi sistemis garda poziciur sistemasac uwodeben.
arapoziciuri sistemis magaliTia romauli sistema.gaixseneT romaul sistemaSi 1-dan 40-mde ricxvebis Caweris wesebi.40-isa da masze meti ricxvebis Casawerad gamoiyeneba sxva niSnebic:
50 - L, 100 - C, 500 - D, 1000 - M.
magaliTebi: 1. VIII=5+1+1+1=8
2. XXIV=10+10+4=24
3. LXXVI=50+10+10+5+1=76.
• gaixseneT aTobiT sistemaSi naturaluri ricxvis CanawerSi Tanrigebis saxelwodebebi da warmoadgineT gaSlili formiT 3 785 698.
erTeulebis, aTaseulebis, milionebis klasebis da saTanado Tanrigebis gaxsenebaSi es Canaweric dagexmarebaT:
3 7 8 5 6 9 8
milionebis klasi
aTaseulebis klasi
erTeulebis klasi
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magaliTi 1 magaliTi 2
• gaixseneT damrgvalebis wesi da daamrgvaleT es ricxvi aTi aTaseulebamde, aseulebamde.
magaliTad, am ricxvis umaRles Tanrigamde damrgvalebiT miiReba:
4 000 000.
• daasaxeleT umciresi naturaluri ricxvi.• SeiZleba Tu ara udidesi naturaluri ricxvis dasaxeleba?
axla e. w. `giganturi~ ricxvebis saxelwodebebi gavixsenoT:
1 000 000 = 1 milions 1 000 milioni = 1 miliards 1 000 miliardi = 1 trilions 1 000 trilioni = 1 kvadrilions 1 000 kvadrilioni = 1 kvintilions
• warmoadgineT TiToeuli am ricxvis (miliardis, trilionis, kvadrilionis, kvintilionis) aTobiTi Canawerebi.
1-iTa da mis marjvniv miwerili nulebiT Sedgenili ricxvebi specialuri aR-niSvnis gamoyenebiT mokled Caiwereba. magaliTad, milioni (1 000 000) SeiZleba war mo vadginoT eqvsi Tanamamravlis namravlis saxiT, romelTagan TiToeuli aris 10;
1 000 000 = 10⋅10⋅10⋅10⋅10⋅10. es ricxvi ase Caiwereba: 106. 106 xarisxia, misi fuZe aris 10, 6 _ xarisxis maCvenebelia.analogiurad, nebismieri ricxvis Tavis Tavze ramdenjerme namravli xarisxis
saxiT Caiwereba:
5⋅5⋅5⋅5=54, vkiTxulobT: 5 meoTxe xarisxSi.SevTanxmdeT: raime ricxvis pirveli xarisxi TviT es ricxvia: a1=a.
vipovoT 24.
24=2⋅2⋅2⋅2=16.
4-jer
vipovoT 37.
37=3⋅3⋅3⋅3⋅3⋅3⋅3=2 187.
7-jer
* *
SevajamoT: saswavlo weli naturaluri ricxvebiTa da am ricxvebis Caweris formebis gaxsenebiT daviwyeT; gavixseneT ricxvis xarisxi, magaliTad, 103=10⋅10⋅10, aq 103 _ xarisxia, 10 _ fuZe, 3 _ xarisxis maCvenebeli.
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a
SearCieT swori pasuxi
warmovidginoT sastumro, romlis oTaxebi naturaluri ricx ve bi Taa
gada nom ri li: 1, 2, 3, ...
maSasadame, warmovidginoT iseTi sastumro, romelSic, ro gorc am SemTx
vevaSi vit yviT, usasrulod bevri oTaxia. vTqvaT, yve la oTaxi `dakavebulia~.
saRamos sas tumros kidev erTi stu ma ri ew via. Tavis ufali oTaxi ar aris
- mimarTavs mas sas tum ros admi nis t ra to ri. saubarSi sastumros direqtori
erT ve ba da ga mo sa vals ad vilad poulobs: stumari nomer 1 oTaxidan gadaviy
va noT no mer 2Si, stumari nomer 2dan - nomer 3Si da a. S. axali stu mari
ki nomer 1 oTaxSi gan vaTavsoT...~. miax loebiT aseTi ma ga liTis ganxilviT
cdilobda di di germaneli maTematikosi da vid hil berti msmenelTaTvis natu
ra luri ricx ve bis Tvisebebis Tval sa Cinod axsnas.
* * *
ricxvebis Caweris wesi - Tvlis sistemebi - uZveles dros Ca mo ya lib
da. mi ax lo ebiT 3 000 wlis win egviptelebi, faqtobrivad, aTo biT sistemas
iyenebdnen. ma ga liTad, 321s isini ase Cawerdnen:
|
aq | erTeulia, - aTeuli, -aseuli. romauli sistema miaxloebiT
Zveli welT aRricxviT 500 wels Se iqm na.
zogierT qveyanaSi ricxvebi asoebiT Caiwereboda. saqarTveloSi am wess
XVIII sau kunemdec ki misdevdnen. Zvel berZnul da qarTul anbanebSi pirveli
cxra aso ga mosaxavs erTeulebs: 1, 2, 3, 4, 5, 6, 8, 9; asoebis nawili
- aTeulebs. aris aseulebis da aTaseulebis aRmniSvneli asoebic. magali
Tad, a=1, b=2, c=3, d=4; a=1, b=2, i=10, r=100, C=1000 (Zvel qarTul
anbanSi 38 aso niSania, Tanamedrove anbanSi ar ixmareba xuTi aso niSani:
H, Y, U, Q, H. isini, Sesabamisad, gamosaxavs ricxvebs: 8, 60, 400, 7 000,
40 000).
romaul sistemaSi, Tu raime mocemuli cifris zemoT `-~ niSnakia gamo
saxuli, maSin am cif riT 1 000jer gazrdili ricxvi Caiwereba.
magaliTad,
V=5·1 000=5 000, X=10·1 000=10 000.
1 nuli 1) naturaluri ricxvia
2) ar aris naturaluri ricxvi 3) umciresi naturaluri ricxvia
4) 1-is momdevno naturaluri ricxvia.
s
10
2 5-is momdevno naturaluri ricxvia 1) 7 2) 4 3) 6 4) 10.
3 CawereT gaSlili formiT 360 463.
1) 30 000+6 000+400+60+3 2) 36 000+4 000+60+3
3) 300 000+60 000+400+60+3 4) 300 000+6 000+400+60+3.
4 umciresi naturaluri ricxvia 1) erTi 2) nuli 3) ori 4) sami.
5 udidesi naturaluri ricxvi, romlis aTobiTi Canaweri sami cifriT warmoidgineba (anu samniSnaa), aris
1) 1 000 2) 999 3) 100 4) 899.
6 im naturaluri ricxvebidan, romelTa aTobiTi Canaweri oTxi ci-friT warmoidgineba (anu oTxniSnaa), udidesi ricxvi gaSlili saxiT ase Caiwereba:
1) 9·1 000+9·100+9·10+9 2) 1 000+100+10+9
3) 9·100+9·10+9 4) 9·10 000+9·1 000+9.
7 udidesi naturaluri ricxvi 1) trilionia 2) miliardia
3) ar arsebobs 4) milionia.
8 96-gverdiani wignis gverdebis gadasanomrad (gverdebis gadanomrva iwyeba 1-dan) saWiroa
1) 96 cifri 2) 192 cifri 3) 174 cifri 4) 183 cifri.
9 nuli 1) oTxniSna ricxvia 2) erTniSna ricxvia 3) xuTniSna ricxvia 4) orniSna ricxvia.
10 XXXIV=
1) 26 2) 36 3) 34 4) 35.
11 sityva `miliardi~ xmarebaSi safrangeT-prusiis omis dam Tav re bisas - 1871 wels Semovida. romel saukuneSi Semovida xmarebaSi sit yva `mil-iardi~?
1) XVIII 2) XIX 3) XXI 4) XVII.
12 CamoTvlili Sefasebebidan SearCieT 927+328 jamis saukeTeso Se-faseba:
1) aTas samasze metia 2) aTas samasze naklebia 3) 1400-ze metia 4) 1400-ze naklebia.
13 romel saukuneSi gaimarjva didgoris brZolaSi qar Tul ma laS qar-ma?
1) XI saukuneSi 2) XII saukuneSi 3) X saukuneSi 4) IX saukuneSi.
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14 CVL=
1) 100+(50-5) 2) 100+(5+50) 3) 100-(5+50) 4) (100+5)-50.
15 42 368 aTobiT sistemaSi SeiZleba ase CavweroT:
1) 4·103+2·102+3·10+6+8 2) 4·104+2·103+3·102+6·10+8 3) 10000·44+1000·23+100·32+10·6+8 4) 42·103+36·102+8.
16 7·104+5·102+3·10= 1) 753 2) 7053 3) 7530 4) 70530.
17 131=
1) 169 2) 1 3) 0 4) 13.
18 kubis gverdis sigrZe 5 sm-ia. 1 sm-iani wibos mqone ramdeni kubisgan Sedgeba es kubi?
1) 5 2) 25
3) 125 4) 625.
19 12006=
1) 2005 2) 0 3) 1 4) 5.
20 7·7·7·5·5=
1) 75 2) 55 3) 7·54 4) 73·52.
21 ipoveT naturaluri ricxvi, romlis kvadrati TviT es ricxvia. 1) 1 2) 2 3) 0 4) 3.
22 CawereT 81 xarisxis saxiT, romlis fuZe aris 3.
1) 33 2) 34 3) 35 4) 53.
23 mocemulia naturaluri ricxvi: 81 452. • ra cifria am ricxvis aTeulis TanrigSi? • ra cifria am ricxvis aseulis TanrigSi? • ra cifria am ricxvis aTaseulis TanrigSi?
• sul ramdeni erTeulia am ricxvSi? • sul ramdeni aTaseulia am ricxvSi? • sul ramdeni aseulia am ricxvSi? • sul ramdeni aTeulia am ricxvSi?
24 dawereT naturaluri ricxvi, romelic Seicavs: a) 73 aTeulsa da 5 erTeuls; b) 41 aseulsa da 15 aTeuls; g) 100 aseulsa da 100 aTeuls.
25 moiZieT msoflios amJamindeli mosax le obis odenoba da CawereT igi sityvebiTa da cifrebiT.
amoxseniT amocanebi
12
26 dawereT xuTas meTxuTmete naturaluri ricxvi. warmoadgineT igi gaSlili formiT. ramdeni erTeuli aklia am ricxvs 10 000-mde?
27 stadionze gamarTul koncerts 27 148 mayu-rebeli da es w ro. daamrgvaleT es ricxvi:
a) aTeulebamde; b) aseulebamde; g) aTaseulebamde.
28 SeadgineT udidesi SesaZlo ricxvi, romlis cifrebi ar meordeba da es cifrebia:
a) 7, 5, 9 da 3; b) 5, 6, 8, 4 da 0; g) 2, 4, 7, 5 da 9.
29 SeadgineT umciresi SesaZlo naturaluri ricxvi, romlis cifrebi ar meordeba da es cifrebia:
a) 6, 1, 3, 5 da 4; b) 1, 5, 7 da 0; g) 9, 3, 6, 0 da 8.
30 msoflios okeaneTagan udidesia wynari okeane. misi zedapiris far-Tobi cnobarSi asea miTiTebuli: 178 684 (aTasi km2). gamosaxeT es sidide kvadratul kilometrebSi da waikiTxeT miRebuli ricxvi.
31 50 wels miRweuli adamianis gulis ce maTa saerTo ricxvi miaxloe-biT ori miliard xu Ta si milionia. CawereT es ricxvi aTobiT siste-maSi.
32 adamianis Zilis saSualo xangrZlivoba yovel xuT weliwadSi mi-axloebiT cameti aTas samas oTxmocdasami saaTia. CawereT es ricxvi aTobiT sistemaSi.
33 dawereT udidesi eqvsniSna ricxvi. ramdeni aTeulia am ricxvSi? ramdeni aseulia am ricxvSi? ra ricxvi unda mivumatoT am ricxvs, rom miviRoT umciresi SvidniSna ricxvi?
34 56212 ricxvi kalkulatoris daxmarebiT gamoTvaleT:
5 6 2 1 × = ?
miRebuli ricxvi daamrgvaleT: a) aTaseulebamde; b) milionebamde; g) umaRles Tanrigamde.
35 1958 wels aRmoaCines didi istoriuli mniSvne l o bis mqone eg vipturi papirusi. is mecnierebma Zveli wel TaRricxvis 1650 wliT daaTariRes. am or TariRs Soris miaxloebiT ramdeni sa uku nea?
36 vTqvaT, n aris 1-ze meti naturaluri ricxvi. dawereT:
a) n-is wina da momdevno naturaluri ricxvebi; b) n+2-is wina da momdevno naturaluri ricxvebi.
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37 gadaiwereT da SeavseT winadadebebSi gamotovebu-li ricxvebi. gamoiyeneT ara buli da romauli numera-cia. mefe daviT . . . (aRmaSenebeli) daibada . . . wels. is . . . wlisa gamefda. . . . saukuneSi saqa r T ve los aRorZineba daviTis saxels ukavSirdeba. . . . wels did gor Si didi gamarjvebis Semoqmedi mefe daviTia.
38 CawereT aTobiT da romaul sistemebSi: a) Tqveni dabadebis weli; b) skolaSi Tqveni Sesvlis weli.
39 Ca wereT aTobiT da romaul sistemebSi saqarTvelos damo uki deb-lobis ga mo cxa debisa da aRdgenis wlebi.
40 CawereT aTobiT da romaul sistemebSi weli, roca adamianma pirve-lad dadga fexi mTvareze.
41 aTobiT sistemaSi Cawerili ricxvi warmoadgineT gaSlili formiT:
a) 357; b) 230203; 42 samniSna ricxvis aseulebisa da aTeulebis cifrebi (anu am ricxvis
aTobiT Ca na werSi aseulisa da aTeulis Tanrigis cifrebi), Sesabamisad, aris 3 da 2. ram deniT Semcirdeba es ricxvi, Tu misi aseulebisa da aTeulebis cifrebs gadavanacvlebT?
43 ramdeni iseTi oTxniSna ricxvi arsebobs, romelic iwyeba 1-iT da iyofa 10-ze?
44 amowerilia TanamimdevrobiT Semdegi naturaluri ricxvebi: 1, 4, 9, 16, 25, 36.
ra kanonzomierebas amCnevT am ganlagebaSi? miuwereT ricxvebs imave ka non zo mi ere biT kidev xuTi naturaluri ricxvi.
45 amowerilia TanamimdevrobiT Semdegi naturaluri ricxvebi: 1, 8, 27, 64, 125, 216.
dasviT wina amocanis analogiurad kiTxvebi da upasuxeT maT.
46 ipoveT naturaluri ricxvi, romlis kvadrati aris: a) 9; b) 64; g) 144; d) 625.
47 212 warmoadgineT xarisxis saxiT, romlis fuZea:
a) 4; b) 8; g) 64.
48 ipoveT naturaluri ricxvi, romlis: a) kvadrati aris 81; b) meoTxe xarisxi aris 81.
49 36 warmoadgineT xarisxis saxiT, romlis fuZea: a) 9; b) 27.
14
50 babuam zurabs, roca erTi wlis gaxda, 1 wigni aCuqa da yovel Semdeg dabadebis dReze saCuqrad aZlevda orjer met wigns, vidre - winaze.
a) ramdeni wigni aCuqa zurabs babuam, roca igi 6 wlis gaxda?
b) babuas naCuqari ramdeni wigni eqneboda 6 wlis zurabs?
51 kubis wibo aris a. dawereT: a) kubis zedapiris farTobis (eqvsive waxnagis farTobebis jami) ga-
mosaTvleli gamosaxuleba;
b) kubis moculobis gamosaTvleli gamosaxuleba;
g) ramdenjer gaizrdeba kubis moculoba, Tu mis wibos gavzrdiT 3-jer?
d) ramdenjer gaizrdeba kubis zedapiris farTobi, Tu mis yovel wibos gav zr diT 3-jer?
Sesavsebia cxrili. mas pirobiTad SeiZleba `bolo cifris cxrili~ vuwodoT.
boloxarisxis cifrimaCvenebeli
0 1 2 3 4 5 6 7 8 9
1 8
2 9
3 0 4 9
45 2
6
pirvel svetSia ricxvebis xarisxis maCveneblebi: 1, 2, 3, 4, 5, 6; pirvel stri qon Si - ricxvebis bolo cifrebi. magaliTad, Tu ricxvis erTeulebis cifri aris 4, maSin am ricxvis mesame xarisxis bolo ci-fric 4-ia. Sesabamisi svetisa da Sesabamisi stri qo nis (cxrilSi feri-Taa moniSnuli) gadakveTaze vwerT 4-s.
• SeavseT `bolo cifris~ cxrili,
• ra kanonzomierebis aRmoCena SeiZleba?
• cxrils daumateT meSvide da merve striqonebi da SeamowmeT, Tu marTldeba Tqven mier aRmoCenili kanonzomiereba am SemTxvevaSic.
• ra aris 416-is bolo cifri? 9131-is bolo cifri?
rv
eul
Si!
j
15
1. 2 naturaluri ricxvebis gayofa. gayofadobis niSnebi
1) vTqvaT, 80 wigns Tanabrad vanawilebT yuTebSi da TiToeul yuTSi 16 wigni Tavsdeba. saWiro yuTebis raodenobas gayofiT vpoulobT:
80:16=5.
TiToeul yuTSi 16 wignia, 5 yuTSi iqneba 16⋅5=80 wigni.gvaqvs ricxvi 80 da ricxvi 16, moiZebna ricxvi _ 5, iseTi, rom 80=16⋅5;
vityviT _ 80 iyofa 16-ze, 80:16=5, 80 gasayofia, 16 _ gamyofi, 5 _ ganayofi.
gayofis zogierTi wesi Semdegi cxriliT warmovadginoT:
nebismieri ricxvis 1-ze gayofiTTviT es ricxvi miiReba.
9:1=9 a:1=a
Tu naturalur ricxvs TavisTavze gavyofT, miviRebT 1-s.
9:9=1 a:a=1
nulis naturalur ricxvzegayofiT nuli miiReba.
0:5=0 0:a=0
nulze gayofa ar SeiZleba.
axla kidev ramdenime Tviseba warmovadginoT.
1. Tu erTi ricxvi iyofa meo-reze da meore iyofa mesameze, ma-Sin pirveli ricxvi iyofa mesameze.
aseTi msjelobiT ufro rTuli SemTxvevebis ganxilvasac SeZlebT.
2. Tu ori ricxvidan TiToeuli iyofa erTsa da imave ricxvze, ma-Sin mocemuli ori ricxvis jamica da sxvaobac iyofa am ricxvze.
• ricxvebis moqmedebebis ra Tvi-sebebi gamoviyeneT aRniSnuli tolo-bebis dawerisas?
3. Tu mocemuli ori ricx-vidan erT-erTi iyofa mesame ricxvze, meore ar iyo fa am mesame ricxvze, maSin moce-muli ori ricxvis jamica da sxvaobac ar iyofa am me same ricxvze.
magaliTad, 555 iyofa 111-ze: 555=5·111; 111 iyofa 37-ze: 111=3·37. vRebulobT:
555=5·111=5·(3·37)=(5·3)·37=15·37,
anu 555 iyofa 37-ze.
magaliTad, 120 iyofa 4-ze: 120=30·4; 36-ic iyofa 4-ze: 36=9·4. vRebulobT:
120+36=30·4+9·4=(30+9)·4=39·4,
anu 120+36 jami (156) iyofa 4-ze.
magaliTad, 144 iyofa 36-ze, 48 ar iyofa 36-ze. martivad SeamowmebT, rom 144+48 (anu 192) ar iyofa 36-ze (es Sede-gi SeiZleba aseTi msjelobiTac mivi-RoT: 144+48 rom iyofodes 36-ze, maSin, me-2 Tvisebis Tanaxmad, 36-ze gaiyofoda (144+48)-144=48, rac ar aris swori. am-rigad, 144+48 ar iyofa 36-ze).
16
magaliTi 1
2) vTqvaT, axla 82 wigni gasanawilebelia yuTebSi da TiToeul yuTSi SeiZleba 16 wignis moTavseba. sul mcire ramdeni yuTi dagvWirdeba?
amocanis amosaxsnelad 82 unda gavyoT 16-ze. gayofisas naSTi miiReba:
82=16⋅5+2.
82-is 16-ze gayofisas miviReT: ganayofi _ 5 (zogjer mas arasrul ganayofsac uwodeben), naSTi _ 2. naSTi naklebia gamyofze. am naSTis gamo, amocanaSi dasmul kiTxvaze pasuxis gasacemad, cxadia, ganayofi erTiT unda gavzardoT _ sul mcire 6 yuTi iqneba saWiro (5 yuTs vavsebT, me-6-Si ki ganTavsdeba 2 wigni).
axla weriTi gayofis wesi gavixsenoT:
1479 gavyoT 7-ze.
1 4 7 9 7
1 4 211 da naSTi 2
7
7
9
7
2 _ naSTi
maSasadame, 1479=7⋅211+2
211 ganayofia, 2 _ naSTi;
naSTi gamyofze naklebia _ 2<7.
3) axla gayofadobis niSnebi CamovayaliboT _ 2-ze, 5-ze, 10-ze, 3-ze da 9-ze gayofadobis niSnebi.
• Tu ricxvi 0-iT bolovde-ba (vgulisxmobT, rom ricxvis aTobiTi Canaweri bolovdeba 0-iT), maSin es ricxvi 10-ze iyofa.
• Tu ricxvi 0-iT an 5-iT bolovdeba, maSin es ricxvi iyofa 5-ze.
magaliTad, 3570 iyofa 10-ze, radgan 3570=357·10.
3571 ar iyofa 10-ze, radgan 3571=3570+1 da 3570 iyofa 10-ze, 1 ki ar iyofa 10-ze.
magaliTad, 2400 iyofa 5-ze, radgan 2400=240·10 da 10 iyofa 5-ze. 2405-c iyofa 5-ze, radgan 2405=2400+5.
2407 ar iyofa 5-ze, radgan 2407=2405+2 da 2 ar iyofa 5-ze.
• Tu ricxvi nuliT ar bolovdeba, maSin es ricxvi ar iyofa 10-ze.
•Tu ricxvi ar bolovdeba arc 0-iT, arc 5-iT, maSin es ricxvi ar iyofa 5-ze.
• 0, 2, 4, 6, 8 cifrebs luwi cifrebi ewodeba; 1, 3, 5, 7, 9 cifrebs - kenti cifrebi.
Tu naturaluri ricxvi iyofa 2-ze, am ricxvs luwi ricxvi ewodeba. am-rigad, nebismieri luwi ricxvi 2k saxisaa, k naturaluri ricxvia.
a)
b)
g)
17
• Tu ricxvis cifrebis jami iyofa 3-ze, ma-Sin es ricxvi iyofa 3-ze. wi-naaRmdeg SemTxvevaSi, anu Tu cifrebis jami ar iyofa 3-ze, maSin ricxvi ar iyofa 3-ze.
• Tu ricxvi luwi ci-friT bolovdeba, maSin is luwia (iyofa 2-ze).
• Tu ricxvi kenti cif-riT bolovdeba, maSin is kentia (ar iyofa 2-ze).
magaliTad, 4 luwi cifria, masTan erTad luwia 324 ricxvic.
magaliTad, 5 kenti cifria, kentia 325 ricxvic.
magaliTad, 3123 iyofa 3-ze, radgan 3+1+2+3=9 da 9 iyofa 3-ze.
3124 ar iyofa 3-ze (3+1+2+4=10 da 10 ar iyofa 3-ze).
4275 ricxvis cifrebis jami (igulisxmeba saTanado TanrigebSi er-Teulebis odenobaTa jami — 4-is, 2-is, 7-isa da 5-is jami) - 18 iyofa 9-ze da Tavad 4275 ricxvic iyofa 9-ze - 4275=475·9.
es faqti SeiZleba ase davasabuToT:
4275=4·1000+2·100+7·10+5=4(999+1)+2(99+1)+7(9+1)+5=
=(4·999+2·99+7·9)+(4+2+7+5).
frCxilebiT gamoyofili jamebi iyofa 9-ze, maSasadame, 4275 iyo-fa 9-ze.
4276 ar iyofa 9-ze, radgan 4276=4275+1, pirveli Sesakrebi iyofa 9-ze, meore - ar iyofa.
Tu ricxvi ar iyofa 2-ze, am ricxvs kenti ricxvi ewodeba.
• sazogadod, Tu raime ricxvis cifrebis jami iyofa 9-ze, maSin es ricxvi iyofa 9-ze.
• Tu mocemuli ricxvis cifrebis jami ar iyo-fa 9-ze, maSin arc mocemuli ricxvi iyofa 9-ze.
• wina SemTxvevis analogiurad, warmoadgineT 3123 gaSlili formiT da SeamowmeT misTvis Camoyalibebuli winadadebis WeSmaritoba.
SevajamoT: gavixseneT naturaluri ricxvebis gayofis Tvisebebi. Camoya-libebulia 2-ze, 5-ze, 10-ze, 3-ze da 9-ze gayofadobis niSnebi _ ra SemTxvevaSi iyofa mocemuli ricxvi 2-ze, 5-ze, 10-ze, 3-ze an 9-ze da ra SemTxvevaSi _ ar iyofa.
d)
e)
18
1 Tu a, b da c naturaluri ricxvebia da a=b⋅c, maSin 1) a ar iyofa c-ze 2) a ar iyofa b-ze 3) a iyofa b-ze.
2 Tu a nebismieri naturaluri ricxvia, maSin a:a=
1) 2 2) 1 3) 3 4) 0.
3 radgan 36 iyofa 3-ze, 17 ar iyofa 3-ze, amitom 36+17
1) iyofa 3-ze 2) ar iyofa 3-ze 3) iyofa 17-ze 4) iyofa 36-ze.
4 Tu raime a ricxvi iyofa 3-ze, b ricxvic iyofa 3-ze, maSin 1) a+b iyofa 3-ze 2) a+b ar iyofa 3-ze 3) a-b ar iyofa 3-ze 4) a⋅b ar iyofa 3-ze.
5 vTqvaT, n nebismieri naturaluri ricxvia, maSin yoveli luwi naturaluri ricxvi (ricxvi, romelic 2-ze iyofa) SeiZleba ase CavweroT
1) 2n+1 2) 2n-1 3) 2n+3 4) 2n.
6 vwerT yvela SesaZlebel oTxniSna ricxvs, romlis cifrebi SeiZleba iyos mxolod 0, 1, 2 an 3 (cifrebi SeiZleba gameordes). ramdeni aseTi ricxvis dawera SeiZleba?
1) 256 2) 192 3) 81 4) 36.
7 Tu naturaluri x ricxvis 7-ze gayofiT miRebuli naSTia 2, maSin (x+3)-is 7-ze gayofisas miiReba naSTi:
1) 3 2) 1 3) 5 4) 4.
8 ipoveT umciresi naturaluri ricxvi, romelic 3-ze, 5-ze da 7-ze gayo fisas naSTSi gvaZlevs 1-s da metia TiToeul am ricxvze.
1) 105 2) 104 3) 103 4) 106.
9 mocemul ricxvebSi SearCieT ricxvi, romelic iyofa 5-ze.
1) 5723 2) 2375 3) 3752 4) 7352.
10 2, 7, 5 da 3 cifrebiT, gameorebis gareSe, SeiZleba SevadginoT 24 oTxniSna ricxvi. maTSi luwi ricxvebis odenobaa
1) 6 2) 8 3) 12 4) 3.
11 nebismieri luwi naturaluri ricxvis aTobiTi Canaweri bolovdeba 1) 2-iT 2) 2-iT an 4-iT 3) 2-iT, 4-iT, 6-iT an 8-iT 4) raime luwi cifriT.
a
SearCieT swori pasuxi
19
12 yoveli ricxvi, romelic iyofa 15-ze, gaiyofa 1) 2-zec 2) 3-zec da 5-zec
3) 5-zec da 10-zec 4) 30-zec.
13 yoveli kenti naturaluri ricxvi SeiZleba ase CavweroT:
1) 3k, k naturaluri ricxvia 2) 2k+1, k naturaluri ricxvia 3) 2k-1, k naturaluri ricxvia 4) 4k-1, k naturaluri ricxvia.
14 Semdegi ricxvebidan romeli iyofa 3-ze?
1) 72426 2) 12127 3) 82528 4) 12118.
15 dasaxelebuli ricxvebidan 9-ze iyofa 1) 9127 2) 9128 3) 9125 4) 9126.
16 2n+1 kenti ricxvis momdevno kenti ricxvia 1) 3n+1 2) 2n+2 3) 2(2n+1) 4) 2n+3.
17 n da a naturaluri ricxvebia. cnobilia, rom 2n+a aris luwi ricx-vi. maSin aucileblad a aris
1) luwi 2) kenti 3) 2-is toli 4) orniSna luwi ricxvi.
18 nikas imdenive moneta aqvs, ramdenic vaxtangs. nikas mxolod 10-TeT-rianebi aqvs, vaxtangs - mxolod 20-TeTrianebi. orTaves erTad am mon-etebiT 1 lari da 50 TeTri ugrovdeba. ramdeni moneta aqvs TiToeuls?
19 mzias imdenive 20-TeTriani moneta aqvs, ramdenic - 5-TeTriani. am mo ne tebiT mas 2 lari ugrovdeba. ramdeni 20-TeTriani moneta aqvs mzias?
20 lelas ori moneta aqvs. misi Tanxa 15 TeTria. erT-erTi moneta ar aris 5-TeTriani. ra Rirebulebisaa es moneta?
21 eqvsi momdevno luwi ricxvis jamia 1326. ipoveT am ricxvebidan umci-resi.
22 dawereT umciresi 1-ze meti naturaluri ricxvi, romelic: a) iyofa 13-zec da 4-zec;
b) rogorc 13-ze, ise 4-ze gayofisas naSTSi iZleva 1-s.
23 sawyobSi Semotanili yuTebis ode-noba ar aRema teba 50-s. yuTebs Tu sam-sa mad davawyobT, erTi yuTi dagvrCeba; Tu - eqvs-eqvsad - kvlav 1 yuTi dagv-r Ceba; Tu yuTebs Svid-Svidad davawy-obT, isev 1 yuTi dagvrCeba. ramdeni yuTia Semotanili sawyobSi?
amoxseniT amocanebi
20
24 Svidi moswavle damalobanas TamaSobs. isini wrewirze gan lag d nen da gadawyvites: pirveli is ixuWeba, romelic gaTvlisas 25-e aRmoCndeba. SeTanxmdnen: gaTvlas jaba iwyebs.
ja bam daiwyo gaTvla: 1, 2, 3, ... nikam ar daucada Tvlis dam Tav rebas da wamoiZaxa - me vixuWebi. rogor mixvda nika, rom igi 25-e aR moC n de-boda? ra nomeria nikas adgili?
25 ra naSTi miiReba nakiani wlis dReebis ricxvis 7-ze gayofisas?
26 iyo Tu ara 2011 wlis pirveli ianvari da 2010 wlis pirveli ianvari kviris erTi da igive dReebi?
27 kalendris gamouyeneblad daadgineT: ra dRe iyo 2011 wlis pirve-li ianvari. gaiTvaliswineT, rom 2010 wlis 1 ianvari iyo paraskevi.
28 dasaxelebulia xarisxebi: 41, 42, 43, ..., 429. ramdenia maT Soris iseTi ricxvi, romlis 10-ze gayofisas miRebuli naSTia:
a) 4; b) 6; g) 8?
29 dasaxelebulia xarisxebi 71, 72, 73, ..., 745. ramdenia maT Soris iseTi ricxvi, romlis 10-ze gayofisas miRebuli naSTia:
a) 1; b) 9; g) 3; d) 7; e) 6?
30 vTqvaT, a da b (a>b) naturaluri ricxvebis 5-ze gayofisas erTi da igive naSTi miiReba. iyofa Tu ara a-b sxvaoba 5-ze? pasuxi daasabuTeT.
31 samma biWma savarjiSod CogburTis burTebis nawili Ta nabrad gainawila. danarCeni burTebi cxra go go naSic Tanabrad ganawilda. SeiZleba Tu ara, rom am biWebsa da gogonebs 20 burTi hqondes?
32 gigam maRaziaSi 18 erTnairi rveuli, ori erTnairi albomi, erTi kalami da ramdenime erTnairi fanqari iyida. fanqari 4 TeTri Rirs, kalami - 80 TeTri. gamyidvelma mas 5 lari 57 TeTri uangariSa. gigam
miTiTeba. gaiTvaliswineT, rom aranakian wlebSi dReebis ricx-via 365 da igi 7-ze gayofisas naSTSi iZleva 1-s.
miTiTeba. gamoTvlis gaiolebis mizniT kviris TiToeul dRes mianiWeT nomeri: orSabaTi - 1, samSabaTi - 2, oTxSabaTi - 3, xuT-SabaTi - 4, paraskevi - 5, SabaTi - 6, kvira - 7.
21
sTxova gamyidvels dae zus tebina Rirebu-leba, radgan mixvda, rom gamyidveli cde-boda.
rogor SeZlo gigam gamoTvlebis Cau-tareblad Secdomis aRmoCena?
33 dawereT eqvsi samniSna ricxvi, romelTagan TiToeulis cifrebia 1, 2 da 3 (arc erT ricxvSi cifrebi ar meordeba). iyofa Tu ara dawerili ricxvebi:
a) 3-ze; b) 6-ze; g) 9-ze?
34 daasaxeleT udidesi eqvsniSna ricxvi, romelic iyofa a) 2-ze; b) 3-ze; g) 5-ze; d) 9-ze; e) 10-ze.
35 ipoveT 1-dan 13-mde (CaTvliT) ricxvebis namravlis bolo ori cif-ri.
1 · 2 · 3 ·...·13=...36 ramdeni nuliT bolovdeba naturaluri ricxvebis namravli:
a) 1-dan 15-mde (CaTvliT)? b) 1-dan 30-mde (CaTvliT)?
37 123*4* CanawerSi *-ebis nacvlad CasviT iseTi cifrebi, rom mi Re bu-li eqvsniSna ricxvi gaiyos 45-ze.
38 mocemulia ricxvebi:
1·105+1; 8·107+1; 3·104+2·102+1; 2·10n+1. romeli maTgani iyofa 3-ze?
22
magaliTi 1
magaliTi 2
1. 3 martivi da Sedgenili ricxvebi.udidesi saerTo gamyofi da umciresi saerTo jeradi
1) gavixsenoT, rom mocemuli naturaluri ricxvis gamyofi aris natural-uri ricxvi, romelzec mocemuli naturaluri ricxvi iyofa.
magaliTad, 8-is gamyofebia: 1, 2, 4 da 8;
42-is gamyofebia: 1, 2, 3, 6, 7, 14, 21, 42.
19-is gamyofebia: 1, 19.
1 arc martivi ricxvia da arc Sedgenili.
1-dan 20-mde martivi ricxvebia: 2, 3, 5, 7, 11, 13, 17, 19.
1-dan 20-mde (CaTvliT) Sedgenili ricxvebia: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
amrigad,
yoveli naturaluri ricxvi aris an 1, an martivi, an Sedgenili ricxvi.
damtkicebulia, rom arsebobs uamravi martivi ricxvi.
yoveli Sedgenili ricxvi SeiZleba warmovadginoT martivi ricxvebis namrav-lis saxiT, anu davSaloT martiv mamravlebad. magaliTad, 28=2·2·7; 22=2·11; 81=3·3·3·3.
Tu xarisxis cnebasac gamoviyenebT, miviRebT
28=22·7, 81=34.
ganvixiloT martiv mamravlebad daSlis sxvadasxva wesi.
naturalur ricxvs, romelsac mxolod ori gamyofi aqvs, martivi ricxvi ewodeba.
naturalur ricxvs, romelsac orze meti gamyofi aqvs, Sedgenili ricxvi ewodeba.
davSaloT martiv mamravlebad 54. SeiZleba gamoviyenoT `mamravlebis xe~. vi-wyebT 54-is raime ori gamyofis namravlad warmodgeniT:
54
6 × 9 maSasadame, 54=2·33.
2 × 3 × 3 × 3
davSaloT martiv mamravlebad 56. axla viyenebT martivi gamyofebis svetad Camoweris wess:
56 2
28 2
14 2 maSasadame, 56=23·7.
7 7
1
23
magaliTi 3 magaliTi 4
a da b ricxvebis udidesi saerTo gamyofi am ricxvebis saerTo gamyofebs Soris udidesi ricxvia.
ricxvebs, romlebsac saerTo martivi gamyofi ara aqvs ur Ti erT mar ti vi (Tana-martivi) ricxvebi ewodeba.
vipovoT usg(180; 128).
180 2 128 2
90 2 64 2
45 3 32 2
15 3 16 2
5 5 8 2
1 4 2
2 2
1
vipovoT usg(56; 45).
56 2 45 3
28 2 15 3
14 2 5 5
7 7 1
1
2) 36-is gamyofebia: 1, 2, 3, 4, 6, 9, 12, 18, 36;
48-is gamyofebia: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
36-isa da 48-is saerTo gamyofebia: 1, 2, 3, 4, 6, 12.
saerTo gamyofebs Soris udidesi 12-ia. 12 aris 36-isa da 48-is udidesi saerTo gamyofi.
a da b ricxvebis udides saerTo gamyofs ase aRvniSnavT: usg(a; b).
maSasadame, usg(36; 48)=12.
ganvixiloT ori ricxvis udidesi saerTo gamyofis povnis magaliTebi.
180=2.2.3.3.5,
128=2.2.2.2.2.2.2,
usg(180;128)=2.2=4.
56=2.2.2.7,
45=3.3.5.
56-sa da 45-s ara aqvs saerTo martivi gamyofi. maTi saerTo gamyofia mxo lod 1:
usg(56; 45)=1.
56 da 45 urTierTmartivi ricxvebia.
yoveli ori gansxvavebuli martivi ricxvi urTierTmartivi ricxvebia. mag-aliTad, usg(17; 23)=1; 17 da 23 urTierTmartivi ricxvebia.
usg(8, 15)=1; 8 da 15-ic urTierTmartivi ricxvebia.
3) gavixsenoT, rom raime a naturaluri ricxvis jeradi aris ricxvi, rome-lic a-ze iyofa, anu a ricxvisa da raime naturaluri ricxvis namravlia.
12-is jeradebia 12, 24, 36, 48, 60, 72, 84, 96, 108, 120,...
18-is jeradebia: 18, 36, 54, 72, 90, 108, 126,...
12-isa da 18-is saerTo jeradebia: 36, 72, 108,...
12-isa da 18-is saerTo jeradebi SeiZleba ase CavweroT: 36n, n naturaluri ricxvia. am ricxvebs Soris umciresi miiReba, roca n=1.
36 aris 12-isa da 18-is umciresi saerTo jeradi. vwerT: usj(12; 18)=36.
24
magaliTi 5
magaliTi 6
a da b ricxvebis umciresi saerTo jeradi ase Caiwereba: usj(a; b);
a, b da c ricxvebisa ki - usj(a; b; c).
mocemuli naturaluri ricxvebis umciresi saerTo jeradi aris um-ciresi ricxvi, romelic iyofa TiToeul mocemul ricxvze.
vipovoT 72-is da 108-is umciresi saerTo jeradi.
I xerxi: amovweroT am ricxvebis jeradebi.
72-is jeradebia: 72, 144, 216, 288, 360, 432, ...
108-is jeradebia: 108, 216, 324, 432, ...
saerTo jeradebia: 216, 432, ...
usj(72; 108)=216.
II xerxi: davSaloT mocemuli ricxvebi martiv mamravlebad:
72=2.2.2.3.3; 108=2.2.3.3.3.
axla amovwerT erT-erTi ricxvis, magaliTad, 72-is yvela martiv gamyofs (um ci resi saerTo jeradi, cxadia, yvela am martiv gamyofs Seicavs) da am namravls `Se vav sebT~ 108-is im martivi gamyofebiT, romlebic ar Sedis an naklebi ode no biTaa 72-is daSlaSi:
usj(72; 108)=2.2.2.3.3.3=8.27=216.
vipovoT usj (18; 24; 30).
18=2.3.3,
24=2·2·2·3, usj (18; 24; 30)=23·32·5=360.
30=2.3.5.
am wesis mixedviT kargad Cans, rom urTierTmartivi ricxvebis umciresi saerTo jeradi am ricxvebis namravlia.
Tqven, albaT, ukve SeniSneT, rom Tu p martivi mamravli a ricxvis martiv mamravlebad daSlaSi α xarisxSia, b-s daSlaSi β- xarisxSia, β≥α, maSin a-sa da b-s umcires saerTo jeradSi p aris β xarisxSi.
axla sami ricxvis umcires saerTo jeradze SevCerdeT.
SevajamoT: yoveli naturaluri ricxvi an martivi ricxvia, an Sedgeni-li ricxvia, an 1-is tolia. yoveli Sedgenili ricxvi SeiZleba davSa loT martiv mamravlebad.
ganvsazRvreT naturaluri ricxvebis udidesi saerTo gamyo fi da umciresi saerTo jeradi. warmovadgineT maTi povnis maga liTebi.
25
martivi ricxvebis povnis sakiTxi uZvelesi droidan izidavda maTema-
tikosebs. jer kidev didma berZenma maTematikosma evklidem daasabuTa, rom
arsebobs uamravi mar tivi ricxvi.
maTematikis frangi moyvarulis mersenis (1588-1648) saxels atarebs spe-
cialuri tipis martivi ricxvebi, romlebic 2p-1 formuliT gamoisaxeba. Tu
p=2, 3, 5 an 7, maSin 2p-1 martivi ricxvia; magram p=11-sTvis vRebulobT:
211-1=2047=23·89,
es ki Sedgenili ricxvia.
kompiuterebis gamoyenebamde napovni iyo mersenis martivi ricxvi, romelic
p=257-s Seesabameba. 1952 wlidan am ZiebaSi kompiuterebic CarTes da kidev
araerT aseT martiv ricxvs miakvlies. magaliTad, 1965 wels, `sarekordo~
aRmoCnda p=11 213-is Sesabamisi ricxvi - mersenis 23-e martivi ricxvi.
XX saukunis bolo meoTxedidan did martiv ricxvebs mniSvnelovani praq-
tikuli gamoyeneba aqvs. Cven amaze SemdgomSi uTuod visaubrebT.
2005 wlisTvis napovnia mersenis 42-e martivi ricxvi 225 964 951-1, romelic
aTobiT sistemaSi 8 milioni cifriT Caiwereba.
* * *
gTavazobT udidesi saerTo gamyofis povnis kidev erT wess, romelsac
evklides algoriTms uwodeben. es algoriTmi naSTian gayofas efuZneba.
vTqvaT, a>b, maSin, rogorc viciT, gvaqvs, a=bq+r, 0≤r<b.
am tolobidan advilad davaskvniT:
usg (a; b)=usg (b; r). (1)
(1) tolobis gaTvaliswinebiT vipovoT 904-isa da 162-is udidesi saerTo
gamyofi. evklides algoriTmi am ricxvebisTvis ase Caiwereba:
904=162.5+94,
162=94.1+68,
94=68.1+26,
68=26.2+16,
26=16.1+10,
16=10.1+6,
10=6.1+4,
6=4.1+2,
4=2.2.
(1) tolobis gamo gvaqvs:
usg (904; 162)=usg (162; 94)=usg (94; 68)=...= usg (10; 6)=usg (6; 4)=usg (4; 2)=2.
maSasadame, usg(904; 162)=2.
amrigad, mocemuli ori ricxvidan met ricxvs vyofT naklebze. Semdeg am
nakleb ricxvs vyofT wina gayofisas miRebul naSTze (Tu es naSTi nuli ar
aris) da am process vagrZelebT uka nasknel aranulovan naSTamde. swored es
ukanaskneli aranulovani naSTi aris mocemuli ricx vebis udidesi saerTo
gamyofi.
s
26
* * *
gTavazobT formulas, romelic erTmaneTTan akavSirebs ori ricxvis udides
sa er To gamyofsa da amave ricxvebis umcires saerTo jerads:
usj (a; b) =
ab .
usg (a; b)
magaliTad, vTqvaT, a=140, b=110. usg (140; 110)=10; usj (a; b)=1540.
ab=140·110=10·1540=usg (140; 110) · usj (140; 110).
moviyvanoT dasabuTebis sqema. vTqvaT, raime p martivi ricxvi a ricxvis mar-
tiv mamravlebad daSlaSi a xarisxSia, b ricxvis martiv mamravlebad daSlaSi
- b xarisxSi. maSin, cxadia, ab namravlSi es ricxvi a+b xarisxSia. vTqvaT,
a≤b. maSin p ricxvi usg (a; b)-Si Sedis a xarisxSi, usj(a; b)-Si - b xarisxSi,
maT namravlSi - (a+b) xarisxSi (aq SeiZleba moxdes, rom a=0).
aqedan SeiZleba davaskvnaT, rom ab=usj (a; b) · usg (a; b).
SearCieT swori pasuxi:
1 Tu 37x martivi ricxvia, maSin x=
1) 2 2) 0 3) 3 4) 1.
2 ricxvi 1
1) Sedgenili ricxvia 2) martivi ricxvia 3) arc martivi ricxvia da arc Sedgenili 4) martivi ricxvicaa da Sedgenilic.
3 umciresi kenti martivi ricxvia 1) 3 2) 2 3) 5 4) 7.
4 udidesi orniSna martivi ricxvia 1) 98 2) 97 3) 96 4) 99.
5 udidesi orniSna Sedgenili ricxvia 1) 98 2) 97 3) 100 4) 99.
6 ramdeni gamyofi aqvs yovel martiv ricxvs?
1) ori 2) sami 3) erTi 4) oTxi.
7 Tu Sedgenili ricxvi ori gansxvavebuli martivi ricxvis namravlia, maSin mas aqvs mxo lod
1) ori gamyofi 2) sami gamyofi 3) oTxi gamyofi 4) erTi gamyofi.
a
SearCieT swori pasuxi
27
8 Tu Sedgenili ricxvi raime martivi ricxvis kubia, maSin misi gamyofebis odenobaa
1) ori 2) sami 3) oTxi 4) xuTi.
9 CamoTvlili winadadebebidan romelia swori?
1) nebismieri ori martivi ricxvis jami martivi ricxvia 2) nebismieri ori martivi ricxvis jami Sedgenili ricxvia 3) nebismieri ori martivi ricxvis namravli martivi ricxvia 4) nebismieri ori martivi ricxvis namravli Sedgenili ricxvia.
10 usg(48; 96)=
1) 48 2) 96 3) 24 4) 32.
11 usg(14; 7)=
1) 14 2)7 3) 1 4) 28.
12 usg(62; 63)=
1) 2 2) 62 3) 63 4) 1.
13 SearCieT araswori debuleba.
1) yoveli Sedgenili ricxvi iSleba martiv mamravlebad 2) ori naturaluri ricxvi urTierTmartivia, Tu maTi udidesi
saerTo gamyofi aris 1
3) Tu naturaluri ricxvi 0-iT bolovdeba, maSin mas ara aqvs mar-tivi gamyofi
4) martiv mamravlebad daSlisas SeiZleba gamoviyenoT gayofadobis niSani.
14 ipoveT usg(45; 72).
1) 3 2) 9 3) 18 4) 8.
15 SearCieT swori debuleba.
1) nebismieri ori luwi ricxvi urTierTmartivia 2) yoveli ori ricxvi - luwi da kenti, urTierTmartivia 3) arsebobs ori luwi martivi ricxvi 4) yoveli ori momdevno naturaluri ricxvi urTierTmartivia.
16 Tu usg(a; b)=a, maSin 1) a metia b-ze 2) b iyofa a-ze 3) b ar iyofa a-ze.
17 Tu a metia b-ze da a iyofa b-ze, maSin a da b ricxvebis umciresi saerTo jeradia
1) b 2) a 3) 2a 4) 3a.
18 Tu raime a da b naturaluri ricxvebis udidesi saerTo gamyofi aris 1, maSin usj(a; b)=
1) 2ab 2) a+b 3) ab+1 4) ab.
19 ipoveT usj(42; 16).
1) 2 2) 336 3) 72 4) 168.
28
20 123454321 iyofa 11111-ze. am ricxvebis umciresi saerTo jeradia 1) 11111 2) 123454321 3) 1111 4) 246908642.
21 usj(17; 3)=
1) 17 2) 3 3) 1 4) 51.
22 ori ricxvis umciresi saerTo jeradi 1) naklebia TiToeul mocemul ricxvze 2) ar aris naklebi arcerT mocemul ricxvze 3) naklebia mocemuli ricxvebidan udidesze.
23 ra umciresi sigrZe SeiZleba hqondes Toks, rom misi daWra SeiZle-bodes 6 m sigrZis naWrebadac da 4 m sigrZis naWrebadac.
1) 24 m 2) 12 m 3) 18 m 4) 48 m.
24 316-isa da
524-is umciresi saerTo mniSvnelia
1) 24 2) 16 3) 48 4) 96.
25 5, 6 da 7-is umciresi saerTo jeradia:
1) 420 2) 210 3) 1 4) 7.
26 4, 10, 12 da 15-is umciresi saerTo jeradia:
1) 4 2) 3 3) 30 4) 60.
27 CamoTvlili ricxvebidan 5-is jeradi ar aris 1) 28-1 2) 36-1 3) 56-103.
28 im martivi ricxvebis odenoba, romlebic naklebia 10-ze, aris 1) 3 2) 2 3) 4 4) 5.
29 Tu x-is 6-ze gayofisas miiReba ganayofi 8 da naSTi 1, maSin x=
1) 47 2) 48 3) 50 4) 49.
30 Tu ori naturaluri ricxvis jami kentia, maSin am ricxvebis namravli 1) kentia 2) luwia
3) iyofa 5-ze da ar iyofa 10-ze
4) SeiZleba luwic iyos da kentic.
31 sworia Tu ara, rom:
a) yoveli kenti ricxvi 3-is jeradia;
b) yoveli kenti ricxvi martivi ricxvia;
g) yoveli martivi ricxvi kentia?
32 daasaxeleT martivi ricxvebi, romlebic metia 10-ze da naklebia 40-ze.
33 arsebobs Tu ara luwi ricxvi, romelic gansxvavebulia 2-sgan da martivia?
amoxseniT amocanebi
29
34 daSaleT martiv mamravlebad:
a) 800, b) 216, g) 343, d) 1024,
1728; 1575; 9225; 1001.
35 warmoadgineT martivi ricxvis xarisxis an martivi ricxvebis xarisxebis namravlis saxiT:
a) 64; b) 200; g) 144; d) 256; e) 333; v) 348; z) 512; T) 100 000; i) 4096; k) 250 000; l) 750 000.
36 ramdenjer gvxvdeba 1-dan 100-mde (CaTvliT) ricxvebis Ca na we rebSi
cifri 9?
37 cnobilia, rom ricxvi iyofa 5-ze da mas mxolod sami gamyofi aqvs. ipoveT es ricxvi.
38 cnobilia, rom ricxvi 21-ze iyofa, 9-ze ki - ara; mas sul eqvsi gamyofi aqvs. ipoveT es ricxvi.
39 vTqvaT, x naturaluri ricxvi 2-ze metia. Sedgenilia Tu martivi misi wina da momdevno ricxvebis namravli? pasuxi daasabuTeT.
40 m metia n naturalur ricxvze 14-iT. SeadareT A da B:
a B Sedarebis Sedegi
m-10 n A>B
m n+20
m-10 n+10
m-7 n+7
(m+n)-is 2-ze gayofisas miRebuli naSTi
0
(3m+5n)-is 2-ze gayofisas miRebuli naSTi
1
m-is 7-ze gayofisas miRe-buli naSTi
n-is 7-ze gayofisas miRebuli naSTi
(m-n)-is 7-ze gayofisas miRebuli naSTi
0
41 ipoveT:
a) usg(120; 150); b) usg(124; 93); g) usg(46; 69);
d) usg(24; 48); e) usg(62; 31); v) usg(36; 21).
42 sworia Tu ara: usg(33; 198)=66 ? pasuxi daa sa buTeT.
rv
eul
Si!
30
43 mzrunvelobas moklebulTaTvis saaxalwlo ama na Te bisTvis SearCies 322 man da rini, 230 vaSli da sxva da sxva tkbileuli. ra udidesi odenobis amanaTis Sed ge na SeiZleba ise, rom amanaTebSi mandarinis erTi da igive odenoba iyos da vaSlic Tanabrad ganawildes?
44 vTqvaT, usg (a; 3b)=2. maSin SesaZlebelia, Tu ara, rom:
a) a iyos martivi ricxvi, b) b iyos kenti ricxvi,
g) b iyos martivi ricxvi, d) a iyos b-s toli,
e) b iyos Sedgenili?
45 p martivi ricxvia. ramdeni gamyofi aqvs: a) p2-s, b) p3-s?
46 p da q martivi ricxvebia, p≠q. ramdeni gamyofi aqvs pq2-s?
47 dawereT raime ricxvi, romelsac mxolod sami gamyofi aqvs. ramdeni ase Ti ricxvi arsebobs?
48 p da q martivi ricxvebia, p>q da q>2. ipoveT: a) usg (p, q); b) usg (p2, pq);
g) usg (p-q, 2); d) usg (q+1, 2p).
49 kbilaniani ori diskos CaWidebis adgili moni Snu-lia. erT-erT diskoze 60 kbilanaa, meoreze - 28. Ti-Toeuli diskos ramdeni mob ru ne bis Semdeg da emTxveva maTi CaWidebis adgili moniSnuls meorejer?
50 vipovoT umciresi ricxvi, romelic 4-ze gayofisas naSTSi iZleva 3-s da amave naSTs iZleva 5-ze da 7-ze gayofis SemTxvevaSic.
51 ipoveT umciresi naturaluri ricxvi, romelic 437 da
541 wiladebze
ga yo fisas mTel ricxvebs gvaZlevs.
52 vTqvaT, p martivi ricxvia. ra SeiZleba iyos a da p naturaluri
ricxvebis udidesi sa er To gamyofi da umciresi saerTo jeradi?
53 ori ricxvis udidesi saerTo gamyofi aris 9, maTi nam rav li 3645-ia, am ricx vebidan umciresi 5-is jeradia. ipoveT es ricxvebi.
54 sami sportsmeni moZraobas iwyebs erTdroulad sam wriul bilikze A wertilidan. yoveli maTgani Tavis bilikze moZraobs 5 m/wm siCqariT. bilikebis sigrZeebia 400 m, 600 m da 900 m. moZ ra obis dawyebidan pirvelad ra drois Semdeg Sexvdeba samive erT maneTs?
55 vipovoT umciresi ricxvi, romlis 5-ze gayofisas naSTia 2, 8-ze gayofisas - 5, 10-ze gayofisas - 7, 15-ze gayofisas - 12.
31
56 daasaxeleT umciresi ricxvi, romlis aTobiT CanawerSi monawileobs yvela cifri 0-dan 9-mde da romelic 36-is jeradia.
57 vTqvaT, ricxvis aTobiT CanawerSi mxolod 0 da 6 cifrebi gvxvdeba. SeiZleba Tu ara, es ricxvi iyos raime naturaluri ricxvis kvadrati? pasuxi daasabuTeT.
58 oTxniSna ricxvi iyofa 12-ze. misi cifrTa jami 3-ia. ramdeni aseTi ricxvi arsebobs?
vTqvaT, Tqvenma klasma gaimarjva erT-erT satelevizio konkursSi da miiRo jil do - TiToeul moswavleze 1250 lari (samogzaurod).
romel qveyanas airCevdiT safrangeTs Tu italias? gaiTvaliswi-neT, rom gam gzav reba sagzuriT xdeba. sagzuris fasSi Sedis mxolod satransporto xarjebi, kvebis da gidis momsaxureba. gasaTvaliswine-belia Semdegi monacemebic:
• orkviriani 1 sagzuris fasi 950 laria.
• Tqveni klasis sami moswavle flobs ita li urs.
• 5 Rame Tqven unda gaataroT italielTa oja xebSi, danarCeni - axalgazrdul sa zaf xulo banakebSi.
• 5 dRis ganmavlobaSi gaCerdeb-iT ger ma niaSi.
• Tqveni mogzaurobis program-aSi da ma te biT Sedis arCevanis ga-keTeba (sa misa, Sem de gi oTxidan):
• erT-erT restoranSi sadili (100 lari erT moswavleze).
• romis daTvaliereba (70 la ri erT mos wav leze).
• vatikanis daTvaliereba (100 la ri).
• 25 lariani suveniri.
• samkviriani 1 sagzuris fasi 1050 laria.
• Tqveni klasis yvela moswavle amJa mad swavlobs franguls.
• sastumroSi an axalgazrdebis sa er To sacxovrebelSi mogiwevT ga-Ce re ba.
• 4 dRe gaCerdebiT germaniaSi.
• Tqveni mogzaurobis program-aSi da ma te biT Sedis arCevanis ga-keTeba (samisa, Sem degi xuTi dan):
• erT-erT restoranSi sadili (80 la ri erT moswavleze).
• naxevari dRiT mogzauroba pa -rizSi (20 lari erT moswavle ze).
• parizis dis ne ilen dis daTva li-ereba (80 lari erT mos wav le ze).
• 15 lariani suveniri.
• eifelis koSkidan parizis da-Tva li e reba (30 lari erT moswav-leze).
italia safrangeTi
j
32
SeecadeT klasSi sajarod axsnaT, ram ganapiroba Tqveni arCevani.
geografiis saxelmZRvaneloSi (an sxva cnobarebSi) moipoveT informacia sasaaTo zolebis Sesaxeb. upasuxeT kiTxvebs:
romel sasaaTo zolSia Tqven mier Ser-Ceuli qalaqi (romi an parizi)?
romel sasaaTo zolSia Tbilisi? Tu adgilobrivi droiT 9 saaTze gamoemgza-vrebiT Tqven mier samogzaurod SerCeuli qalaq-idan, romel saaTze iqnebiT TbilisSi (adgilo-brivi droiT)? gaiTvaliswineT mgzavrobis droc. moqmedebs Tu ara TbilisSi amJamad zoluri dro? romeli mezobeli qveynis dedaqalaqSi ar mo-qmedebs amJamad zolu ri dro?
33
1. 4 simravle. qvesimravle
maTematikur termins – simravles – qarTul enaSi araerTi sinonimi moeZebneba. magaliTad: erToblioba, krebuli, nakrebi, gundi, fara, rema. gaixseneT da moiyvaneT am sityvaTa xmarebis mag-aliTebi. yvela am SemTxvevaSi SeiZleboda `sim-ravlis~ xma re bac. magaliTad, skolis moswavleTa simravle, fexburTelTa simravle, cxenebis sim-rav le.
ori cxenis erTobliobaze aravin ityvis: cxe-nebis jo giao. Tumca SeiZleba gan vi xi loT sim rav-le, romelic mxo lod ori cxe nisgan Sedgeba.
simravlis magaliTebia: caze varskvlavebis si-mrav le, na tu ra lur ricxvTa simravle, cif rebis simravle, qarTuli an banis asoe bis simravle, la-Tinuri anbanis asoebis sim ravle, sko lis bib-lioTekaSi wignebis simravle.
simravlis dasaxelebisas zustad unda iyos garkveuli is obieqtebi, rom-lebic simravles ekuTvnis. am obieqtebs simravlis elementebi ewo deba. erTi da igive obi eqti ar SeiZleba warmogvidges, rogorc simravlis sxva da sxva el-ementi.
winadadeba: `a aris A simravlis elementi~ – ase Caiwereba: vkiTxulobT: `a ekuTvnis A simravles~.
winadadeba `a ar aris A simravlis elementi~ – ase Caiwereba:
vTqvaT, zari daireka da saklaso oTaxSi aravin darCa. maSin vityviT, rom saklaso oTaxSi moswav-leTa simravle carieli simravlea. carieli sim-ravle vuwodoT simravles, ro melsac arc erTi elementi ar ekuTvnis.
carieli simravle erTaderTia. is ∅ simboloTi aRvniSnoT.
zogierTi simravle sakmaod xSirad aris xolme dasaxelebuli. amitom maT-Tvis specialuri aRniSvnebia SerCeuli; magaliTad, n naturalur ricxvTa sim-ravles aRniSnavs.
• waikiTxeT Canawerebi, romelia maTgan swori?
5∈n, 12∈n, 0,5∈n.
simravle sasrulia, Tu SesaZlebelia misi yvela elementis CamoTvla. magaliTad, 12-is gamyofebis simravle SeiZleba ase CavweroT: {1, 2, 3, 4, 6, 12}.
es simravle 6 elementisgan Sedgeba, eqvselementiani simravlea.carieli simravlec sasruli simravlea. am simravlis elementebis odenoba
nulia.
a A
a A
34
sim ravles, romelic sasruli simravle ar aris, usasrulo simravle ewodeba. n usasru-lo simravlea. is SeiZleba ase CavweroT:
aq miTiTebulia simravlis mxolod ramdenime elementi.Tu A da B simravleebi erTi da imave elementebisgan Sedge-
ba, maSin maT toli simravleebi ewodeba; Sesabamisad, vwerT:
magaliTad, vTqvaT, A={2; 3; 5}, B={5; 2; 3}. maSin A=B.
Tu raime C da D simravleebi toli ar aris, vwerT:
Tu A simravlis yoveli elementi B simravlis elementicaa, maSin A-s ewo-deba B-s qvesimravle. Sesabamisad, vwerT:
• vTqvaT, A aris erTi naturaluri ricxvis gamyofebis simravle, B _ meore naturaluri ricxvis. ra SemTxvevaSi SeiZleba Sesruldes piroba A⊂B?
• aris Tu ara yoveli simravle Tavisi Tavis qvesimravle?
SevajamoT: gavecaniT simravlis magaliTebs. mocemuli simravlis qvesim ravle aris simravle, romlis yoveli elementi mocemul si mravlesac ekuTvnis. cariel sim rav les arc erTi elementi ar ekuTvnis; carieli simravle ne bis mieri simravlis qve simravlea.
N= { 1; 2 ; 3; ...}
A B=
C D≠
A B⊂ B A⊃
simravleTa Teoriis fuZemdeblad germaneli maTematiko-
si georg kantori (1845-1918) iTvleba. gansakuTrebiT mniS-
vnelovania misi mecnieruli Sromebi usasrulo sim rav le Ta
TeoriaSi. georg kantori simravlis cnebis aseT aRweras
iZleva: `simravle garkveuli da erTmaneTisgan garCevadi
sagnebis nebismieri erTobliobaa~.
bevri cnobili maTematikosi simravleTa Teorias mTeli
maTematikis safuZvlad mi iCnevs. simravleTa Teoriis bev-
ri cneba da aRniSvna maTematikis TiTqmis yvela na wilSi
gamoiyeneba. am cnebebsa da aRniSvnebs Cvenc xSirad vi ye nebT.georg kantori
an
s
35
1 Tu a aris A simravlis elementi, maSin vwerT:
1) a∉A 2) a∈A 3) a⊃A 4) A∈a.
2 Tu a ar aris A simravlis elementi, maSin vwerT:
1) a∉A 2) A∈a 3) a⊃A 4) a∈A.
3 mocemulia: a) 3∈n; b) 0∈n; g) 12∈n; d) 1∈n. maTgan WeSmaritia:
1) a), b) da d) 2) mxolod a) 3) mxolod d) 4) a) da d).
4 vTqvaT, Nm-iT aRvniSneT pirveli m naturaluri ricxvis simravle.
magaliTad, N5={1, 2, 3, 4, 5}. mocemulia: a) n3⊂n; b) n⊂n
7; g) n
8⊂n;
d) N9=n. maTgan WeSmaritia:
1) d) 2) b) 3) a) da g) 4) mxolod a).
5 A={0,6; 1; 5} da B={77; 5;
35} ricxvTa simravleebis Sesaxeb da sa xe le-
bu lia winadadebebi: a) A≠B; b) A⊂B; g) B⊂A; d) A=B. maTgan WeSmaritia:
1) a) 2) mxolod b) 3) mxolod d) 4) b), g) da d).
6 vTqvaT, A={1}, maSin 1) 1∈A 2) 1=A 3) A⊂1 4) 1∉A.
7 orniSna ricxvebis simravlis is qvesimravle, romlis elementebi 11-is jeradi yvela SesaZlebeli orniSna ricxvebia, aris
1) {11; 99} 2) {11; 66; 99} 3) {11; 22; 44; 66; 99} 4) {11; 22; 33; 44; 55; 66; 77; 88; 99}.
8 vTqvaT, A aris 3-is jeradi orniSna ricxvebis simravle, B ki – 6-is jeradi orniSna ricxvebis simravle. maSin
1) A aris B-s qvesimravle 2) B aris A-s qvesimravle 3) A da B simravleebi tolia.
9 weliwadis Tveebis simravleSi gamoyaviT im Tveebis qvesimravle, romlis dRe ebis ricxvi aris 31.
10 ganixileT Tqveni klasis moswavleTa simravle. anbanis erTsa da ima-ve asoze daw yebuli gvaris mqone moswavleebiT erTi qvesimravle Seq-meniT. ganixileT 33-ive aso. miRebuli qvesimravleebidan:
a) romelia carieli simravle,
b) romeli Seicavs yvelaze met elements?
a
SearCieT swori pasuxi
amoxseniT amocanebi
36
11 me-10 amocanaSi ganixileT moswavleTa saxelis pirveli asos mixed-viT Sed ge ni li qvesimravleebi da daadgineT:
• romeli qvesimravlea carieli,
•romel qvesimravles ekuTvniT Tqven da kidev romeli moswavleebi ekuTvnian am qvesimravles?
12 Tqveni klasis yvela im moswavlis simravle, romlis gvari `m~ asoTi iwyeba SeiZleba ase Caiweros:
{x| x – Tqveni klasis moswavle, romlis gvari iwyeba `m~ asoTi}. amave wesiTaa Cawerili qvemoT moyvanili simravleebic. ipoveT sim-
ravlis yvela elementi:
a) {x| x – wlis Tve, romelic iwyeba `i~ asoTi};
b) {x| x– evropuli saxelmwifo, romlis dasaxeleba iwyeba `S~ asoTi};
g) {x| x – Tqveni skolis maTematikis maswavlebeli};
d) {x| x – 10-ze naklebi da 3-is jeradi naturaluri ricxvi};
e) {(x;y)| x, y – naturaluri ricxvebi, romlebisTvisac xy=12}.
13 CawereT simravle ori sxvadasxva wesiT: a) simravle, romlis elementebia 24-is yvela gamyofi;
b) simravle, romlis elementebia 19-is yvela gamyofi;
g) simravle, romlis elementebia 59-is yvela gamyofi;
d) simravle, romlis elementebia 18-is yvela gamyofi;
TiToeul SemTxvevaSi daasaxeleT simravleSi elementebis raode-noba.
14 CawereT simravle, romlis elementebia:
a) 50-ze naklebi martivi ricxvebi;
b) 50-ze naklebi luwi ricxvebi;
• aqvs Tu ara am simravleebs saerTo elementebi?
15 vTqvaT, A luwi ricxvebis simravlea, B ki – kenti ricxvebis sim-ravle. SeavseT cxrili nimuSis mixedviT.
16 aris Tu ara Semdegi simravleebi sasruli? dadebiTi pasuxis SemTx-vevaSi mi uTiTeT am simravleSi Semavali elementebis raodenoba: a) n; b) n
3; g) n
6; d) 5-ze meti da 9-ze naklebi naturaluri ricxvebis sim-
ravle.
miTiTeba. a) {x| x aris 24-is gamyofi} an {1; 2; 3; 4; 6; 8; 12; 24}.
miTiTeba. aq gamoyenebulia 4 amocanaSi SemoRebuli aRniSvnebi.
3 0 2 4 5 6 7 9 12
3∉A
3∈B rv
eul
Si!
37
17 vTqvaT, n∈n, m∈n da n<m. wertilebis nacvlad CasviT saWiro niSani (⊃, ⊂, =):
a) n...nn; b) n
n...n
m;
g) nm...n.
18 ipoveT m, Tu cnobilia, rom nm-1
simravle cxra elementisgan Sedge-ba.
19 cnobilia, rom k∈n da nk⊂n
7. ipoveT k-s mniSvnelobebis simravle.
20 daasaxeleT m da n ricxvTa yvela SesaZlo wyvili, Tu cnobilia, rom m∈n. n∈n da n
m⊂n
n⊂n
6.
21 vTqvaT, A aris yvela orniSna ricxvis simravle. daasaxeleT A-s raime qve sim ravle, Sedgenili yvela im orniSna ricxvisgan, romlebic 3-ze gayofisas erTsa da imave naSTs iZleva:
a) ramdeni aseTi qvesimravle arsebobs,
b) ras udris erTi romelime aseTi qvesimravlis nebismieri ori el-ementis sxva obis 3-ze gayofisas miRebuli naSTi?
22 vTqvaT, A aris 30-is yvela gamyofis simravle, B ki – 42-is yvela gamyofis simravle. ipoveT:
a) A da B-s yvela saerTo elementis simravle;
b) im elementebis simravle, romlebic A da B simravleebidan erT-erTs mainc ekuTvnis;
g) A-s yvela im elementis simravle, romlebic B-s ar ekuTvnis;
d) B-s yvela im elementis simravle, romlebic A-s ar ekuTvnis.
1 212121:21=
1) 111 2) 1101 3) 10101 4) 1011.
2 mocemulia ricxvebi: 712, 329, 89, 954. maTgan 9-ze iyofa 1) 329 2) 89 3) 954 4) 712.
3 CamoTvlili ricxviTi gamosaxulebebidan romlis mniSvneloba iyofa 10-ze?
1) 25.21-9.18 2) 219.9-7.213 3) 219.9+7.213 4) 25.21+19.18.
4 147*2 iyofa 7-ze. ra cifria Secvlili * niSnakiT? 1) 2 2) 3 3) 4 4) 5.
g
SearCieT swori pasuxi
38
5 maswavlebelma 20 rveuli gaaswora, rac rveulebis saerTo rao de-
no bis 45-ia. ramdeni rveuli darCa gasasworebeli
maswavlebels?
6 erT dRes levanma wignis 37 waikiTxa, meore
dRes _ darCenili gverdebis naxevari, mesame dRes _ 32 gverdi da amiT wignis wakiTxva daasrula. ramdengverdiania wigni?
7 cnobilia, rom a umciresi samniSna ricxvia, romlis 2-ze gayofisas mi iReba naSTi 1; 3-ze gayofisas miiReba naSTi 2; 4-ze gayofisas _ naS Ti 3; 5-ze gayofisas _ naSTi 4; 6-ze gayofisas _ naSTi 5. ipoveT a.
8 mudmivi siCqariT moZravma velosipedistma gzis naxevari 3 saaTSi gaiara, meore naxevari ki _ 2 saaTSi, radgan siCqare gazarda 4 km/sT-iT. ra manZili gaiara velosipedistma sul?
amoxseniT amocanebi
39
1. 5 simravleTa gaerTianeba da TanakveTa.venis diagramebi
vTqvaT, erT-erTi klasis moswavleTa Soris zogierTi oqtom ber Sia dabade-buli – maTi simravle aRvniSnoT A asoTi. vTqvaT, am klasis moswavleTa So-ris zogierTis gvari `g~ asoTi iwyeba _ maTi simravle aRvniSnoT B asoTi. am skolis moswavleTa simravle aRvniSnoT M-iT.
vTqvaT, A da B simravleebs saerTo elementebi aqvs (ax-seniT, ras niS navs es). Tu am simravleebs pirobiTad wreebiT gamovsaxavT, ro gorc amas ingliseli ma Te ma tikosi jon veni akeTebda, maSin es wreebi ikveTeba.
A da B simravleebi venis diagramiTaa gamosaxuli. am sim rav le ebis saer-To na wi li gaferadebulia, mas A da B simravleebis TanakveTa ewodeba – es aris mocemuli klasis oq tomberSi dabadebuli yvela im moswavleTa sim ravle, romelTa gva ri `g~ asoTi iw yeba. maSasadame,
A da B sim rav leebis TanakveTa aris am sim rav leebis yvela saerTo ele-mentis sim rav le, mas ase aRvniSnavT: A∩B.
yvela im moswavlis simravle, romelic an oqtomberSi daibada, an romlis gvari `g~ asoTi iwyeba, qmnis A da B sim-ravleebis gaer Tiane bas. suraTze A da B simravleebis ga-erTianebaa gaferadebuli. am sim rav les ase aRvniSnavT: A∪B.
A∪B simravle aris yve la im elementis sim ravle, romelic A da B sim-ravleebidan erT-erTs mainc ekuT vnis.
axla gavixsenoT, rom A da B simravleebi M simravlis qve-sim rav le ebia. venis diagramiT es sami simravle ase gamois-axeba:
vTqvaT, aRniSnul klasSi oqtomberSi dabadebuli moswav-le ar aris, maSin A=∅ (A carieli simravlea).
ris tolia am SemTxvevaSi ∅∩B, ∅∪B? imedia ar gagiWird-ebaT am kiTxvebze pasuxebis gacema.
• vTqvaT, A aris n naturaluri ricxvis gamyofebis simravle. B aris m naturaluri ricxvis gamyofebis simravle. A da B simravleebiT warmoadgineT simravle, romlis udidesi elementia usg (m, n).
• vTqvaT, C aris n naturaluri ricxvis jeradebis simravle. D aris m naturaluri ricxvis jeradebis simravle. sasrulia Tu usasrulo es simravleebi? C da D simravleebiT warmoadgineT simravle, romlis umciresi elementia usj (m; n).
SevajamoT: gavecaniT simravleebis TvalsaCinod gamosaxvis xerxs – venis di ag ramebs. venis diagramebiT warmovadgineT ori simravlis gaerTianeba da Ta na kve Ta.
40
ingliseli maTematikosi jon veni XIX saukunis miwurulsa da XX saukunis
da saw yis Si moRvaweobda. venis diagramebis saSualebiT bevri debuleba, ro-
melic sim rav le ebs exeba, TvalsaCinod warmogvidgeba.
magaliTad, Tu A⊂B da B⊂C, maSin A⊂C.
es debuleba suraTze TvalsaCinodaa warmodgenili.
simravleebis wris an raime sxva figuriT gamosaxvis
ideas ufro adrec iyenebdnen. ma ga liTad, Sveicarieli maTema-
tikosi leonard eileri (1707-1783) da germaneli vil helm
laibnici (1646-1716). Tum ca, am meTodis dawvrilebiT gadmocema TvalsaCino
ilus traciebiT jon vens ekuTvnis. ami tom aRniSnul sqemebs metwilad `venis
diagramebs~ uwodeben (Tumca, SesaZloa, `eiler-venis diagramebic~ SegxvdeT).
1 Tu A=∅ da B raime simravlea, maSin A∩B=
1) B 2) ∅ 3) A∪B.
2 Tu A=∅ da B raime simravlea, maSin A∪B=
1) A 2) ∅ 3) B.
3 yoveli A da B simravleebisTvis, Tu A⊂B, maSin A∩B=
1) A 2) B 3) ∅.
4 yoveli A da B simravleebisTvis, Tu A⊂B, maSin A∪B=
1) A 2) B 3) ∅.
5 suraTze daStrixulia 1) A∪B 2) A∩B 3) A 4) B.
6 suraTze daStrixulia 1) A∩B 2) A 3) B 4) A∪B.
7 vTqvaT, L – luw ricxvTa simravlea, K – kent ricxvTa simravle. ga mov sa xoT es simravleebi venis diagramiT.
1) 2) 3) 4)
8 suraTze gamosaxuli A da B sim rav le ebis Tana-kveTa aris
1) A simravle 2) B simravle 3) ∅.
s
aSearCieT swori pasuxi
41
9 vTqvaT, A aris naturaluri ricxvebis 5-ze gayofis Sedegad miRe-buli naS Te bis simravle, B – 7-ze gayofis Sedegad miRebuli naSTebis simravle. maSin A∩B=
1) {0; 1; 2; 3; 4; 5; 6; 7} 2) {1; 2; 3; 4}
3) {1; 2; 3; 4; 5; 6; 7} 4) {0; 1; 2; 3; 4}.
10 mocemulia ori simravle:
A={1; 2; 7; 9; 10; 12; 15} da B={0; 3; 5; 6; 7; 8; 9; 11}. a) ipoveT A∪B da A∩B;
b) ipoveT yoveli C simravle, romelic orive mocemuli simravlis qve simravlea;
g) yoveli aseTi C simravlisTvis ipoveT C∩(A∩B).
11 suraTze A, B da C simravleebia gamosaxuli. daxazeT eqvsi ase Ti diagrama Tqvens rveulSi da mimdevrobiT da StrixeT:
a) A∪B; b) A∩B; g) A∩C; d) B∩C; e) (A∪B)∪C; v) A∩(B∩C); z) A∪(B∩C).
12 vTqvaT, A aris im orniSna ricxvebis simravle, romlebic iyofa 3-ze, B – im orniSna ricxvebis simravle, romlebic iyofa 7-ze. gamosaxeT es simravle ebi da maTi TanakveTa venis diagramebiT. ramdeni elementia am TanakveTaSi?
13 ipoveT 210-isa da 231-is martiv gamyofTa simravleebis TanakveTa.
14 ipoveT 36-is gamyofebis simravlisa da 40-is gamyofebis simravlis TanakveTa.
15 vTqvaT, A aris im orniSna ricxvebis simravle, romelTa 3-ze gayo-fisas naST ia 1, B – im orniSna ricxvebis simravle, romelTa 3-ze gayo-fisas naST Sia 2. ipoveT A∪B. daaxasiaTeT es simravle.
16 Tqveni klasis im moswavleebis simravle, romelTac xatva exerxebaT, iyos A, B iyos maTi simravle, vinc kargad mReris, C ki – cekvis moyvar-ulTa simravle. gamosaxeT erToblivad es simravleebi venis diagramiT.
17 vTqvaT, • yvelas, vinc M bankSi muSaobs, hyavs sakuTari avtomobili. avtomobilebis mflobelTa simravle aRniSneT A-Ti, M bankis Tanam-
Sromlebis simravle – B-Ti. gamosaxeT es simravleebi venis diagramiT da ipoveT
a) A∩B; b) A∪B.
18 mocemulia simravleebi A={1; 21; 121; 15; 52; 512}, B={16; 2; 5}, C={1; 21; 15}, D={21; 15; 52}, E={121; 512}, F={12; 15; 25}, K={21; 121; 512; 52},
Q={512}. ro melia am simravleebidan A-s qvesimravle? sworia, Tu ara, rom:
a) B⊂A; b) C⊂A; g) D⊂A; d) Q⊂E; e) E⊂F; v) A⊂Q.
amoxseniT amocanebi
42
1 vTqvaT, A1 aris im naturaluri ricxvebis simravle, romelTa 5-ze
gayofisas naSTSi 1 miiReba, A2 – im naturaluri ricxvebis simravle,
romelTa 5-ze gayofisas naSTSi 2 miiReba, A3 – im naturaluri ricx-
vebis simravle, romelTa 5-ze gayofisas naSTSi 3 miiReba, A4 – im natu-
raluri ricxvebis simravle, romelTa 5-ze gayofisas naSTSi 4 miiReba. • aqvs Tu ara am simravleebs saerTo elementi? • arsebobs Tu ara naturaluri ricxvi, romelic arc erT CamoTv-
lil sim rav le Si ar Sedis?
• rogor SeiZleba davaxasiaToT simravle A1∪A
2∪A
3∪A
4 (simravle im
ricxvebisa, romlebic an A1-s ekuTvnis, an A
2-s, an A
3-s, an A
4-s)?
• rogor SeiZleba davaxasiaToT is naturaluri ricxvebi, romle-bic arc erT CamoTvlil simravleSi ar Sedis? ra aRniSvnas SeurCev-diT am ricxvTa simravles?
• warmoadgineT CamoTvlili simravleebi da naturalur ricxvTa simravle venis diagramebiT.
2 SeadgineT wina amocanis analogiuri amocana, roca 5-is nacvlad gvaqvs raime naturaluri n (n>2, n≠5) ricxvi da amoxseniT es amocana.
1 daalageT ricxvebi zrdis mixedviT:
a) 56; 1
12; 1,25; 1
23; 4,6;
25. b)
12; 1,7; 0;
13; 2,3; 6,5;
23.
2 klasSi moswavleTa 60% gogonaa; biWebis odenobaa 8. a) klasis moswavleTa ramdeni procentia biWi?
b) ipoveT klasis moswavleTa 10%;
g) ramdeni moswavlea klasSi?
3 20-moswavlian klasSi gogonebis odenoba aris biWebis odenobis 25%.
a) ramdenjer metia am klasSi biWebis odenoba gogonebis odeno-baze?
b) ramdeni gogonaa am klasSi?
4 wignis gverdebis gadanomvras dasWirda 2292 cifri. ramdeni danom-rili furcelia wignSi? (gverdebis gadanomvra iwyeba 1-dan).
5 gamoTvaleT 445 : ( 12
25-415 )+ 15
16 415 .
j
g
43
1. 6 geometriuli figurebi
1) wertili, wrfe, monakveTi, sxivi. wrfeebis urTierTganlagebis SemTxvevebi
Tqven ukve icnobT mraval geo-metriul figuras, amitom ar gagi-WirdebaT gaixsenoT im figuraTa dasaxelebebi, romlebic su raT zea gamosaxuli: kuTxe, sam kuTxe di, wre, kvadrati.
yoveli geometriuli figura wer ti lebisgan Sedgeba _ wertilTa nebismieri simravle geometriuli figuraa.
• gaixseneT _ ra Tvisebis wertilTa simravlea wrewiri?• ra Tvisebis wertilTa simravlea wre?suraTze brtyeli figurebia gamosaxuli — TiToeuli figuris yoveli wer-
tili erTsa da imave sibrtyes ekuTvnis.
sibrtyeze, an mis nawilze mainc, warmodgenas gviqmnis, maga li Tad, magidis, fanjris minis, uZravi wylis zedapiri.
sibrtye SemousazRvreli, usasrulo figuraa. suraTze mxo lod misi nawilia gamosaxuli. un da warmovidginoT, rom sibrtye am nawilis yvela mimarTulebiT usasrulo `gagr-Ze le biT~ miiReba.
am suraTze zogierTi sivrculi figuraa gamosaxuli.
davasaxeloT isini:
kubi, marTkuTxa paralelepipedi, piramida, cilindri, konusi, birTvi.
am suraTze monakveTia gamosaxuli — AB monakve-Ti. A da B wertilebi AB monakveTis boloebia. am bo-loebis mqone mxolod erTi monakveTi arsebobs da is
suraTzea gamo sa xu li. es monakveTi SeiZleba aRvniSnoT rogorc AB, ise — BA saxis CanaweriT.
wertilebi, romlebic monakveTs ekuTvnis da ar emTxveva bo loebs, monakve-Tis Siga wer ti lebia.
C wertili AB monakveTis Siga wertilia. igi A-sa da B-s Sorisaa.
rveulis furcelze monakveTi umjobesia saxazavis saSualebiT gamosaxoT (gaavloT).
44
saxlis saZirkvlis Cayrisas, ezos Se mo-Rob visas da sxva miznebisTvis xSirad gvWir-de ba mo nakveTis gamosaxva. winaswar moniSnul wer tilebSi CavarWobT or palos da maTze gav Wi mavT Toks. To kis gaswvriv carcis, sa-Re bavis an raime wvetiani xelsawyos sa Sua-le biT mov niS navT monakveTis gamosaxulebas. am SemTxvevaSi xSi rad vambobT — swori xazi gavavleTo.
vTqvaT, gvsurs SedarebiT grZeli mo-nakveTis ga mosaxva. gvaqvs paloebis sak-marisi odenoba, mag ram Tokis sigrZe ar gvyofnis.
daakvirdiT suraTs da aRwereT, rogor xdeba monakveTis gamosaxva am SemTxvevaSi.
am suraTze B wertili AC monakveTis Siga wertilia. SeiZleba vTqvaT: AC monakveTi miviReT AB-s `gagrZele-biT~ B wertilis mxares. AB monakveTis usazRvrod gagrZelebiT B-s mxares miiReba sxivi — AB sxivi.
A wertili AB sxivis saTavea. BA sxi vis saTave B wertilia.
AB-s gagrZelebiT usazRvrod orive mxares mi-iReba wrfe — AB wrfe, romelic A da B wertile-bze gadis. raime sibr t yis yovel or wertilze erTaderTi wrfe gadis. es wrfe am sib rtyeze mdebareobs (sibrtyeze Zevs).
aq A da B wertilebze gamavali wrfe a asoTi aRvniSneT.
• a wrfe wertilTa simravlea, A da B wertilebi am simrav les ekuTvnis. gaixseneT da CawereT es faq-
ti (elementi ekuT v nis simravles).
2) wrfeebis da sibrtyeebis urTierTganlagebis SemTxvevebi
suraTze erT sibrtyeze mdebare sami wrfea gamosax-uli: a, b da c. amasTanave, a da c wrfeebi ikveTeba, isini gadamkveTi wrfeebia. M wertili a da c wrfeebis saerTo wertilia — wrfeTa gadakveTis wertilia.
aris Tu ara b da c gadamkveTi wrfeebi? Tu aris, ra wertilSi ikveTeba es wrfeebi?
Tu erT sibrtyeze mdebare a da b wrfeebs saerTo wertili ara aqvs, maSin am wrfeebs paraleluri wrfeebi ewodeba da vwerT: a||b.
45
suraTze erT sibrtyeze mdebare a wrfe da M wertilia gamosaxuli, M∉a, anu, M wer tili ar ekuT-vnis a wrfes. M wertilze amave sibrtyeze mdebare ramdenime wrfea gavlebuli. maTgan mxolod er Ti SeiZleba iyos a wrfis paraleluri. yvela sxva wrfe a wrfes kveTs.
sibrtyis wertilze, romelic mocemul wrfes ar ekuTvnis gadis am wrfis pa ra leluri erTaderTi wrfe.
maSasadame, ori wrfe paraleluria, Tu isini erT si-brtyeze mdeba re o bs da maT saerTo wertili ara aqvs.
suraTze TqvenTvis nacnobi sivrculi figura — marT-kuTxa para le lepipedia gamosaxuli. gavixsenoT, rom AA
1,
AB da AD monakveTebi mi si wiboebia. daasaxeleT sxva wiboe-bi. A, A
1, B, C
1 am paralelepi pedis wveroebia. daasaxeleT
sxva wveroebi. AA1D
1D marT kuTxedi mi si erT-erTi waxnagia.
daasaxeleT sxva marTkuTxedebi — waxna ge bi.
warmovidginoT AA1 da BB
1 wrfeebi. pirveli maTgani A da
A1 wer tilebze gadis, meore — B da B
1-ze. es wrfeebi erT
sibrtyeze mdebareobs. amasTanave, am wrfeebs saerTo wertili ara aqvs, isini paraleluri wrfeebia: AA
1||BB
1. AA
1 da B
1D
1 wrfeebsac ara aqvs saerTo wer tili,
magram isini sxvadasxva sib r tyeze mdebareobs — isini paraleluri wrfeebi ar aris — acdenili wrfeebia. da asaxeleT sxva wrfeebi, romlebic paraleluri ar aris da erT sibrtyeze ar mdeba reobs. warmovidginoT sibrtyeebi, romle-bic ABCD da A
1B
1C
1D
1 wax nagebze gadis. am sibrtyeebs saerTo wertili ara aqvs
— isini paraleluri sib rtye ebia. ABCD da AA1B
1B waxnagebze gamavali sibrty-
eebi ki ikveTeba — isini AB wrfeze ik ve Te ba.
D1C
1 wrfe AA
1B
1B sibrtyis paralelur sibrtyezea, D
1C
1 wrfes ara aqvs saer-
To wertili AA1B
1B-sTan, D
1C
1 paraleluria AA
1B
1B sibrtyis, xolo D
1B
1 wrfe
kveTs AA1B
1B sibrtyes B
1 wertilSi.
m||n
SevajamoT: gavixseneT zogierTi brtyeli da sivr cu li figura. ganvi xileT wrfe-ebisa da sibrtyeebis ur Ti erT gan-la gebis Se mTxve vebi. magaliTad, a wrfes ara aqvs saerTo wertili α si-brtyesTan, a pa ra le luria α-si, a||α. b wrfe ki kveTs β sibrtyes M wer-tilSi.
46
Zveli welTaRricxvis IV-III saukuneebSi moRvawe didi
berZeni ge o metri evklide ase `gansazRvravda~ wer tils: `wer ti li aris is, rasac na wi lebi ara aqvs~.
zogierTi maTematikuri cnebis Se sa bamisi qarTuli
termi ne bis Se mo Re basa da damkvidrebaze zrunva Tbi li sis
saxelmwifo uni ver sitetis da ar sebis pirveli dRe ebidanve
daiw yo. saq mes sa Ta veSi gio r gi niko laZe (1888-1931) Caud-ga. am muSobis Sedegia ara marto maTe ma tikuri terminebis (wrfe, wer tili, wrewiri, wiri, ...), aramed sxva araerTi terminis Se mo Re bac. ma ga liTad: wneva, deni, tan var jiSi, mTa msvle lo ba da a. S.
1. ramdeni saerTo wertili SeiZleba hqondes or wrfes?
2. ra SemTxvevaSi ewodeba or wrfes gadamkveTi wrfeebi?
3. ra geometriuli figuraa gadamkveTi wrfeebis saerTo nawili?
4. ra geometriuli figuraa monakveTi, sxivi? aRwereT maTi saerTo da ganmasxvavebeli Tvisebebi.
5. ra ewodeba wertils, romliTac aris sxivi SemosazRvruli?
6. sworia Tu ara, rom • sxivi aris wrfe, romelsac aqvs saTave da bolo; • sxivi aris monakveTi, romelsac mxolod erTi bolo aqvs; • monakveTi aris wrfe, romelic ori wertiliTaa SemosazRvruli; • monakveTi wrfis nawilia.
7. SeiZleba Tu ara AB monakveTi ase CavweroT: BA monakveTi?
8. suraTze AB sxivia gamosaxuli. SeiZleba Tu ara is asec CavweroT: BA sxivi.
9. furcelze moniSnulia C da D wertilebi. vTqvaT, mariamma gaavlo CD wrfe, ninikom _ DC wrfe. sxvadasxvaa es wrfeebi?
evklide
giorgi nikolaZe
s
47
1 risgan Sedgeba yoveli geometriuli figura?
1) monakveTisgan 2) wertilebisgan 3) wrfeebisgan.
2 vTqvaT, N-iT raime wertilia aRniSnuli, a asoTi — wrfe. ras niS-navs Canaweri: N∈a?
1) a wrfe ar gadis N wertilze 2) N wrfe gadis a wertilze 3) a wrfe gadis N wertilze 4) N ar aris a wrfis wertili.
3 vTqvaT, M wertilia, b — wrfe. ras niSnavs Canaweri: M∉b?
1) M wrfe gadis b wertilze 2) M aris b wrfis wertili 3) b wrfe ar gadis M wertilze 4) b wrfe gadis M wertilze.
4 nebismier or wertilze gadis 1) ori wrfe 2) sami wrfe
3) uamravi wrfe 4) erTaderTi wrfe.
5 ori wrfis gadakveTis wertilia is wertili, romelic 1) mxolod erT-erT wrfes ekuTvnis 2) mocemul wrfeTagan arc erTs ar ekuTvnis 3) orive mocemul wrfes ekuTvnis.
6 Tu M wertili marTkuTxa paralelepipedis AA
1 wibos ekuTvnis, maSin is ekuTv nis
1) A1B
1C
1D
1 waxnags
2) ABCD waxnags 3) AA
1B
1B da AA
1D
1D waxnagebs
4) BB1C
1C da DD
1C
1C waxnagebs.
7 warmodgenili marTkuTxa paralelepipedis A wve rodan `gamo su li~ wiboebia
1) AA1, AB, AC da AD
2) AA1, AB da AD
3) mxolod AB da AD
4) mxolod AA1 da AD.
8 suraTze gamosaxuli geo met riuli figurebidan ro me lia brtyeli?
a
SearCieT swori pasuxi
amoxseniT amocanebi
48
9 suraTze gamosaxulia a wrfe da wertilebi. daasaxeleT wertilebi, romlebic:
a) a wrfes ekuTvnis; b) a wrfes ar ekuTvnis.
10 gaavleT a wrfe, aRniSneT M da K wertilebi, romlebic ekuTvnis a wrfes, E da F wertilebi, romlebic ar ekuTvnis a wrfes.
11 sibrtyeze moniSneT M da N wertilebi. gaavleT a wrfe ise, rom M∈a da N∉a. b wrfe ki gaavleT ise, rom M∈b da N∈b.
12 suraTis mixedviT: a) daasaxeleT wrfe, ro-melic gadis A da B wertileb ze;
b) da a sa xeleT wrfe, romelic gadis A wertil-ze, B wertilze ki — ara;
g) daasaxeleT wertilebi, romlebic ar ekuTv-nis a wrfes;
d) daasaxeleT wertilebi, romlebic ar ekuT-vnis arc a da arc b wrfes;
e) daasaxeleT wertilebi, romlebic am wrfeTagan erT-erTs mainc ekuTvnis;
v) daasaxeleT wertilebi, romlebic am wrfeTagan mxolod erTs ekuTvnis;
z) wina kiTxvebis pasuxebis mixedviT SeavseT cxrili:
13 SeasruleT naxazi: sibrtyeze moniSneT A da B wertilebi. gaavleT a wrfe, romelic gadis A wertilze, B-ze ki — ara. gaavleT b wrfe, romelic B wertilze gadis, A-ze ki — ara. gaavleT c wrfe, romelic gadis A da B wertilebze.
14 sibrtyeze moniSneT A, B, C wertilebi da gaavleT a, b, c wrfeebi ise, rom Sesruldes Semdegi pirobebi: A∈a, B∈a, B∈b, C∈b, A∈c, C∈c.
15 vTqvaT, mocemulia a da b sxvadasxva wrfeebi. SeiZleba Tu ara am wrfeebs ori an meti saerTo wertili hqondes?
ar ekuTvnis a wrfes
ar ekuTvnis b wrfes
ar ekuTvnis arc a-s da arc b-s
erT-erT wrfes mainc ekuTvnis
mxolod erT wrfes ekuTvnis
rv
eul
Si!
49
16 suraTze marTkuTxa paralelepipedia gamo-saxu li. aq moniSnul wertilTagan romeli:
a) ekuTvnis ABCD marTkuTxedis sibrtyes;
b) ar ekuTvnis ABCD marTkuTxedis sibrtyes;
g) ekuTvnis A1B
1C
1D
1 marTkuTxedis sibrtyes;
d) ar ekuTvnis AA1B
1B marTkuTxedis si-
brtyes?
17 warmodgenili marTkuTxa paralelepipedis moni Snul wertilebze gamaval wrfe Tagan rome-li:
a) mdebareobs ABCD marTkuTxedis sibrtye-ze;
b) kveTs ABCD marTkuTxedis sibrtyes;
g) mdebareobs AA1C
1C marTkuTxedis sibrtyeze;
d) kveTs BB1D
1D marTkuTxedis sibrtyes?
18 warmodgenili suraTis mixedviT ipoveT: a) moniSnul wertilebze gamaval parale-
lur wrfeTa ramdenime wyvili;
b) moniSnul wertilebze gamaval acdenil wrfe Ta ramdenime wyvili;
g) wrfe, romelzec ikveTeba ABCD da AA1B
1B
marTkuTxedebis sibrtyeebi;
d) wrfe, romelzec ikveTeba AA1C
1C da
BB1D
1D marTkuTxedebis sibrtyeebi;
e) paralelur sibrtyeTa sami wyvili;
v) gadamkveT sibrtyeTa sami wyvili gadakveTis wrfeebis miTiTebiT.
19 suraTze gamosaxuli piramidis mixedviT SeavseT cxrili.
20 suraTze marTkuTxa paralelepipedia gamo-saxuli. da asa xe leT:
a) ABCD marTkuTxedis sibrtyis paraleluri sami wrfe;
b) BB1D
1D marTkuTxedis sibrtyis paraleluri
wrfe;
acdenil wrfeTa wyvilebi
saerTo wertilis mqonewrfeTa wyvilebi
im sibrtyeTa wyvili, romlebic ikveTeba BC wrfeze
im sibrtyeTa wyvili, romlebic ikveTeba AM wrfeze
rv
eul
Si!
50
g) AB1C
1D marTkuTxedis sibrtyis paraleluri ori wrfe;
d) AD wrfis paraleluri sami sibrtye;
e) sibrtye, romelic DC1 wrfis paraleluria da gadis AB wiboze;
v) sibrtye, romelic DD1 wrfis paraleluria da gadis AA
1 wiboze;
z) sibrtye, romelic BC wrfis paraleluria da gadis B1D wrfeze.
21 ramdennairad SeiZleba suraTze warmodgenili AB wrfis dasaxeleba?
22 ramdennairad SeiZleba suraTze warmo d ge-nili AB wrfis dasaxeleba?
23 wrfeze moniSnuli 10 wertili asoebiTaa aRniSnuli. ramdennairad SeiZleba am wrfis dasaxeleba?
24 ramdennairad SeiZleba am AB sxivis dasa-xeleba?
25 ramdeni sxvadasxva sxivia gamosaxuli su-raTze?
