Lub hauv paus daim ntawv yog cov lus piav qhia algebraic uas muaj lub cim ntawm lub hauv paus square (lossis lub voos xwmfab hauv paus lossis siab dua). Daim ntawv no feem ntau tuaj yeem sawv cev rau ob tus lej uas muaj tus nqi zoo ib yam txawm hais tias lawv yuav tshwm sim sib txawv thaum xub thawj siab ib muag (piv txwv li, 1/(sqrt (2) - 1) = sqrt (2) +1). Yog li ntawd, peb xav tau "tus qauv txheem" rau hom ntawv no. Yog tias muaj ob nqe lus, ob qho tib si hauv tus qauv qauv, uas pom sib txawv, lawv tsis zoo ib yam. Mathematicians pom zoo tias tus qauv tsim ntawm daim duab plaub ua tau raws cov hauv qab no:
- Zam kev siv cov feem faib
- Tsis txhob siv lub zog faib ua feem
- Zam kev siv cov hauv paus hauv cov lej
- Tsis muaj qhov sib npaug ntawm ob daim ntawv hauv paus
- Cov lej hauv qab lub hauv paus tuaj yeem tsis muaj hauv paus ntxiv lawm
Ib qho siv tau ntawm qhov no yog hauv kev xaiv ntau qhov kev xeem. Thaum koj pom cov lus teb, tab sis koj cov lus teb tsis zoo ib yam li cov kev xaiv muaj, sim ua kom yooj yim rau hauv tus qauv qauv. Txij li cov neeg tsim cov lus nug feem ntau sau cov lus teb hauv cov qauv qauv, ua tib yam nrog koj cov lus teb kom phim lawv. Hauv cov lus nug sau ntawv, cov lus txib xws li "ua kom yooj yim rau koj cov lus teb" lossis "ua kom yooj yim rau txhua lub hauv paus" txhais tau tias cov tub ntxhais kawm yuav tsum ua cov theem hauv qab no kom txog thaum lawv ua tau raws li cov qauv txheej txheem saum toj no. Cov kauj ruam no tseem tuaj yeem siv los daws qhov sib npaug, txawm hais tias qee hom kev ua zauv yooj yim dua los daws cov qauv uas tsis yog tus qauv.
Kauj ruam
Kauj Ruam 1. Yog tias tsim nyog, tshuaj xyuas cov cai rau kev ua haujlwm cov hauv paus hniav thiab nthuav tawm (ob qho tib si yog sib npaug - cov hauv paus yog lub zog ntawm kev faib ua feem) raws li peb xav tau hauv txheej txheem no
Kuj tshuaj xyuas cov cai kom yooj yim rau ntau tus lej thiab cov ntaub ntawv tsim nyog raws li peb yuav xav tau kom yooj yim rau lawv.
Txoj Kev 1 ntawm 6: Squares Zoo Tshaj
Kauj Ruam 1. Yooj yim rau txhua lub hauv paus uas muaj cov xwm txheej zoo meej
Ib lub xwmfab zoo tshaj plaws yog cov khoom ntawm tus lej ntawm nws tus kheej, piv txwv li 81, uas yog cov khoom lag luam ntawm 9 x 9. Txhawm rau ua kom yooj yim ib lub xwmfab zoo meej, tsuas yog tshem cov hauv paus plaub fab thiab sau cov hauv paus plaub ntawm tus lej.
- Piv txwv, 121 yog lub xwmfab zoo tshaj plaws vim tias 11 x 11 sib npaug 121. Yog li, koj tuaj yeem yooj yim lub hauv paus (121) txog 11, los ntawm kev tshem lub hauv paus cim.
- Txhawm rau ua kom cov kauj ruam no yooj yim dua, koj yuav tsum nco ntsoov kaum ob qhov ua tau zoo tshaj plaws: 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, 6 x 6 = 36, 7 x 7 = 49, 8 x 8 = 64, 9 x 9 = 81, 10 x 10 = 100, 11 x 11 = 121, 12 x 12 = 144
Kauj Ruam 2. Ua kom yooj yim tag nrho cov hauv paus hniav uas muaj cov ntsuas zoo meej
Lub voos xwmfab zoo meej yog cov khoom ntawm muab tus lej los ntawm nws tus kheej ob zaug, piv txwv li 27, uas yog cov khoom ntawm 3 x 3 x 3. Txhawm rau kom yooj yim hauv paus daim ntawv ntawm lub voos xwmfab zoo meej, tsuas yog tshem cov hauv paus cag thiab sau cov hauv paus cag ntawm tus lej.
Piv txwv, 343 yog lub voos xwmfab zoo meej vim nws yog cov khoom ntawm 7 x 7 x 7. Yog li lub hauv paus cube ntawm 343 yog 7
Txoj Kev 2 ntawm 6: Hloov Fractions rau Roots
Los yog hloov lwm txoj hauv kev ib puag ncig (nws pab qee zaum), tab sis tsis txhob sib xyaw lawv hauv tib nqe lus li hauv paus (5) + 5^(3/2). Peb yuav xav tias koj xav siv daim ntawv hauv paus thiab peb yuav siv cov cim hauv paus (n) rau cov hauv paus cag thiab sqrt^3 (n) rau lub hauv paus cube.
Kauj Ruam 1. Coj ib qho mus rau lub zog ntawm cov feem thiab hloov nws mus rau hauv paus daim ntawv, piv txwv li x^(a/b) = hauv paus rau b lub zog ntawm x^a
Yog tias lub hauv paus square yog nyob hauv daim ntawv fraction, hloov nws mus rau daim ntawv tsis tu ncua. Piv txwv, square hauv paus (2/3) ntawm 4 = hauv paus (4)^3 = 2^3 = 8
Kauj Ruam 2. Hloov cov zauv tsis zoo rau feem feem ntau, piv txwv li x^-y = 1/x^y
Cov mis no tsuas yog siv rau qhov nthuav tawm thiab tsim nyog qhov nthuav tawm. Yog tias koj tab tom cuam tshuam nrog daim ntawv zoo li 2^x, tsis txhob hloov nws, txawm tias qhov teeb meem qhia tias x tuaj yeem yog ib feem lossis tus lej tsis zoo
Kauj ruam 3. Ua ke tib pab pawg thiab ua kom yooj yim rau qhov tsim nyog tau txais daim ntawv.
Txoj Kev 3 ntawm 6: Tshem Tawm Fractions hauv Cov Hauv Paus
Cov qauv txheej txheem xav kom lub hauv paus yog tus lej.
Kauj Ruam 1. Saib tus lej hauv qab lub hauv paus cag yog tias nws tseem muaj ib feem
Yog tseem,…
Kauj Ruam 2. Hloov mus rau ib feem uas muaj ob lub hauv paus siv lub hauv paus tus kheej (a/b) = sqrt (a)/sqrt (b)
Tsis txhob siv tus kheej no yog tus lej tsis zoo, lossis yog nws yog qhov txawv txav uas yuav tsis zoo. Hauv qhov no, yooj yim feem ua ntej
Kauj Ruam 3. Ua kom yooj yim txhua qhov xwm txheej zoo ntawm qhov tshwm sim
Ntawd yog, hloov sqrt (5/4) rau sqrt (5)/sqrt (4), tom qab ntawd yooj yim rau sqrt (5)/2.
Kauj Ruam 4. Siv lwm txoj hauv kev yooj yim xws li ua kom yooj yim sib faib, sib txuas cov ntsiab lus sib luag, thiab lwm yam
Txoj Kev 4 ntawm 6: Muab Cov Hauv Paus Sib Ntxiv
Kauj Ruam 1. Yog tias koj tab tom muab ib daim ntawv cag los ntawm lwm qhov, ua ke ob qho hauv ib lub hauv paus cag siv cov qauv:
sqrt (a)*sqrt (b) = sqrt (ab). Piv txwv li, hloov hauv paus (2)*hauv paus (6) rau hauv paus (12).
- Tus kheej saum toj no, sqrt (a)*sqrt (b) = sqrt (ab), siv tau yog tias tus lej hauv qab kos npe ntawm sqrt tsis zoo. Tsis txhob siv cov mis no thaum a thiab b tsis zoo vim tias koj yuav ua yuam kev ntawm ua sqrt (-1)*sqrt (-1) = sqrt (1). Cov lus ntawm sab laug yog sib npaug rau -1 (lossis tsis tau txhais yog tias koj tsis siv cov lej sib txawv) thaum cov lus ntawm sab xis yog +1. Yog tias a thiab/lossis b tsis zoo, ua ntej "hloov" lub cim zoo li sqrt (-5) = i*sqrt (5). Yog tias daim ntawv hauv qab lub hauv paus kos npe yog qhov sib txawv uas nws lub cim tsis paub los ntawm cov ntsiab lus lossis tuaj yeem ua qhov zoo lossis tsis zoo, tso nws raws li nws yog rau lub sijhawm tam sim no. Koj tuaj yeem siv tus kheej ntiag tug ntau dua, sqrt (a)*sqrt (b) = sqrt (sgn (a))*sqrt (sgn (b))*sqrt (| ab |) uas siv rau txhua tus lej tiag a thiab b, tab sis feem ntau cov mis no tsis pab ntau vim nws ntxiv qhov nyuaj rau siv sgn (kos npe) ua haujlwm.
- Tus neeg no yog siv tau yog tias cov hauv paus hniav muaj tib lub zauv piav qhia. Koj tuaj yeem muab cov cag sib txawv sib txawv xws li sqrt (5)*sqrt^3 (7) los ntawm kev hloov lawv mus rau tib lub hauv paus cag. Txhawm rau ua qhov no, hloov pauv ib puag ncig ib ntus rau ib feem: sqrt (5) * sqrt^3 (7) = 5^(1/2) * 7^(1/3) = 5^(3/6) * 7 ^(2/6) = 125^(1/6) * 49^(1/6). Tom qab ntawd siv txoj cai sib npaug los sib npaug ob mus rau cov hauv paus cag ntawm 6125.
Txoj Kev 5 ntawm 6: Tshem Tawm Cov Square Factor los ntawm Root
Kauj Ruam 1. Txheeb xyuas cov hauv paus hniav tsis zoo rau qhov tseem ceeb
Ib qho tseem ceeb yog tus lej uas thaum sib tw los ntawm lwm tus lej ua tus lej - piv txwv li, 5 thiab 4 yog ob yam ntawm 20. Txhawm rau txhawm rau txhawm rau cov hauv paus hniav tsis zoo, sau txhua yam ntawm tus lej (lossis ntau li ntau tau, yog tias tus lej loj dhau) kom txog thaum koj pom lub xwmfab zoo meej.
Piv txwv, sim nrhiav txhua yam ntawm 45: 1, 3, 5, 9, 15, thiab 45. 9 yog qhov tseem ceeb ntawm 45 thiab tseem yog lub xwmfab zoo meej (9 = 3^2). 9 x5 = 45
Kauj Ruam 2. Tshem tawm txhua qhov sib npaug uas zoo sib xws los ntawm hauv paus hauv paus
9 yog lub xwmfab zoo tshaj plaws vim tias nws yog cov khoom ntawm 3 x 3. Siv 9 tawm ntawm cov hauv paus cag thiab hloov nws nrog 3 nyob rau hauv ntej ntawm cov hauv paus plaub fab, tawm hauv 5 sab hauv lub hauv paus cag. Yog tias koj "muab" 3 rov qab rau hauv cov hauv paus hauv paus, sib npaug los ntawm nws tus kheej kom ua 9, thiab yog tias koj sib npaug 5 nws rov qab 45. 3 cov hauv paus hniav ntawm 5 yog txoj hauv kev yooj yim ntawm kev qhia lub hauv paus ntawm 45.
Ntawd yog, sqrt (45) = sqrt (9*5) = sqrt (9)*sqrt (5) = 3*sqrt (5)
Kauj Ruam 3. Nrhiav lub xwmfab zoo tshaj plaws hauv qhov sib txawv
Cov hauv paus plaub fab ntawm ib lub xwmfab yog | a |. Koj tuaj yeem yooj yim qhov no tsuas yog "a" yog tias paub qhov sib txawv yog qhov zoo. Lub hauv paus cag ntawm a rau lub zog ntawm 3 thaum tawg mus rau lub hauv paus cag ntawm lub sijhawm sib npaug a - nco ntsoov tias cov neeg nthuav tawm ntxiv thaum peb muab ob tus lej rau lub zog ntawm a, yog li cov plaub npaug sib npaug sib npaug a peb zog.
Yog li ntawd, lub xwmfab zoo tshaj plaws nyob rau hauv daim ntawv cubed yog ib lub xwmfab
Kauj Ruam 4. Tshem tawm qhov sib txawv uas muaj cov xwm txheej zoo meej los ntawm cov hauv paus cag
Tam sim no, siv ib lub xwmfab los ntawm cov hauv paus cag thiab hloov nws mus rau | a |. Daim ntawv yooj yim ntawm cov hauv paus a rau lub zog ntawm 3 yog | a | hauv paus a.
Kauj Ruam 5. Muab cov ntsiab lus sib npaug thiab ua kom yooj yim rau txhua lub hauv paus ntawm cov txiaj ntsig kev suav
Txoj Kev 6 ntawm 6: Rationalize tus Denominator
Kauj Ruam 1. Cov qauv txheej txheem xav kom tus lej faib ua ib tus lej (lossis ntau tus lej yog tias nws muaj qhov sib txawv) ntau li ntau tau
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Yog tus lej suav nrog ib lo lus hauv qab lub hauv paus kos npe, xws li […]/hauv paus (5), tom qab ntawd muab ob tus lej thiab tus lej los ntawm lub hauv paus kom tau […]*sqrt (5)/sqrt (5)*sqrt (5) = […]*hauv paus (5)/5.
Rau cov hauv paus cag lossis siab dua, muab cov cag los tsim nyog kom cov lej sib faib yog qhov tsim nyog. Yog tias tus lej yog lub hauv paus^3 (5), muab tus lej thiab tus lej faib ua sqrt^3 (5)^2
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Yog tias tus lej suav nrog suav nrog ntxiv lossis rho tawm ob lub hauv paus plaub fab xws li sqrt (2) + sqrt (6), muab cov lej sib faib thiab cov lej los ntawm lawv cov ntawv sib txuas, uas yog tib daim ntawv tab sis nrog cov cim rov qab. Tom qab ntawd […]/(hauv paus (2) + hauv paus (6)) = […] (hauv paus (2) -root (6))/(hauv paus (2) + hauv paus (6)) (hauv paus (2) -root (6)). Tom qab ntawd siv tus lej tus lej rau qhov sib txawv ntawm ob lub xwmfab [(a + b) (ab) = a^2-b^2] txhawm rau txiav txim siab tus lej, kom yooj yim (sqrt (2) + sqrt (6)) (sqrt (2) -sqrt (6)) = sqrt (2)^2 -sqrt (6)^2 = 2-6 = -4.
- Qhov no kuj siv rau cov lej xws li 5 + sqrt (3) vim tias txhua tus lej yog cov hauv paus ntawm lwm cov lej. [1/(5 + sqrt (3)) = (5-sqrt (3))/(5 + sqrt (3)) (5-sqrt (3)) = (5-sqrt (3))/(5^ 2-sqrt (3)^2) = (5-sqrt (3))/(25-3) = (5-sqrt (3))/22]
- Txoj kev no tseem siv rau kev ntxiv cov hauv paus hniav xws li sqrt (5) -sqrt (6)+sqrt (7). Yog tias koj muab lawv faib ua (sqrt (5) -sqrt (6))+sqrt (7) thiab muab sib ntxiv los ntawm (sqrt (5) -sqrt (6))-sqrt (7), cov lus teb tsis yog nyob rau hauv daim ntawv xav, tab sis tseem nyob hauv a+b*hauv paus (30) qhov twg a thiab b twb yog cov lej muaj txiaj ntsig. Tom qab ntawd rov ua cov txheej txheem nrog cov conjugates a+b*sqrt (30) thiab (a+b*sqrt (30)) (a-b*sqrt (30)) yuav yog qhov tseeb. Hauv qhov tseeb, yog tias koj tuaj yeem siv qhov ua kom yuam kev no kom tshem tawm ib lub hauv paus cim hauv tus lej, koj tuaj yeem rov ua nws ntau zaus kom tshem tag nrho cov hauv paus hniav.
- Txoj hauv kev no tseem tuaj yeem siv rau cov lej uas muaj lub hauv paus siab dua, xws li lub hauv paus plaub ntawm 3 lossis lub hauv paus thib xya ntawm 9. Muab cov zauv sib faib thiab cov zauv los ntawm cov zauv sib faib. Hmoov tsis zoo, peb tsis tuaj yeem tau txais qhov sib txuas ntawm tus lej thiab nws nyuaj rau ua qhov ntawd. Peb tuaj yeem nrhiav cov lus teb hauv phau ntawv algebra ntawm tus lej lej, tab sis kuv yuav tsis nkag mus rau qhov ntawd.
Kauj Ruam 2. Tam sim no tus lej yog nyob hauv daim ntawv xav tau, tab sis tus zauv zoo li tsis meej pem
Tam sim no txhua yam koj yuav tsum tau ua yog sib npaug nws los ntawm kev sib txuas ntawm cov lej. Mus tom ntej thiab sib npaug raws li peb yuav muab cov lej sib npaug. Txheeb xyuas seb puas muaj cov ntsiab lus twg tuaj yeem raug tshem tawm, yooj yim dua, lossis ua ke, yog ua tau.
Kauj Ruam 3. Yog tias tus lej lej yog tus lej tsis zoo, muab ob tus lej thiab cov lej faib ua -1 kom nws zoo
Lub tswv yim
- Koj tuaj yeem tshawb online rau cov chaw uas tuaj yeem pab ua kom yooj yim hauv paus cov ntawv. Tsuas yog ntaus qhov sib npaug nrog lub hauv paus kos npe, thiab tom qab nias Enter, cov lus teb yuav tshwm sim.
- Rau cov lus nug yooj yim, koj yuav tsis siv txhua qib hauv kab lus no. Rau cov lus nug nyuaj dua, koj yuav tsum siv ob peb kauj ruam ntau dua ib zaug. Siv cov "yooj yim" cov kauj ruam ob peb zaug, thiab txheeb xyuas seb koj cov lus teb haum rau cov qauv txheej txheem uas peb tau tham txog ua ntej. Yog tias koj cov lus teb nyob hauv tus qauv mis, koj ua tiav; tab sis yog tsis yog, koj tuaj yeem tshawb xyuas ib qho ntawm cov kauj ruam saum toj no los pab koj ua tiav.
- Feem ntau cov ntawv xa mus rau "pom zoo tus qauv mis" rau daim ntawv ntawm cov hauv paus kuj siv rau cov lej sib txawv (i = hauv paus (-1)). Txawm hais tias nqe lus muaj "i" hloov lub hauv paus, zam cov lej uas tseem muaj tus i ntau li ntau tau.
- Qee cov lus qhia hauv kab lus no xav tias txhua lub hauv paus yog plaub fab. Cov hauv paus ntsiab lus zoo ib yam siv rau cov hauv paus ntawm kev muaj hwj chim siab dua, txawm hais tias qee qhov (tshwj xeeb tshaj yog kev txiav txim siab tus lej) tuaj yeem nyuaj ua haujlwm nrog. Txiav txim siab rau koj tus kheej tias koj xav tau lub cev zoo li cas, xws li sqr^3 (4) lossis sqr^3 (2)^2. (Kuv tsis nco qab tias daim ntawv twg feem ntau pom zoo hauv phau ntawv).
- Qee cov lus qhia hauv kab lus no siv lo lus "tus qauv txheem" los piav qhia "daim ntawv tsis tu ncua". Qhov sib txawv yog tias tus qauv qauv tsuas lees txais daim ntawv 1+sqrt (2) lossis sqrt (2) +1 thiab txiav txim siab rau lwm daim ntawv raws li tsis yog tus qauv; Plain form xav tias koj, tus nyeem ntawv, ntse txaus kom pom qhov "zoo sib xws" ntawm ob tus lej no txawm hais tias lawv tsis zoo ib yam hauv kev sau ntawv ('tib yam' txhais tau tias hauv lawv cov lej lej (ua haujlwm sib ntxiv), tsis yog lawv cov cuab yeej algebraic (hauv paus) (2) yog lub hauv paus tsis-tsis zoo ntawm x^2-2)). Peb cia siab tias cov nyeem yuav nkag siab txog qhov tsis saib xyuas me ntsis hauv kev siv cov ntsiab lus no.
- Yog tias ib qho ntawm cov ntsiab lus zoo li tsis meej pem lossis tsis sib xws, ua txhua kauj ruam uas tsis meej thiab tsis sib xws, thiab tom qab ntawd xaiv qhov twg los xij uas koj xav tau.
