Kvadrátno iracionálno števílo (redkeje tudi kvadrátni súrd ) je v matematiki algebrsko iracionalno število , ki je rešitev kakšne kvadratne enačbe z racionalnimi koeficienti. Ker se lahko iz kvadratne enačbe ulomke poniči z množenjem obeh strani z njihovima skupnima imenovalcema , se lahko reče, da je kvadratno iracionalno število koren kvadratne enačbe:
k x 2 + m x + n = 0 {\displaystyle kx^{2}+mx+n=0\!\,} s celimi koeficienti k {\displaystyle k\,} , m {\displaystyle m\,} in n {\displaystyle n\,} in z od nič različno diskriminanto m 2 − 4 k n {\displaystyle m^{2}-4kn\,} . Kvadratna iracionalna števila so oblike:
c , ( c > 1 ) {\displaystyle {\sqrt {c}},\qquad (c>1)\!\,} za cela števila c deljiva brez kvadrata . Vsako kvadratno iracionalno število pa se lahko v splošnem zapiše kot:
a ± b c d , a , b , c , d ∈ Z , ( a , b > 0 , c > 1 , d ≠ 0 , d | a 2 − c ) , {\displaystyle {\frac {a\pm b{\sqrt {c}}}{d}},\qquad a,b,c,d\in \mathbb {Z} ,\qquad (a,b>0,c>1,d\neq 0,d|a^{2}-c)\!\,,} kjer c {\displaystyle c\,} ni popolni kvadrat .
To pomeni, da je moč njihove množice enaka množici urejenih trojic celih števil, in je zaradi tega števno neskončna .
Kvadratna iracionalna števila z danim c {\displaystyle c\,} tvorijo obseg , ki se imenuje kvadratni obseg .
Verižni ulomki kvadratnih iracionalnih števil
uredi
Enočlene oblike
uredi
Kvadratna iracionalna števila so posebna števila, še posebej v povezavi z verižnimi ulomki . Za vsa in edino za kvadratna iracionalna števila je razvoj v verižni ulomek periodičen . Na primer števila deljiva brez kvadrata:
2 = 1 , 4142 … = [ 1 ; 2 , … ] , {\displaystyle {\sqrt {2}}=1,4142\ldots =[1;2,\ldots ]\!\,,} 3 = 1 , 7320 … = [ 1 ; 1 , 2 , … ] , {\displaystyle {\sqrt {3}}=1,7320\ldots =[1;1,2,\ldots ]\!\,,} 5 = 2 , 2360 … = [ 2 ; 4 , … ] , {\displaystyle {\sqrt {5}}=2,2360\ldots =[2;4,\ldots ]\!\,,} 6 = 2 , 4494 … = [ 2 ; 2 , 4 , … ] , {\displaystyle {\sqrt {6}}=2,4494\ldots =[2;2,4,\ldots ]\!\,,} 7 = 2 , 6457 … = [ 2 ; 1 , 1 , 1 , 4 , … ] , {\displaystyle {\sqrt {7}}=2,6457\ldots =[2;1,1,1,4,\ldots ]\!\,,} 10 = 3 , 1622 … = [ 3 ; 6 , … ] , {\displaystyle {\sqrt {10}}=3,1622\ldots =[3;6,\ldots ]\!\,,} 11 = 3 , 3166 … = [ 3 ; 3 , 6 , … ] , {\displaystyle {\sqrt {11}}=3,3166\ldots =[3;3,6,\ldots ]\!\,,} 13 = 3 , 6055 … = [ 3 ; 1 , 1 , 1 , 1 , 6 , … ] , {\displaystyle {\sqrt {13}}=3,6055\ldots =[3;1,1,1,1,6,\ldots ]\!\,,} 14 = 3 , 7416 … = [ 3 ; 1 , 2 , 1 , 6 … ] , {\displaystyle {\sqrt {14}}=3,7416\ldots =[3;1,2,1,6\ldots ]\!\,,} 15 = 3 , 8729 … = [ 3 ; 1 , 6 , … ] . {\displaystyle {\sqrt {15}}=3,8729\ldots =[3;1,6,\ldots ]\!\,.} ali števila deljiva s kvadratom, ki niso kvadratna števila (OEIS A051144 ):
8 = 2 2 = 2 , 8284 … = [ 2 ; 1 , 4 , … ] , {\displaystyle {\sqrt {8}}=2{\sqrt {2}}=2,8284\ldots =[2;1,4,\ldots ]\!\,,} 12 = 2 3 = 3 , 4641 … = [ 3 ; 2 , 6 , … ] , {\displaystyle {\sqrt {12}}=2{\sqrt {3}}=3,4641\ldots =[3;2,6,\ldots ]\!\,,} 18 = 3 2 = 4 , 2426 … = [ 4 ; 4 , 8 , … ] , {\displaystyle {\sqrt {18}}=3{\sqrt {2}}=4,2426\ldots =[4;4,8,\ldots ]\!\,,} 20 = 2 5 = 4 , 4721 … = [ 4 ; 2 , 8 , … ] . {\displaystyle {\sqrt {20}}=2{\sqrt {5}}=4,4721\ldots =[4;2,8,\ldots ]\!\,.} Vsi verižni ulomki kvadratnih korenov števil, ki niso popolni kvadrati, imajo posebno obliko periodičnosti, palindromni niz števk:
prazen za števila oblike c = n 2 + 1 ; n > 0 {\displaystyle c=n^{2}+1;\ n>0\!\,} (OEIS A002522 ): 2 {\displaystyle {\sqrt {2}}\!\,} , 5 {\displaystyle {\sqrt {5}}\!\,} , 10 {\displaystyle {\sqrt {10}}\!\,} , 17 {\displaystyle {\sqrt {17}}\!\,} , 26 {\displaystyle {\sqrt {26}}\!\,} , 37 {\displaystyle {\sqrt {37}}\!\,} , 50 {\displaystyle {\sqrt {50}}\!\,} , 65 {\displaystyle {\sqrt {65}}\!\,} , ..., od katerih so praštevila (OEIS A002496 ): 2 {\displaystyle {\sqrt {2}}\!\,} , 5 {\displaystyle {\sqrt {5}}\!\,} , 17 {\displaystyle {\sqrt {17}}\!\,} , 37 {\displaystyle {\sqrt {37}}\!\,} , 101 {\displaystyle {\sqrt {101}}\!\,} , 197 {\displaystyle {\sqrt {197}}\!\,} , 257 {\displaystyle {\sqrt {257}}\!\,} , ... in sestavljena (OEIS A134406 ): 10 {\displaystyle {\sqrt {10}}\!\,} , 26 {\displaystyle {\sqrt {26}}\!\,} , 50 {\displaystyle {\sqrt {50}}\!\,} , 65 {\displaystyle {\sqrt {65}}\!\,} , 82 {\displaystyle {\sqrt {82}}\!\,} , 122 {\displaystyle {\sqrt {122}}\!\,} , 145 {\displaystyle {\sqrt {145}}\!\,} , 170 {\displaystyle {\sqrt {170}}\!\,} , ...Za ta števila tako velja:
c = [ a 0 ; 2 a 0 ¯ ] . {\displaystyle {\sqrt {c}}=[a_{0};{\overline {2a_{0}}}]\!\,.} na primer 1 za 3 {\displaystyle {\sqrt {3}}\,} , 1,1,1 za 7 {\displaystyle {\sqrt {7}}\,} , 1,2,1 za 14 {\displaystyle {\sqrt {14}}\,} , ki mu sledi dvakratnik vodilnega celega števila. Praštevila, ki niso oblike n 2 + 1 {\displaystyle n^{2}+1\,} , imajo neprazen niz (OEIS A070303 ): 3 {\displaystyle {\sqrt {3}}\!\,} , 7 {\displaystyle {\sqrt {7}}\!\,} , 11 {\displaystyle {\sqrt {11}}\!\,} , 13 {\displaystyle {\sqrt {13}}\!\,} , 19 {\displaystyle {\sqrt {19}}\!\,} , 23 {\displaystyle {\sqrt {23}}\!\,} , 29 {\displaystyle {\sqrt {29}}\!\,} , 31 {\displaystyle {\sqrt {31}}\!\,} , 41 {\displaystyle {\sqrt {41}}\!\,} , 43 {\displaystyle {\sqrt {43}}\!\,} , 47 {\displaystyle {\sqrt {47}}\!\,} , 53 {\displaystyle {\sqrt {53}}\!\,} , 59 {\displaystyle {\sqrt {59}}\!\,} , 61 {\displaystyle {\sqrt {61}}\!\,} , 67 {\displaystyle {\sqrt {67}}\!\,} , ...V splošnem tako velja:
c = [ a 0 ; a 1 , a 2 , … , a 2 , a 1 , 2 a 0 ¯ ] . {\displaystyle {\sqrt {c}}=[a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}]\!\,.} Od zgornjih števil, katerih niz je prazen, so deljiva s kvadratom (OEIS A124809 ):
50 = 7 , 0710 … = [ 7 ; 2 ⋅ 7 ¯ ] , {\displaystyle {\sqrt {50}}=7,0710\ldots =[7;{\overline {2\cdot 7}}]\!\,,} 325 = 18 , 0277 … = [ 18 ; 2 ⋅ 18 ¯ ] , {\displaystyle {\sqrt {325}}=18,0277\ldots =[18;{\overline {2\cdot 18}}]\!\,,} 1025 = 32 , 0156 … = [ 32 ; 2 ⋅ 32 ¯ ] , {\displaystyle {\sqrt {1025}}=32,0156\ldots =[32;{\overline {2\cdot 32}}]\!\,,} 1445 = 38 , 0131 … = [ 38 ; 2 ⋅ 38 ¯ ] , {\displaystyle {\sqrt {1445}}=38,0131\ldots =[38;{\overline {2\cdot 38}}]\!\,,} itd.
Števila, katerih perioda se začne:
z 2 (OEIS A065005 ): 2 {\displaystyle {\sqrt {2}}\!\,} , 6 {\displaystyle {\sqrt {6}}\!\,} , 12 {\displaystyle {\sqrt {12}}\!\,} , 19 {\displaystyle {\sqrt {19}}\!\,} , 20 {\displaystyle {\sqrt {20}}\!\,} , 29 {\displaystyle {\sqrt {29}}\!\,} , 30 {\displaystyle {\sqrt {30}}\!\,} , ..., s 3 (OEIS A065006 ): 11 {\displaystyle {\sqrt {11}}\!\,} , 28 {\displaystyle {\sqrt {28}}\!\,} , 40 {\displaystyle {\sqrt {40}}\!\,} , 53 {\displaystyle {\sqrt {53}}\!\,} , 69 {\displaystyle {\sqrt {69}}\!\,} , 86 {\displaystyle {\sqrt {86}}\!\,} , 87 {\displaystyle {\sqrt {87}}\!\,} , ..., s 4 (OEIS A065007 ): 5 {\displaystyle {\sqrt {5}}\!\,} , 18 {\displaystyle {\sqrt {18}}\!\,} , 39 {\displaystyle {\sqrt {39}}\!\,} , 52 {\displaystyle {\sqrt {52}}\!\,} , 68 {\displaystyle {\sqrt {68}}\!\,} , 85 {\displaystyle {\sqrt {85}}\!\,} , 105 {\displaystyle {\sqrt {105}}\!\,} , ..., s 5 (OEIS A065008 ): 27 {\displaystyle {\sqrt {27}}\!\,} , 67 {\displaystyle {\sqrt {67}}\!\,} , 104 {\displaystyle {\sqrt {104}}\!\,} , 1255 {\displaystyle {\sqrt {1255}}\!\,} , 174 {\displaystyle {\sqrt {174}}\!\,} , 201 {\displaystyle {\sqrt {201}}\!\,} , 231 {\displaystyle {\sqrt {231}}\!\,} , ..., s 6 (OEIS A065009 ): 10 {\displaystyle {\sqrt {10}}\!\,} , 38 {\displaystyle {\sqrt {38}}\!\,} , 84 {\displaystyle {\sqrt {84}}\!\,} , 103 {\displaystyle {\sqrt {103}}\!\,} , 148 {\displaystyle {\sqrt {148}}\!\,} , 173 {\displaystyle {\sqrt {173}}\!\,} , 230 {\displaystyle {\sqrt {230}}\!\,} , ..., s 7 (OEIS A065010 ): 51 {\displaystyle {\sqrt {51}}\!\,} , 124 {\displaystyle {\sqrt {124}}\!\,} , 200 {\displaystyle {\sqrt {200}}\!\,} , 229 {\displaystyle {\sqrt {229}}\!\,} , 329 {\displaystyle {\sqrt {329}}\!\,} , 366 {\displaystyle {\sqrt {366}}\!\,} , 447 {\displaystyle {\sqrt {447}}\!\,} , ..., z 8 (OEIS A065011 ): 17 {\displaystyle {\sqrt {17}}\!\,} , 66 {\displaystyle {\sqrt {66}}\!\,} , 147 {\displaystyle {\sqrt {147}}\!\,} , 172 {\displaystyle {\sqrt {172}}\!\,} , 260 {\displaystyle {\sqrt {260}}\!\,} , 293 {\displaystyle {\sqrt {293}}\!\,} , 405 {\displaystyle {\sqrt {405}}\!\,} , ..., z 9 (OEIS A065012 ): 83 {\displaystyle {\sqrt {83}}\!\,} , 199 {\displaystyle {\sqrt {199}}\!\,} , 328 {\displaystyle {\sqrt {328}}\!\,} , 365 {\displaystyle {\sqrt {365}}\!\,} , 534 {\displaystyle {\sqrt {534}}\!\,} , 581 {\displaystyle {\sqrt {581}}\!\,} , 735 {\displaystyle {\sqrt {735}}\!\,} , ... Dvočlene oblike
uredi
Druga kvadratna iracionalna števila, kjer c {\displaystyle c\,} ni kvadratno število:
( 1 + 2 ) / 2 = 1 , 2071 … = [ 1 ; 4 , 1 , … ] , {\displaystyle (1+{\sqrt {2}})/2=1,2071\ldots =[1;4,1,\ldots ]\!\,,} ( 1 + 3 ) / 2 = 1 , 3660 … = [ 1 ; 2 , 1 , … ] , {\displaystyle (1+{\sqrt {3}})/2=1,3660\ldots =[1;2,1,\ldots ]\!\,,} ( 1 + 5 ) / 2 = 1 , 6180 … = [ 1 ; 1 , … ] ≡ [ 1 ; 1 ¯ ] {\displaystyle (1+{\sqrt {5}})/2=1,6180\ldots =[1;1,\ldots ]\equiv [1;{\overline {1}}]\!\,} (število zlatega reza ),( 1 + 2 ) / 3 = 0 , 8047 … = [ 0 ; 1 , 4 , 8 ¯ ] , {\displaystyle (1+{\sqrt {2}})/3=0,8047\ldots =[0;1,{\overline {4,8}}]\!\,,} ( 1 + 3 ) / 3 = 0 , 9106 … = [ 0 ; 1 , 10 , 5 ¯ ] , {\displaystyle (1+{\sqrt {3}})/3=0,9106\ldots =[0;1,{\overline {10,5}}]\!\,,} ( 1 + 5 ) / 3 = 1 , 0786 … = [ 1 ; 12 , 1 , 2 , 2 , 2 , 1 ¯ ] , {\displaystyle (1+{\sqrt {5}})/3=1,0786\ldots =[1;{\overline {12,1,2,2,2,1}}]\!\,,} ( 1 + 2 ) / 5 = 0 , 4828 … = [ 0 ; 2 , 1 , 4 ¯ ] , {\displaystyle (1+{\sqrt {2}})/5=0,4828\ldots =[0;2,{\overline {1,4}}]\!\,,} ( 1 + 3 ) / 5 = 0 , 5464 … = [ 0 ; 1 , 1 , 4 , 1 , 7 ¯ ] , {\displaystyle (1+{\sqrt {3}})/5=0,5464\ldots =[0;1,{\overline {1,4,1,7}}]\!\,,} ( 1 + 5 ) / 5 = 0 , 6472 … = [ 0 ; 1 , 1 , 1 , 1 , 5 , 22 , 5 ¯ ] , {\displaystyle (1+{\sqrt {5}})/5=0,6472\ldots =[0;1,{\overline {1,1,1,5,22,5}}]\!\,,} ( 1 + 5 ) / 6 = 0 , 5393 … = [ 0 ; 1 , 1 , 5 ¯ ] , {\displaystyle (1+{\sqrt {5}})/6=0,5393\ldots =[0;1,{\overline {1,5}}]\!\,,} ( 1 + 5 ) / 7 = 0 , 4622 … = [ 0 ; 2 , 6 , 7 , 1 , 1 , 1 , 30 , 1 , 1 , 1 , 7 ¯ ] , {\displaystyle (1+{\sqrt {5}})/7=0,4622\ldots =[0;2,{\overline {6,7,1,1,1,30,1,1,1,7}}]\!\,,} ( 1 + 5 ) / 8 = 0 , 4045 … = [ 0 ; 2 , 2 , 8 ¯ ] , {\displaystyle (1+{\sqrt {5}})/8=0,4045\ldots =[0;2,{\overline {2,8}}]\!\,,} ( 1 + 5 ) / 9 = 0 , 3595 … = [ 0 ; 2 , 1 , 3 , 1 , 1 , 3 , 9 ¯ ] , {\displaystyle (1+{\sqrt {5}})/9=0,3595\ldots =[0;2,{\overline {1,3,1,1,3,9}}]\!\,,} ( 1 + 5 ) / 10 = 0 , 3236 … = [ 0 ; 3 , 11 ¯ ] , {\displaystyle (1+{\sqrt {5}})/10=0,3236\ldots =[0;3,{\overline {11}}]\!\,,} ( 2 + 5 ) / 2 = 2 , 1180 … = [ 2 ; 8 , 2 ¯ ] , {\displaystyle (2+{\sqrt {5}})/2=2,1180\ldots =[2;{\overline {8,2}}]\!\,,} ( 42 + 2 ) / 42 = 1 , 0336 … = [ 1 ; 29 , 1 , 2 , 3 , 6 , 3 , 2 , 1 , 58 ¯ ] , {\displaystyle (42+{\sqrt {2}})/42=1,0336\ldots =[1;29,{\overline {1,2,3,6,3,2,1,58}}]\!\,,} ( 42 + 42 ) / 42 = 1 , 1543 … = [ 1 ; 6 , 2 , 12 ¯ ] , {\displaystyle (42+{\sqrt {42}})/42=1,1543\ldots =[1;6,{\overline {2,12}}]\!\,,} ( 4242 + 4242 ) / 4242 = 1 , 0153 … = [ 1 ; 65 , 7 , 1 , 1 , 1 , 8 , 1 , 1 , 1 , 7 , 130 ¯ ] . . . {\displaystyle (4242+{\sqrt {4242}})/4242=1,0153\ldots =[1;65,{\overline {7,1,1,1,8,1,1,1,7,130}}]\!\,...} Če je c {\displaystyle c\,} kvadratno število in d > 1 {\displaystyle d>1\,} , je dano število racionalno , njegov verižni ulomek pa je seveda končen. Na primer:
( 2 + 4 ) / 5 = 4 / 5 = 0 , 8 = [ 0 ; 1 , 4 ] , {\displaystyle (2+{\sqrt {4}})/5=4/5=0,8=[0;1,4]\!\,,} ( 41 + 1764 ) / 42 = 83 / 42 = 1 , 9 761904 ¯ = [ 1 ; 1 , 41 ] . {\displaystyle (41+{\sqrt {1764}})/42=83/42=1,9{\overline {761904}}=[1;1,41]\!\,.} To dejstvo periodičnosti členov verižnih ulomkov sta dokazala Lagrange (1770 ) in Legendre , pred njima pa je obrat dokazal Euler z analizo popolnih količnikov periodičnih verižnih ulomkov – če je ζ pravi periodični verižni ulomek, je ζ kvadratno iracionalno število. Iz samega verižnega ulomka je moč konstruirati kvadratno enačbo s celimi koeficienti, za katere velja ζ.
Splošne oblike
uredi
( 1 + 2 2 ) / 2 = 1 , 9142 … = [ 1 ; 1 , 10 , 1 ¯ ] , {\displaystyle (1+2{\sqrt {2}})/2=1,9142\ldots =[{\overline {1;1,10,1}}]\!\,,}
( 1 + 2 3 ) / 2 = 2 , 2320 … = [ 2 ; 4 , 3 ¯ ] , {\displaystyle (1+2{\sqrt {3}})/2=2,2320\ldots =[2;{\overline {4,3}}]\!\,,}
( 1 + 2 2 ) / 3 = 1 , 2761 … = [ 1 ; 3 , 1 , 1 , 1 , 1 ¯ ] , {\displaystyle (1+2{\sqrt {2}})/3=1,2761\ldots =[{\overline {1;3,1,1,1,1}}]\!\,,}
( 1 + 2 3 ) / 3 = 1 , 4880 … = [ 1 ; 2 , 20 , 2 , 1 , 1 , 4 , 1 ¯ ] , {\displaystyle (1+2{\sqrt {3}})/3=1,4880\ldots =[{\overline {1;2,20,2,1,1,4,1}}]\!\,,}
( 1 + 3 2 ) / 2 = 2 , 6213 … = [ 2 ; 1 , 1 , 1 , 1 , 1 , 3 ¯ ] , {\displaystyle (1+3{\sqrt {2}})/2=2,6213\ldots =[2;{\overline {1,1,1,1,1,3}}]\!\,,}
( 1 + 3 3 ) / 2 = 3 , 0980 … = [ 3 ; 10 , 5 ¯ ] , {\displaystyle (1+3{\sqrt {3}})/2=3,0980\ldots =[3;{\overline {10,5}}]\!\,,}
( 1 + 3 2 ) / 3 = 1 , 7475 … = [ 1 ; 1 , 2 , 1 , 24 , 1 , 2 , 1 , 2 , 12 , 2 ¯ ] , {\displaystyle (1+3{\sqrt {2}})/3=1,7475\ldots =[1;{\overline {1,2,1,24,1,2,1,2,12,2}}]\!\,,}
( 1 + 3 3 ) / 3 = 2 , 0653 … = [ 2 ; 15 , 3 , 2 , 1 , 1 , 30 , 1 , 1 , 2 , 3 ¯ ] , {\displaystyle (1+3{\sqrt {3}})/3=2,0653\ldots =[2;{\overline {15,3,2,1,1,30,1,1,2,3}}]\!\,,}
( 1 + 2 3 ) / 4 = 1 , 1160 … = [ 1 ; 8 , 1 , 1 , 1 , 1 , 1 , 1 ¯ ] , {\displaystyle (1+2{\sqrt {3}})/4=1,1160\ldots =[{\overline {1;8,1,1,1,1,1,1}}]\!\,,}
( 2 + 3 5 ) / 7 = 1 , 2440 … = [ 1 ; 4 , 10 , 4 , 1 , 1 , 2 , 18 , 2 , 1 ¯ ] . . . {\displaystyle (2+3{\sqrt {5}})/7=1,2440\ldots =[{\overline {1;4,10,4,1,1,2,18,2,1}}]\!\,...} Druge oblike
uredi
Poseben primer kvadratnih iracionalnih števil so rešitve Fermat-Pellove enačbe .
Zunanje povezave
uredi