Bevezet\303\251s a matematik\303\241ba J\303\241rai Antal Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak.
<Text-field style="Heading 1" layout="Heading 1">1. Halmazok</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">2. Term\303\251szetes sz\303\241mok</Font></Text-field>
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">3. A sz\303\241mfogalom b\305\221v\303\255t\303\251se</Font></Text-field> restart;
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">3.1. Eg\303\251sz sz\303\241mok</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.1. Oszt\303\241lyoz\303\241s kompatibilit\303\241sa m\305\261velettel.</Font></Text-field> `type/ordpair`:=proc(x) type(x,list) and nops(x)=2 end; Zio2I0kieEc2IkYlRiVGJTMtSSV0eXBlRyUqcHJvdGVjdGVkRzYkRiRJJWxpc3RHRikvLUklbm9wc0dGKUYjIiIjRiVGJUYl iscompbinop:=proc(X::set,E::set(ordpair),f::procedure) local x,xx,y,yy; for x in X do for y in X do if not f(x,y) in X then return false fi; for xx in X do for yy in X do if [x,xx] in E and [y,yy] in E and not [f(x,y),f(xx,yy)] in E then return false fi; od; od; od; od; true; end; X:={0,1,2,3,4,5}; E:={[0,0],[0,3],[3,0],[3,3],[1,1],[1,4],[4,1],[4,4],[2,2],[2,5],[5,2],[5,5]}; f:=(x,y)->irem(x+y,6); iscompbinop(X,E,f); Zio2JSdJIlhHNiJJJHNldEclKnByb3RlY3RlZEcnSSJFR0YmLUYnNiNJKG9yZHBhaXJHRiYnSSJmR0YmSSpwcm9jZWR1cmVHRig2JkkieEdGJkkjeHhHRiZJInlHRiZJI3l5R0YmRiZGJkMkPyZGMkYlSSV0cnVlR0YoPyZGNEYlRjhDJEAkNC1JI2luR0YoNiQtRi82JEYyRjRGJU9JJmZhbHNlR0YoPyZGM0YlRjg/JkY1RiVGOEAkMzMtRj42JDckRjJGM0YqLUY+NiQ3JEY0RjVGKjQtRj42JDckRkAtRi82JEYzRjVGKkZCRjhGJkYmRiY= PCgiIiEiIiIiIiMiIiQiIiUiIiY= PC43JCIiIUYkNyRGJCIiJDckRiZGJDckRiZGJjckIiIiRio3JEYqIiIlNyRGLEYqNyRGLEYsNyQiIiNGMDckRjAiIiY3JEYyRjA3JEYyRjI= Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklaXJlbUclKnByb3RlY3RlZEc2JCwmRiQiIiJGJkYvIiInRiVGJUYl SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.1.2. P<Font encoding="UTF-8">\303\251lda</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.1.3. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.4. Oszt\303\241lyoz\303\241s kompatibilit\303\241sa rel\303\241ci\303\263val.</Font></Text-field> iscomprel:=proc(X::set,E::set(ordpair),R::set(ordpair)) local x,xx,y,yy; for x in X do for y in X do for xx in X do for yy in X do if [x,xx] in E and [y,yy] in E and [x,y] in R and not [xx,yy] in R then return false fi; od; od; od; od; true; end; X:={0,1,2,3}; E:={[0,0],[0,2],[2,0],[2,2],[1,1],[1,3],[3,1],[3,3]}; R:={[0,1],[0,3],[2,1],[2,3]}; iscomprel(X,E,R); Zio2JSdJIlhHNiJJJHNldEclKnByb3RlY3RlZEcnSSJFR0YmLUYnNiNJKG9yZHBhaXJHRiYnSSJSR0YmRis2JkkieEdGJkkjeHhHRiZJInlHRiZJI3l5R0YmRiZGJkMkPyZGMUYlSSV0cnVlR0YoPyZGM0YlRjc/JkYyRiVGNz8mRjRGJUY3QCQzMzMtSSNpbkdGKDYkNyRGMUYyRiotRkA2JDckRjNGNEYqLUZANiQ3JEYxRjNGLzQtRkA2JDckRjJGNEYvT0kmZmFsc2VHRihGN0YmRiZGJg== PCYiIiEiIiIiIiMiIiQ= PCo3JCIiIUYkNyQiIiRGJjckIiIiRig3JCIiI0YqNyRGJEYqNyRGKEYmNyRGKkYkNyRGJkYo PCY3JCIiISIiJDckRiQiIiI3JCIiI0YnNyRGKUYl SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.1.5. P<Font encoding="UTF-8">\303\251lda</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.6. T\303\251tel.</Font></Text-field> type(5,integer); type(-3,integer); type(0,integer); SSV0cnVlRyUqcHJvdGVjdGVkRw== SSV0cnVlRyUqcHJvdGVjdGVkRw== SSV0cnVlRyUqcHJvdGVjdGVkRw== `&~`:=(x,y)->x[1]+y[2]=x[2]+y[1]; `&+`:=(x,y)->[x[1]+y[1],x[2]+y[2]]; `&*`:=(x,y)->[x[1]*y[1]+x[2]*y[2],x[1]*y[2]+y[1]*x[2]]; `&le`:=(x,y)->x[1]+y[2]<=x[2]+y[1]; [7,4]&~[3,0]; [7,4]&+[2,6]; [2,1]&*[2,4]; [3,5]&le[2,3]; Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLywmJkYkNiMiIiJGLiZGJjYjIiIjRi4sJiZGJEYwRi4mRiZGLUYuRiVGJUYl Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJiZGJDYjIiIiRi4mRiZGLUYuLCYmRiQ2IyIiI0YuJkYmRjJGLkYlRiVGJQ== Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJiomJkYkNiMiIiJGLyZGJkYuRi9GLyomJkYkNiMiIiNGLyZGJkYzRi9GLywmKiZGLUYvRjVGL0YvKiZGMEYvRjJGL0YvRiVGJUYl Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlMSwmJkYkNiMiIiJGLiZGJjYjIiIjRi4sJiZGJEYwRi4mRiZGLUYuRiVGJUYl LyIiKEYj NyQiIioiIzU= NyQiIikiIzU= MSIiJyIiKA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.1.7. T<Font encoding="UTF-8">\303\251tel: \342\204\225 be\303\241</Font>gyaz<Font encoding="UTF-8">\303\241sa \342\204\244-be.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.1.8. Az <Font encoding="UTF-8">eg\303\251sz sz\303\241mok rendez\303\251</Font>se.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.1.9. Az <Font encoding="UTF-8">eg\303\251sz sz\303\241mok szorz\303\241</Font>sa.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.1.10. Az <Font encoding="UTF-8">eg\303\251sz sz\303\241mok sz\303\241</Font>m<Font encoding="UTF-8">\303\255</Font>t<Font encoding="UTF-8">\303\263</Font>g<Font encoding="UTF-8">\303\251pes \303\241br\303\241</Font>zol<Font encoding="UTF-8">\303\241</Font>sa.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.11. Hatv\303\241nyoz\303\241s eg\303\251sz kitev\305\221vel.</Font></Text-field> (a^(-1))^5; a^(m+n); expand(%); (a*b)^5; (a^n)^m; combine(%,power) assuming integer; KiQpSSJhRzYiIiImISIi KUkiYUc2IiwmSSJtR0YkIiIiSSJuR0YkRic= KiYpSSJhRzYiSSJtR0YlIiIiKUYkSSJuR0YlRic= KiYpSSJhRzYiIiImIiIiKUkiYkdGJUYmRic= KSlJImFHNiJJIm5HRiVJIm1HRiU= KUkiYUc2IiomSSJuR0YkIiIiSSJtR0YkRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.1.12. P<Font encoding="UTF-8">\303\251lda</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.13. Gy\305\261r\305\261k.</Font></Text-field> isgrupoid:=proc(G::set,f::procedure) local x,y; for x in G do for y in G do if not f(x,y) in G then return false fi; od; od; true; end; Zio2JCdJIkdHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRig2JEkieEdGJkkieUdGJkYmRiZDJD8mRi1GJUkldHJ1ZUdGKD8mRi5GJUYxQCQ0LUkjaW5HRig2JC1GKkYsRiVPSSZmYWxzZUdGKEYxRiZGJkYm neutral:=proc(G::set,f::procedure) local x,y,s,S; if not isgrupoid(G,f) then return NULL fi; for x in G do s:=true; for y in G do if f(x,y)<>y or f(y,x)<>y then s:=false; break; fi; od; if s then return x fi; od; NULL end; G:={a,b,c};neutral(G,(x,y)->y);neutral(G,(x,y)->y); neutral({0,1,2},(x,y)->irem(x+y,3)); Zio2JCdJIkdHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRig2JkkieEdGJkkieUdGJkkic0dGJkkiU0dGJkYmRiZDJUAkNC1JKmlzZ3J1cG9pZEdGJjYkRiVGKk9JJU5VTExHRig/JkYtRiVJJXRydWVHRihDJT5GL0Y6PyZGLkYlRjpAJDUwLUYqNiRGLUYuRi4wLUYqNiRGLkYtRi5DJD5GL0kmZmFsc2VHRihbQCRGL09GLUY4RiZGJkYm PCVJImFHNiJJImJHRiRJImNHRiQ= IiIh issemigroup:=proc(G::set,f::procedure) local x,y,z; if not isgrupoid(G,f) then return false fi; for x in G do for y in G do for z in G do if f(x,f(y,z))<>f(f(x,y),z) then return false fi; od; od; od; true end; issemigroup({a,b,c},(x,y)->x); issemigroup({true,false},(x,y)-> x implies y); Zio2JCdJIkdHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRig2JUkieEdGJkkieUdGJkkiekdGJkYmRiZDJUAkNC1JKmlzZ3J1cG9pZEdGJjYkRiVGKk9JJmZhbHNlR0YoPyZGLUYlSSV0cnVlR0YoPyZGLkYlRjk/JkYvRiVGOUAkMC1GKjYkRi0tRio2JEYuRi8tRio2JC1GKjYkRi1GLkYvRjZGOUYmRiZGJg== SSV0cnVlRyUqcHJvdGVjdGVkRw== SSZmYWxzZUclKnByb3RlY3RlZEc= isgroup:=proc(G::set,f::procedure) local x,y,n,i; if not isgrupoid(G,f) then return false fi; if not issemigroup(G,f) then return false fi; n:=neutral(G,f); if n=NULL then return false fi; for x in G do i:=false; for y in G do if f(x,y)=n and f(y,x)=n then i:=true; break fi; od; if i=false then return false fi; od; true; end; isgroup({0,1,2},(x,y)->irem(x+y,3)); Zio2JCdJIkdHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRig2JkkieEdGJkkieUdGJkkibkdGJkkiaUdGJkYmRiZDKEAkNC1JKmlzZ3J1cG9pZEdGJjYkRiVGKk9JJmZhbHNlR0YoQCQ0LUksaXNzZW1pZ3JvdXBHRiZGNkY3PkYvLUkobmV1dHJhbEdGJkY2QCQvRi9JJU5VTExHRihGNz8mRi1GJUkldHJ1ZUdGKEMlPkYwRjg/JkYuRiVGREAkMy8tRio2JEYtRi5GLy8tRio2JEYuRi1GL0MkPkYwRkRbQCQvRjBGOEY3RkRGJkYmRiY= SSV0cnVlRyUqcHJvdGVjdGVkRw== iscommutative:=proc(G::set,f::procedure) local x,y; if not isgrupoid(G,f) then return false fi; for x in G do for y in G do if f(x,y)<>f(y,x) then return false fi; od; od; true; end; iscommutative({0,1,2},(x,y)->irem(x+y,3)); Zio2JCdJIkdHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRig2JEkieEdGJkkieUdGJkYmRiZDJUAkNC1JKmlzZ3J1cG9pZEdGJjYkRiVGKk9JJmZhbHNlR0YoPyZGLUYlSSV0cnVlR0YoPyZGLkYlRjhAJDAtRipGLC1GKjYkRi5GLUY1RjhGJkYmRiY= SSV0cnVlRyUqcHJvdGVjdGVkRw== isabeliangroup:=proc(G::set,f::procedure) isgroup(G,f) and iscommutative(G,f) end; iscommutative({0,1,2},(x,y)->irem(x+y,3)); Zio2JCdJIkdHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRihGJkYmRiYzLUkoaXNncm91cEdGJjYkRiVGKi1JLmlzY29tbXV0YXRpdmVHRiZGL0YmRiZGJg== SSV0cnVlRyUqcHJvdGVjdGVkRw== isleftdistributive:=proc(R::set,f::procedure,g::procedure) local x,y,z; if not isgrupoid(R,f) then return false fi; if not isgrupoid(R,g) then return false fi; for x in R do for y in R do for z in R do if g(x,f(y,z))<>f(g(x,y),g(x,z)) then return false fi; od; od; od; true end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRis2JUkieEdGJkkieUdGJkkiekdGJkYmRiZDJkAkNC1JKmlzZ3J1cG9pZEdGJjYkRiVGKk9JJmZhbHNlR0YoQCQ0LUY2NiRGJUYtRjg/JkYvRiVJJXRydWVHRig/JkYwRiVGPz8mRjFGJUY/QCQwLUYtNiRGLy1GKjYkRjBGMS1GKjYkLUYtNiRGL0YwLUYtNiRGL0YxRjhGP0YmRiZGJg== isrightdistributive:=proc(R::set,f::procedure,g::procedure) local x,y,z; if not isgrupoid(R,f) then return false fi; if not isgrupoid(R,g) then return false fi; for x in R do for y in R do for z in R do if g(f(y,z),x)<>f(g(y,x),g(z,x)) then return false fi; od; od; od; true end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRis2JUkieEdGJkkieUdGJkkiekdGJkYmRiZDJkAkNC1JKmlzZ3J1cG9pZEdGJjYkRiVGKk9JJmZhbHNlR0YoQCQ0LUY2NiRGJUYtRjg/JkYvRiVJJXRydWVHRig/JkYwRiVGPz8mRjFGJUY/QCQwLUYtNiQtRio2JEYwRjFGLy1GKjYkLUYtNiRGMEYvLUYtNiRGMUYvRjhGP0YmRiZGJg== isring:=proc(R::set,f::procedure,g::procedure) isabeliangroup(R,f) and issemigroup(R,g) and isleftdistributive(R,f,g) and isrightdistributive(R,f,g) end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRitGJkYmRiYzMzMtSS9pc2FiZWxpYW5ncm91cEdGJjYkRiVGKi1JLGlzc2VtaWdyb3VwR0YmNiRGJUYtLUkzaXNsZWZ0ZGlzdHJpYnV0aXZlR0YmNiVGJUYqRi0tSTRpc3JpZ2h0ZGlzdHJpYnV0aXZlR0YmRjlGJkYmRiY= iscommutativering:=proc(R::set,f::procedure,g::procedure) isring(R,f,g) and iscommutative(R,g) end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRitGJkYmRiYzLUknaXNyaW5nR0YmNiVGJUYqRi0tSS5pc2NvbW11dGF0aXZlR0YmNiRGJUYtRiZGJkYm isringwithunity:=proc(R::set,f::procedure,g::procedure) isring(R,f,g) and neutral(R,g)<>NULL end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRitGJkYmRiYzLUknaXNyaW5nR0YmNiVGJUYqRi0wLUkobmV1dHJhbEdGJjYkRiVGLUklTlVMTEdGKEYmRiZGJg== isringwithunity({0},(x,y)->0,(x,y)->0); SSV0cnVlRyUqcHJvdGVjdGVkRw== X:={a,b,c}; P:=combinat[powerset](X); isring(P,(x,y)->symmdiff(x,y),(x,y)->{}); PCVJImFHNiJJImJHRiRJImNHRiQ= PCo8IjwlSSJhRzYiSSJiR0YmSSJjR0YmPCRGJ0YoPCNGKDwkRiVGKDwjRiU8I0YnPCRGJUYn SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.14. P\303\251lda.</Font></Text-field> iscommutativering(P,(x,y)->symmdiff(x,y),(x,y)->x intersect y); isringwithunity(P,(x,y)->symmdiff(x,y),(x,y)->x intersect y); SSV0cnVlRyUqcHJvdGVjdGVkRw== SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.1.15. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.16. P\303\251ld\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.1.17. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.1.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.1.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.20. Az \303\241ltal\303\241nos disztributivit\303\241s t\303\251tele.</Font></Text-field> A:=sum(a[i],i=1..4); B:=sum(b[j],j=1..5); A*B; expand(%); LComSSJhRzYiNiMiIiJGJyZGJDYjIiIjRicmRiQ2IyIiJEYnJkYkNiMiIiVGJw== LCwmSSJiRzYiNiMiIiJGJyZGJDYjIiIjRicmRiQ2IyIiJEYnJkYkNiMiIiVGJyZGJDYjIiImRic= KiYsKiZJImFHNiI2IyIiIkYoJkYlNiMiIiNGKCZGJTYjIiIkRigmRiU2IyIiJUYoRigsLCZJImJHRiZGJ0YoJkY0RipGKCZGNEYtRigmRjRGMEYoJkY0NiMiIiZGKEYo LEoqJiZJImFHNiI2IyIiIkYoJkkiYkdGJkYnRihGKComRiRGKCZGKjYjIiIjRihGKComRiRGKCZGKjYjIiIkRihGKComRiRGKCZGKjYjIiIlRihGKComRiRGKCZGKjYjIiImRihGKComJkYlRi1GKEYpRihGKComRjxGKEYsRihGKComRjxGKEYwRihGKComRjxGKEY0RihGKComRjxGKEY4RihGKComJkYlRjFGKEYpRihGKComRkJGKEYsRihGKComRkJGKEYwRihGKComRkJGKEY0RihGKComRkJGKEY4RihGKComJkYlRjVGKEYpRihGKComRkhGKEYsRihGKComRkhGKEYwRihGKComRkhGKEY0RihGKComRkhGKEY4RihGKA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.21. Nulloszt\303\263k, integrit\303\241si tartom\303\241ny, rendezett integrit\303\241si tartom\303\241ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.22. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.1.23. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.1.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.1.25. Feladat.</Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">3.2. Racion\303\241lis sz\303\241mok</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.2.1. T\303\251tel.</Font></Text-field> type(5/7,rational); type(0,rational); SSV0cnVlRyUqcHJvdGVjdGVkRw== SSV0cnVlRyUqcHJvdGVjdGVkRw== `&~`:=(x,y)->x[1]*y[2]=y[1]*x[2]; `&+`:=(x,y)->[x[1]*y[2]+x[2]*y[1],x[2]*y[2]]; `&*`:=(x,y)->[x[1]*y[1],x[2]*y[2]]; `&le`:=(x,y)->(y[1]*x[2]-y[2]*x[1])*x[2]*y[2]>=0; [1,5]&~[2,10]; [1,2]&+[2,3]; [1,2]&*[2,3]; [1,2]&le[2,3]; Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLyomJkYkNiMiIiJGLiZGJjYjIiIjRi4qJiZGJkYtRi4mRiRGMEYuRiVGJUYl Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJiomJkYkNiMiIiJGLyZGJjYjIiIjRi9GLyomJkYmRi5GLyZGJEYxRi9GLyomRjVGL0YwRi9GJUYlRiU= Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQqJiZGJDYjIiIiRi4mRiZGLUYuKiYmRiQ2IyIiI0YuJkYmRjJGLkYlRiVGJQ== Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlMSIiISooLCYqJiZGJjYjIiIiRjEmRiQ2IyIiI0YxRjEqJiZGJEYwRjEmRiZGM0YxISIiRjFGMkYxRjdGMUYlRiVGJQ== LyIjNUYj NyQiIigiIic= NyQiIiMiIic= MSIiISIiJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.2.2. <Font encoding="UTF-8">T\303\251tel: \342\204\244 be\303\241</Font>gyaz<Font encoding="UTF-8">\303\241sa \342\204\232-ba.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.2.3. Ferdetest, test, rendezett test.</Text-field> isskewfield:=proc(R::set,f::procedure,g::procedure) local n; n:=neutral(R,f); if n=NULL then return false fi; isring(R,f,g) and isgroup(R minus {n},g) end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRis2I0kibkdGJkYmRiZDJT5GLy1JKG5ldXRyYWxHRiY2JEYlRipAJC9GL0klTlVMTEdGKE9JJmZhbHNlR0YoMy1JJ2lzcmluZ0dGJjYlRiVGKkYtLUkoaXNncm91cEdGJjYkLUkmbWludXNHRig2JEYlPCNGL0YtRiZGJkYm isfield:=proc(R::set,f::procedure,g::procedure) local n; n:=neutral(R,f); if n=NULL then return false fi; isring(R,f,g) and isabeliangroup(R minus {n},g) end; Zio2JSdJIlJHNiJJJHNldEclKnByb3RlY3RlZEcnSSJmR0YmSSpwcm9jZWR1cmVHRignSSJnR0YmRis2I0kibkdGJkYmRiZDJT5GLy1JKG5ldXRyYWxHRiY2JEYlRipAJC9GL0klTlVMTEdGKE9JJmZhbHNlR0YoMy1JJ2lzcmluZ0dGJjYlRiVGKkYtLUkvaXNhYmVsaWFuZ3JvdXBHRiY2JC1JJm1pbnVzR0YoNiRGJTwjRi9GLUYmRiZGJg== &+(0,0):=0; &+(0,1):=1; &+(1,0):=1; &+(1,1):=0; &*(0,0):=0; &*(0,1):=0; &*(1,0):=0; &*(1,1):=1; IiIh IiIi IiIi IiIh IiIh IiIh IiIh IiIi isfield({0,1},(x,y)->x&+y,(x,y)->x&*y); SSV0cnVlRyUqcHJvdGVjdGVkRw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.2.4. P\303\251ld\303\241k.</Font></Text-field> `&+`:=(x,y)->irem(x+y,5); `&*`:=(x,y)->irem(x*y,5); 3&+4; 3&*4; Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklaXJlbUclKnByb3RlY3RlZEc2JCwmRiQiIiJGJkYvIiImRiVGJUYl Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklaXJlbUclKnByb3RlY3RlZEc2JComRiQiIiJGJkYvIiImRiVGJUYl IiIj IiIj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.2.5 T<Font encoding="UTF-8">\303\251tel: \342\204\232 be\303\241</Font>gyaz<Font encoding="UTF-8">\303\241</Font>sa rendezett testbe.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.2.6. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.7. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.8. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.9. Feladat.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">3.3. Val\303\263s sz\303\241mok</Font></Text-field> restart;
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.1. \303\201ll\303\255t\303\241s.</Font></Text-field> i:='i': x:=0: for i from 0 do while (x+1)^2<2*10^(2*i) do x:=x+1: od; x; x:=x*10: od; IiIi IiM1 IiM5 IiRTIg== IiRUIg== IiU1OQ== IiU5OQ== IiZTVCI= IiZVVCI= Iic/OTk= IidAOTk= Iig1VVQi Iig4VVQi IilJQDk5 IilOQDk5 IipdOFVUIg== IipjOFVUIg== IitnTkA5OQ== IitpTkA5OQ== Iiw/YzhVVCI= IixCYzhVVCI= Ii1JaU5AOTk= Ii1QaU5AOTk= Ii5xQmM4VVQi Ii50QmM4VVQi Ii9JUGlOQDk5 Ii9JUGlOQDk5 IjArdEJjOFVUIg== IjA0dEJjOFVUIg== IjEhNHRCYzhVVCI= IjEmNHRCYzhVVCI= IjJdNHRCYzhVVCI= IjJdNHRCYzhVVCI= IjMrJjR0QmM4VVQi IjMvJjR0QmM4VVQi IjRTXTR0QmM4VVQi IjRbXTR0QmM4VVQi IjUhW100dEJjOFVUIg== IjUpW100dEJjOFVUIg== IjYhKVtdNHRCYzhVVCI= IjYhKVtdNHRCYzhVVCI= IjcrKVtdNHRCYzhVVCI= IjcsKVtdNHRCYzhVVCI= Ijg1ISlbXTR0QmM4VVQi Ijg7ISlbXTR0QmM4VVQi IjlnLClbXTR0QmM4VVQi IjlvLClbXTR0QmM4VVQi IjohbywpW100dEJjOFVUIg== IjopbywpW100dEJjOFVUIg== IjshKW8sKVtdNHRCYzhVVCI= IjsoKW8sKVtdNHRCYzhVVCI= IjxxKW8sKVtdNHRCYzhVVCI= 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IlxwKzJAWXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi IlxwLjJAWXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi Il1wSXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== Il1wUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== Il5wIVFxNWkleUN0ISp6dHptPHQhW3BQdj1ucCZ5ISlwNFVzKW8sKVtdNHRCYzhVVCI= Il5wKVFxNWkleUN0ISp6dHptPHQhW3BQdj1ucCZ5ISlwNFVzKW8sKVtdNHRCYzhVVCI= Il9wISlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi Il9wJilRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImBwXSlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImBwXSlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImFwKyYpUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== ImFwLiYpUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== ImJwSV0pUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== ImJwUV0pUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== ImNwIVFdKVFxNWkleUN0ISp6dHptPHQhW3BQdj1ucCZ5ISlwNFVzKW8sKVtdNHRCYzhVVCI= ImNwKFFdKVFxNWkleUN0ISp6dHptPHQhW3BQdj1ucCZ5ISlwNFVzKW8sKVtdNHRCYzhVVCI= ImRwcVFdKVFxNWkleUN0ISp6dHptPHQhW3BQdj1ucCZ5ISlwNFVzKW8sKVtdNHRCYzhVVCI= ImRwdlFdKVFxNWkleUN0ISp6dHptPHQhW3BQdj1ucCZ5ISlwNFVzKW8sKVtdNHRCYzhVVCI= ImVwXShRXSlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImVwYChRXSlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImZwSXZRXSlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImZwTXZRXSlRcTVpJXlDdCEqenR6bTx0IVtwUHY9bnAmeSEpcDRVcylvLClbXTR0QmM4VVQi ImdwU2AoUV0pUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== ImdwVmAoUV0pUXE1aSV5Q3QhKnp0em08dCFbcFB2PW5wJnkhKXA0VXMpbywpW100dEJjOFVUIg== Warning, computation interrupted LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.3.2. A<Font encoding="UTF-8">rkhim\303\251d\303\251szi tulajdons\303\241g.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.3. \303\201ll\303\255t\303\241s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.4. \303\201ll\303\255t\303\241s.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*3.3.5. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.6. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.7. Val\303\263s sz\303\241mok.</Font></Text-field> abs(7.4); abs(-3); abs(0); signum(7.4); signum(-3); signum(0); JCIjdSEiIg== IiIk IiIh IiIi ISIi IiIh floor(3.14); ceil(3.14); ceil(-3.14); IiIk IiIl ISIk Rmod:=proc(x::realcons,y::realcons) if y=0 then x else x-floor(x/y)*y fi; end; Rmod(5,0); Rmod(3.1415,2.78); Zio2JCdJInhHNiJJKXJlYWxjb25zRyUqcHJvdGVjdGVkRydJInlHRiZGJ0YmRiZGJkAlL0YqIiIhRiUsJkYlIiIiKiYtSSZmbG9vckc2JEYoSShfc3lzbGliR0YmNiMqJkYlRi9GKiEiIkYvRipGL0Y3RiZGJkYm IiIm JCIlOk8hIiU= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.3.8. B<Font encoding="UTF-8">\305\221</Font>v<Font encoding="UTF-8">\303\255tett val\303\263s sz\303\241mok.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.3.9. Val<Font encoding="UTF-8">\303\263</Font>s <Font encoding="UTF-8">sz\303\241mok</Font> kerek<Font encoding="UTF-8">\303\255</Font>t<Font encoding="UTF-8">\303\251</Font>se <Font encoding="UTF-8">\303\251</Font>s fixpontos <Font encoding="UTF-8"> \303\241br\303\241</Font>zol<Font encoding="UTF-8">\303\241</Font>sa <Font encoding="UTF-8">sz\303\241</Font>m<Font encoding="UTF-8">\303\255</Font>t<Font encoding="UTF-8">\303\263</Font>g<Font encoding="UTF-8">\303\251pben</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.3.10. Val<Font encoding="UTF-8">\303\263</Font>s <Font encoding="UTF-8">sz\303\241mok</Font> lebeg<Font encoding="UTF-8">\305\221</Font>pontos<Font encoding="UTF-8"> \303\241br\303\241</Font>zol<Font encoding="UTF-8">\303\241</Font>sa <Font encoding="UTF-8">sz\303\241</Font>m<Font encoding="UTF-8">\303\255</Font>t<Font encoding="UTF-8">\303\263</Font>g<Font encoding="UTF-8">\303\251pben</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.11. T\303\251tel: a v</Font>al<Font encoding="UTF-8">\303\263</Font>s <Font encoding="UTF-8">sz\303\241mok</Font> l<Font encoding="UTF-8">\303\251</Font>tez<Font encoding="UTF-8">\303\251</Font>se.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*3.3.12. A val\303\263s sz\303\241mok m\303\241s bevezet\303\251sei.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.13. T\303\251tel: gy\303\266kvon\303\241s</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.3.14. K\303\266vetkem\303\251</Font>ny. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.3.15. A term<Font encoding="UTF-8">\303\251</Font>szetes, az eg<Font encoding="UTF-8">\303\251</Font>sz <Font encoding="UTF-8">\303\251</Font>s a racion<Font encoding="UTF-8">\303\241</Font>lis sz<Font encoding="UTF-8">\303\241</Font>mok bevezet<Font encoding="UTF-8">\303\251</Font>se a val<Font encoding="UTF-8">\303\263</Font>s sz<Font encoding="UTF-8">\303\241</Font>mok seg<Font encoding="UTF-8">\303\255</Font>ts<Font encoding="UTF-8">\303\251</Font>g<Font encoding="UTF-8">\303\251</Font>vel. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.16. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.17. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.20. Feladat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">3.2.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.22. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.23. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.25. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.27. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.28. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.29. Feladat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.2.30. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.31. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.32. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.33. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.34. Feladat: <Font encoding="UTF-8">\303\266r\303\266</Font>knapt<Font encoding="UTF-8">\303\241</Font>r.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.35. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.36. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.2.37. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.38. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.39. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.40. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.41. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.42. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.43. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.44. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.45. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.46. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.47. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*3.2.48. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.2.49. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*3.2.50. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.3.51. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">3.4. Komplex sz\303\241mok</Font></Text-field> restart;
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.1. Komplex sz\303\241mok.</Font></Text-field> `&+`:=proc(z,w) [z[1]+w[1],z[2]+w[2]] end; `&*`:=proc(z,w) [z[1]*w[1]-z[2]*w[2],z[1]*w[2]+z[2]*w[1]] end; [x,y]&+[0,0]; [x,y]&+[-x,-y]; [x,y]&*[1,0]; [x,y]&*[x/(x^2+y^2),-y/(x^2+y^2)]; simplify(%); [0,1]&*[0,1]; Zio2JEkiekc2Ikkid0dGJUYlRiVGJTckLCYmRiQ2IyIiIkYrJkYmRipGKywmJkYkNiMiIiNGKyZGJkYvRitGJUYlRiU= Zio2JEkiekc2Ikkid0dGJUYlRiVGJTckLCYqJiZGJDYjIiIiRiwmRiZGK0YsRiwqJiZGJDYjIiIjRiwmRiZGMEYsISIiLCYqJkYqRixGMkYsRiwqJkYvRixGLUYsRixGJUYlRiU= NyRJInhHNiJJInlHRiQ= NyQiIiFGIw== NyRJInhHNiJJInlHRiQ= NyQsJiomKUkieEc2IiIiIyIiIiwmKiRGJUYpRikqJClJInlHRidGKEYpRikhIiJGKSomRi1GKUYqRi9GKSIiIQ== NyQiIiIiIiE= NyQhIiIiIiE= Complex(3,5); z:=3+5*I; w:=-2-6*I; z*w; Re(z); Im(z); conjugate(z); LCYiIiQiIiIqJiIiJkYkXiNGJEYkRiQ= LCYiIiQiIiIqJiIiJkYkXiNGJEYkRiQ= LCYiIiMhIiIqJiIiJyIiIl4jRidGJ0Yk LCYiI0MiIiIqJiIjR0YkXiNGJEYkISIi IiIk IiIm LCYiIiQiIiIqJiIiJkYkXiNGJEYkISIi z:='z';w:='w'; conjugate(z); conjugate(conjugate(z));conjugate(z+w);expand(%);conjugate(1/z); SSJ6RzYi SSJ3RzYi LUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjSSJ6RzYi SSJ6RzYi LUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjLCZJInpHNiIiIiJJIndHRihGKQ== LCYtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJInpHNiIiIiItRiQ2I0kid0dGKEYp KiQtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJInpHNiIhIiI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.2. P\303\251lda.</Font></Text-field> 64/(3^(1/2)+I); evalc(%); LCQqJiIjayIiIiwmKiQpIiIkI0YlIiIjRiVGJV4jRiVGJSEiIkYl LCYqJiIjOyIiIikiIiQjRiUiIiNGJUYlKiZGJEYlXiNGJUYlISIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.3. Komplex sz\303\241m abszol\303\272t \303\251rt\303\251ke.</Font></Text-field> z:=x+I*y;abs(z);evalc(%);evalc(1/(x+I*y));evalc(conjugate(z)/abs(z)^2); LCZJInhHNiIiIiIqJl4jRiVGJUkieUdGJEYlRiU= LUkkYWJzRyUqcHJvdGVjdGVkRzYjLCZJInhHNiIiIiIqJl4jRilGKUkieUdGKEYpRik= KiQpLCYqJClJInhHNiIiIiMiIiJGKiokKUkieUdGKEYpRipGKiNGKkYpRio= LCYqJkkieEc2IiIiIiwmKiQpRiQiIiNGJkYmKiQpSSJ5R0YlRipGJkYmISIiRiYqKCwkXiNGJkYmRiZGLUYmRidGLkYu LCYqJkkieEc2IiIiIiwmKiQpRiQiIiNGJkYmKiQpSSJ5R0YlRipGJkYmISIiRiYqKCwkXiNGJkYmRiZGLUYmRidGLkYu signum(3+4*I); signum(-5); signum(0); LCYjIiIkIiImIiIiKiYjIiIlRiVGJl4jRiZGJkYm ISIi IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.4.4. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.5. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.6. Feladat.</Text-field> evalc(2/(1-I)/(3+I)); evalc(1/(3+4*I)^2); evalc((2+I)/I/(-3+4*I)); evalc((3^(1/2)+I)/(1-I)/(3^(1/2)-I)); simplify(%); evalc(1/I/(3-2*I)/(1+I)); evalc(1/(1-I)/(1-2*I)/(1+2*I)); LCYjIiIjIiImIiIiKiYjRiZGJUYmXiNGJkYmRiY= LCYjIiIoIiREJyEiIiomIyIjQ0YlIiIiXiNGKkYqRiY= LCYjIiM2IiNEISIiKiYjIiIjRiUiIiJeI0YqRipGKg== LCoqKCMiIiIiIiVGJSwmKiYjRiUiIiNGJSkiIiRGKUYlRiVGKSEiIkYlRitGJUYlKiYjRiUiIilGJUYrRiVGLUYvRi0qJl4jRiVGJSwoKihGJEYlLCZGKEYlRilGJUYlRitGJUYlRi5GJUYvRi1GJUYl LCgqJiMiIiIiIiVGJSkiIiQjRiUiIiNGJSEiIiwmRiRGJSomRiRGJV4jRiVGJUYlRiUqJkYtRiVGJ0YlRiU= LCYjIiIiIiNFISIiKiYjIiImRiVGJF4jRiRGJEYm LCYjIiIiIiM1RiQqJkYjRiReI0YkRiRGJA==
<Text-field style="Heading 3" layout="Heading 3">->3.4.7. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.8. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.9. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.10. Komplex sz\303\241m argumentuma \303\251s trigonometrikus alakja.</Font></Text-field> polar(x+I*y); op(1,%); op(2,%%); polar(3+4*I); evalc(%); argument(3+I*4); LUkmcG9sYXJHSShfc3lzbGliRzYiNiQtSSRhYnNHJSpwcm90ZWN0ZWRHNiMsJkkieEdGJSIiIiomXiNGLUYtSSJ5R0YlRi1GLS1JKWFyZ3VtZW50RzYkRilGJEYq LUkkYWJzRyUqcHJvdGVjdGVkRzYjLCZJInhHNiIiIiIqJl4jRilGKUkieUdGKEYpRik= LUkpYXJndW1lbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSJ4R0YnIiIiKiZeI0YrRitJInlHRidGK0Yr LUkmcG9sYXJHSShfc3lzbGliRzYiNiQiIiYtSSdhcmN0YW5HNiQlKnByb3RlY3RlZEdGJDYjIyIiJSIiJA== LCYiIiQiIiIqJiIiJUYkXiNGJEYkRiQ= LUknYXJjdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMjIiIlIiIk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.11. P\303\251lda.</Font></Text-field> z:=16*sqrt(3)-I*16; polar(z); LCYqJiIjOyIiIikiIiQjRiUiIiNGJUYlKiZGJEYlXiNGJUYlISIi LUkmcG9sYXJHSShfc3lzbGliRzYiNiQiI0ssJComIyIiIiIiJ0YrSSNQaUclKnByb3RlY3RlZEdGKyEiIg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.12. Gy\303\266kvon\303\241s komplex sz\303\241mb\303\263l.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.13. P\303\251lda.</Font></Text-field> z:='z'; i:='i'; w:=16*sqrt(3)-I*16; solve(z^5=w,z); z1:=w^(1/5); r:=abs(w); phi:=argument(w); r^(1/5)*(cos(phi/5+i*2*Pi/5)+I*sin(phi/5+i*2*Pi/5))$i=0..4; evalf(%); solve(z^5=1); map(z->evalf(z*z1),[%]); SSJ6RzYi SSJpRzYi LCYqJiIjOyIiIikiIiQjRiUiIiNGJUYlKiZGJEYlXiNGJUYlISIi Warning, solutions may have been lost KiQpLCYqJiIjOyIiIikiIiQjRiciIiNGJ0YnKiZGJkYnXiNGJ0YnISIiI0YnIiImRic= IiNL LCQqJiMiIiIiIidGJUkjUGlHJSpwcm90ZWN0ZWRHRiUhIiI= NicqJikiI0sjIiIiIiImRicsJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJiMiIigiIzpGJ0kjUGlHRi1GJ0YnRicqJiwkXiNGJ0YnRictSSRjb3NHRixGMEYnISIiRicqJkYkRicsJi1GKzYjLCQqJiMiIiNGNUYnRjZGJ0YnRicqJkY5RictRjtGQEYnRidGJyomRiRGJywmLUYrNiMsJComIyIiJUY1RidGNkYnRidGPComRjlGJy1GO0ZKRidGJ0YnKiZGJEYnLCYqJiNGJ0ZERicpIiIkRlRGJ0Y8KiZGVEYnRjlGJ0Y8RicqJkYkRicsJi1GKzYjLCQqJiNGJ0Y1RidGNkYnRidGJyomRjhGJy1GO0ZlbkYnRjxGJw== NicsJiQiKyJ6ViEqKT4hIioiIiIqJiQiK2UjcDA0IyEjNUYnXiNGJ0YnISIiLCYkIitnR3RNIilGK0YnKiYkIis6NDRGPUYmRidGLEYnRicsJiQiK14nKkcnWyJGJkYtKiYkIis3N0VROEYmRidGLEYnRicsJiQiKzMzMEs8RiZGLSomJCIrKysrKzVGJkYnRixGJ0YtLCYkIis9UUJlVEYrRicqJiQiKyxfSGM+RiZGJ0YsRidGLQ== Warning, solutions may have been lost NiciIiIsKCNGIyIiJSEiIiomRiVGIykiIiYjRiMiIiNGI0YjKigqJkYlRiNeI0YjRiNGIylGLEYrRiMpLCZGKkYjKiRGKUYjRiNGK0YjRiMsKEYlRidGKEYnKihGLkYjRjBGIyksJkYqRiNGM0YnRitGI0YjLChGJUYnRihGJyooLCRGLkYjRiNGMEYjRjZGI0YnLChGJUYnRihGIyooRjpGI0YwRiNGMUYjRic= NycsJiQiKyJ6ViEqKT4hIioiIiIqJiQiK2wjcDA0IyEjNUYnXiNGJ0YnISIiLCYkIitlR3RNIilGK0YnKiYkIis6NDRGPUYmRidGLEYnRicsJiQiK14nKkcnWyJGJkYtKiYkIis3N0VROEYmRidGLEYnRicsJiQiKzIzMEs8RiZGLSomJCIrJyoqKioqKioqKkYrRidGLEYnRi0sJiQiKzpRQmVURitGJyomJCIrLF9IYz5GJkYnRixGJ0Yt LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.14. Az algebra alapt\303\251tele.</Font></Text-field> f:=(x-1)^2*(x-2); f:=expand(f); solve(f,x); solve(x^3=1,x); r:=[%]; KiYpLCZJInhHNiIiIiJGJyEiIiIiI0YnLCZGJUYnRilGKEYn LCoqJClJInhHNiIiIiQiIiJGKComIiIlRigpRiUiIiNGKCEiIiomIiImRihGJUYoRihGLEYt Warning, solutions may have been lost NiUiIiMiIiJGJA== Warning, solutions may have been lost NiUiIiIsJiNGIyIiIyEiIiomKiZGJUYjXiNGI0YjRiMpIiIkRiVGI0YjLCZGJUYnKiYsJEYpRiNGI0YrRiNGJw== NyUiIiIsJiNGIyIiIyEiIiomKiZGJUYjXiNGI0YjRiMpIiIkRiVGI0YjLCZGJUYnKiYsJEYpRiNGI0YrRiNGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.4.15. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.16. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.17. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.20. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.22. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.23. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.25. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.27. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.28. Kvaterni\303\263k.</Font></Text-field> `&+`:=(p,q)->[p[1]+q[1],p[2]+q[2]]; `&*`:=(p,q)->[p[1]*q[1]-conjugate(q[2])*p[2],q[2]*p[1]+p[2]*conjugate(q[1])]; Zio2JEkicEc2IkkicUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJiZGJDYjIiIiRi4mRiZGLUYuLCYmRiQ2IyIiI0YuJkYmRjJGLkYlRiVGJQ== Zio2JEkicEc2IkkicUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJiomJkYkNiMiIiJGLyZGJkYuRi9GLyomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjJkYmNiMiIiNGLyZGJEY3Ri8hIiIsJiomRi1GL0Y2Ri9GLyomRjlGLy1GMzYjRjBGL0YvRiVGJUYl p:=[a+I*b,c+I*d]; p&+[0,0]; p&+[-a-I*b,-c-I*d]; NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm NyQiIiFGIw== p&*[1,0]; [1,0]&*p; q:=[(a-I*b)/(a^2+b^2+c^2+d^2),(-c-I*d)/(a^2+b^2+c^2+d^2)]; p&*q;evalc(%);simplify(%); q&*p;evalc(%);simplify(%); NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm NyQqJiwmSSJhRzYiIiIiKiYsJF4jRidGJ0YnSSJiR0YmRichIiJGJywqKiQpRiUiIiNGJ0YnKiQpRitGMEYnRicqJClJImNHRiZGMEYnRicqJClJImRHRiZGMEYnRidGLComLCZGNUYsKiZGKUYnRjhGJ0YsRidGLUYs NyQsJiooLCZJImFHNiIiIiIqJl4jRihGKEkiYkdGJ0YoRihGKCwmRiZGKComLCRGKkYoRihGK0YoISIiRigsKiokKUYmIiIjRihGKCokKUYrRjNGKEYoKiQpSSJjR0YnRjNGKEYoKiQpSSJkR0YnRjNGKEYoRi9GKComLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjKiYsJkY4Ri8qJkYuRihGO0YoRi9GKEYwRi9GKCwmRjhGKComRipGKEY7RihGKEYoRi8sJiooRiVGKEZCRihGMEYvRigqJkZERigtRj42IyomRixGKEYwRi9GKEYo NyQsKComLCYqJClJImFHNiIiIiMiIiJGKyokKUkiYkdGKUYqRitGK0YrLCpGJkYrRixGKyokKUkiY0dGKUYqRitGKyokKUkiZEdGKUYqRitGKyEiIkYrKiZGMUYrRi9GNkYrKiZGNEYrRi9GNkYrLCoqJiwmKiZGKEYrRjJGK0Y2KiZGLkYrRjVGK0YrRitGL0Y2RisqKEYyRitGKEYrRi9GNkYrKihGNUYrRi5GK0YvRjZGNiomXiNGK0YrLCgqJiwmKiZGLkYrRjJGK0Y2KiZGKEYrRjVGK0Y2RitGL0Y2RisqKEY1RitGKEYrRi9GNkYrKihGMkYrRi5GK0YvRjZGK0YrRis= NyQiIiIiIiE= NyQsJiooLCZJImFHNiIiIiIqJl4jRihGKEkiYkdGJ0YoRihGKCwmRiZGKComLCRGKkYoRihGK0YoISIiRigsKiokKUYmIiIjRihGKCokKUYrRjNGKEYoKiQpSSJjR0YnRjNGKEYoKiQpSSJkR0YnRjNGKEYoRi9GKCooLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjLCZGOEYoKiZGKkYoRjtGKEYoRigsJkY4Ri8qJkYuRihGO0YoRi9GKEYwRi9GLywmKihGLEYoRjBGL0ZBRihGKCooRkNGKEYwRi8tRj42I0YlRihGKA== NyQsJiomLCYqJClJImFHNiIiIiMiIiJGKyokKUkiYkdGKUYqRitGK0YrLCpGJkYrRixGKyokKUkiY0dGKUYqRitGKyokKUkiZEdGKUYqRitGKyEiIkYrKiYsJkYwRitGM0YrRitGL0Y2RisiIiE= NyQiIiIiIiE= z:='z';w:='w';z1:='z1';p:=[z,w];p1:=[z1,w1];p2:=[z2,w2]; p&*(p1&*p2);expand(%);(p&*p1)&*p2;expand(%); SSJ6RzYi SSJ3RzYi SSN6MUc2Ig== NyRJInpHNiJJIndHRiQ= NyRJI3oxRzYiSSN3MUdGJA== NyRJI3oyRzYiSSN3MkdGJA== NyQsJiomSSJ6RzYiIiIiLCYqJkkjejFHRiZGJ0kjejJHRiZGJ0YnKiYtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJI3cyR0YmRidJI3cxR0YmRichIiJGJ0YnKiYtRi42IywmKiZGKkYnRjFGJ0YnKiZGMkYnLUYuNiNGK0YnRidGJ0kid0dGJkYnRjMsJiomRiVGJ0Y3RidGJyomRjxGJy1GLjYjRihGJ0Yn NyQsKiooSSJ6RzYiIiIiSSN6MUdGJkYnSSN6MkdGJkYnRicqKEYlRictSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJI3cyR0YmRidJI3cxR0YmRichIiIqJkkid0dGJkYnLUYsNiMqJkYoRidGL0YnRidGMSooRjNGJ0YpRictRiw2I0YwRidGMSwqKihGJUYnRihGJ0YvRidGJyooRiVGJ0YwRictRiw2I0YpRidGJyomRjNGJy1GLDYjKiZGKEYnRilGJ0YnRicqKEYzRidGL0YnRjhGJ0Yx NyQsJiomLCYqJkkiekc2IiIiIkkjejFHRihGKUYpKiYtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJI3cxR0YoRilJIndHRihGKSEiIkYpSSN6MkdGKEYpRikqJi1GLTYjSSN3MkdGKEYpLCYqJkYnRilGMEYpRikqJkYxRiktRi02I0YqRilGKUYpRjIsJiomRiVGKUY3RilGKSomRjhGKS1GLTYjRjNGKUYp NyQsKiooSSJ6RzYiIiIiSSN6MUdGJkYnSSN6MkdGJkYnRicqKEkid0dGJkYnRilGJy1JKmNvbmp1Z2F0ZUclKnByb3RlY3RlZEc2I0kjdzFHRiZGJyEiIiooRiVGJy1GLTYjSSN3MkdGJkYnRjBGJ0YxKihGM0YnRitGJy1GLTYjRihGJ0YxLCoqKEYlRidGKEYnRjVGJ0YnKihGK0YnRjVGJ0YsRidGMSooRiVGJ0YwRictRi02I0YpRidGJyooRj1GJ0YrRidGN0YnRic= p&*(p1&+p2);expand(%);(p&*p1)&+(p&*p2); (p1&+p2)&*p;expand(%);(p1&*p)&+(p2&*p); NyQsJiomSSJ6RzYiIiIiLCZJI3oxR0YmRidJI3oyR0YmRidGJ0YnKiYtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiMsJkkjdzFHRiZGJ0kjdzJHRiZGJ0YnSSJ3R0YmRichIiIsJiomRiVGJ0YwRidGJyomRjNGJy1GLTYjRihGJ0Yn NyQsKiomSSJ6RzYiIiIiSSN6MUdGJkYnRicqJkYlRidJI3oyR0YmRidGJyomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjSSN3MUdGJkYnSSJ3R0YmRichIiIqJkYxRictRi02I0kjdzJHRiZGJ0YyLCoqJkYlRidGMEYnRicqJkYlRidGNkYnRicqJkYxRictRi02I0YoRidGJyomRjFGJy1GLTYjRipGJ0Yn NyQsKiomSSJ6RzYiIiIiSSN6MUdGJkYnRicqJkYlRidJI3oyR0YmRidGJyomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjSSN3MUdGJkYnSSJ3R0YmRichIiIqJkYxRictRi02I0kjdzJHRiZGJ0YyLCoqJkYlRidGMEYnRicqJkYlRidGNkYnRicqJkYxRictRi02I0YoRidGJyomRjFGJy1GLTYjRipGJ0Yn NyQsJiomSSJ6RzYiIiIiLCZJI3oxR0YmRidJI3oyR0YmRidGJ0YnKiYtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJIndHRiZGJywmSSN3MUdGJkYnSSN3MkdGJkYnRichIiIsJiomRihGJ0YwRidGJyomRjFGJy1GLTYjRiVGJ0Yn NyQsKiomSSJ6RzYiIiIiSSN6MUdGJkYnRicqJkYlRidJI3oyR0YmRidGJyomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjSSJ3R0YmRidJI3cxR0YmRichIiIqJkYsRidJI3cyR0YmRidGMiwqKiZGMEYnRihGJ0YnKiZGMEYnRipGJ0YnKiYtRi02I0YlRidGMUYnRicqJkY5RidGNEYnRic= NyQsKiomSSJ6RzYiIiIiSSN6MUdGJkYnRicqJkYlRidJI3oyR0YmRidGJyomLUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjSSJ3R0YmRidJI3cxR0YmRichIiIqJkYsRidJI3cyR0YmRidGMiwqKiZGMEYnRihGJ0YnKiZGMEYnRipGJ0YnKiYtRi02I0YlRidGMUYnRicqJkY5RidGNEYnRic= j:=[0,1]; j&*j; [z,0]&+([w,0]&*j); NyQiIiEiIiI= NyQhIiIiIiE= NyRJInpHNiJJIndHRiQ= k:=[0,I]; k&*k; i:=[I,0]; i&*i; [a,0]&+([b,0]&*i)&+([c,0]&*j)&+([d,0]&*k); NyQiIiFeIyIiIg== NyQhIiIiIiE= NyReIyIiIiIiIQ== NyQhIiIiIiE= NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm p:=[a+I*b,c+I*d]; evalc([x,0]&*p); evalc(p&*[x,0]); NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm NyQsJiomSSJ4RzYiIiIiSSJhR0YmRidGJyooXiNGJ0YnRiVGJ0kiYkdGJkYnRicsJiomRiVGJ0kiY0dGJkYnRicqKEYqRidGJUYnSSJkR0YmRidGJw== NyQsJiomSSJ4RzYiIiIiSSJhR0YmRidGJyooXiNGJ0YnRiVGJ0kiYkdGJkYnRicsJiomRiVGJ0kiY0dGJkYnRicqKEYqRidGJUYnSSJkR0YmRidGJw== j&*[z,0]; [z,0]&*j; NyQiIiEtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJInpHNiI= NyQiIiFJInpHNiI= i&*j; j&*k; k&*i; j&*i; k&*j; i&*k; NyQiIiFeIyIiIg== NyReIyIiIiIiIQ== NyQiIiEiIiI= NyQiIiEsJF4jIiIiISIi NyQsJF4jIiIiISIiIiIh NyQiIiEhIiI= i:='i'; j:='j'; k:='k'; C2toR4:=q->evalc(Re(q[1])+Im(q[1])*i+Re(q[2])*j+Im(q[2])*k); q:=C2toR4(p); SSJpRzYi SSJqRzYi SSJrRzYi Zio2I0kicUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywqLUkjUmVHRiw2IyZGJDYjIiIiRjUqJi1JI0ltR0YsRjJGNUkiaUdGJUY1RjUqJi1GMTYjJkYkNiMiIiNGNUkiakdGJUY1RjUqJi1GOEY8RjVJImtHRiVGNUY1RiVGJUYl LCpJImFHNiIiIiIqJkkiYkdGJEYlSSJpR0YkRiVGJSomSSJjR0YkRiVJImpHRiRGJUYlKiZJImRHRiRGJUkia0dGJEYlRiU= R4toC2:=q->[q-coeff(q,i)*i-coeff(q,j)*j-coeff(q,k)*k+I*coeff(q,i),coeff(q,j)+I*coeff(q,k)]; R4toC2(q); Zio2I0kicUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsLEYkIiIiKiYtSSZjb2VmZkclKnByb3RlY3RlZEc2JEYkSSJpR0YlRitGMUYrISIiKiYtRi42JEYkSSJqR0YlRitGNkYrRjIqJi1GLjYkRiRJImtHRiVGK0Y6RitGMiomXiNGK0YrRi1GK0YrLCZGNEYrKiZGPEYrRjhGK0YrRiVGJUYl NyQsJkkiYUc2IiIiIiomXiNGJkYmSSJiR0YlRiZGJiwmSSJjR0YlRiYqJkYoRiZJImRHRiVGJkYm qIm:=q->coeff(q,i)*i+coeff(q,j)*j+coeff(q,k)*k;qRe:=q->q-qIm(q); qRe(q); qIm(q); Zio2I0kicUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJi1JJmNvZWZmRyUqcHJvdGVjdGVkRzYkRiRJImlHRiUiIiJGL0YwRjAqJi1GLDYkRiRJImpHRiVGMEY0RjBGMComLUYsNiRGJEkia0dGJUYwRjhGMEYwRiVGJUYl Zio2I0kicUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZGJCIiIi1JJHFJbUdGJUYjISIiRiVGJUYl SSJhRzYi LCgqJkkiYkc2IiIiIkkiaUdGJUYmRiYqJkkiY0dGJUYmSSJqR0YlRiZGJiomSSJkR0YlRiZJImtHRiVGJkYm qconjugate:=q->qRe(q)-qIm(q); qconjugate(q); Zio2I0kicUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSSRxUmVHRiVGIyIiIi1JJHFJbUdGJUYjISIiRiVGJUYl LCpJImFHNiIiIiIqJkkiYkdGJEYlSSJpR0YkRiUhIiIqJkkiY0dGJEYlSSJqR0YkRiVGKSomSSJkR0YkRiVJImtHRiRGJUYp q; qconjugate(q); qconjugate(%); q+qconjugate(q); q-qconjugate(q); LCpJImFHNiIiIiIqJkkiYkdGJEYlSSJpR0YkRiVGJSomSSJjR0YkRiVJImpHRiRGJUYlKiZJImRHRiRGJUkia0dGJEYlRiU= LCpJImFHNiIiIiIqJkkiYkdGJEYlSSJpR0YkRiUhIiIqJkkiY0dGJEYlSSJqR0YkRiVGKSomSSJkR0YkRiVJImtHRiRGJUYp LCpJImFHNiIiIiIqJkkiYkdGJEYlSSJpR0YkRiVGJSomSSJjR0YkRiVJImpHRiRGJUYlKiZJImRHRiRGJUkia0dGJEYlRiU= LCQqJiIiIyIiIkkiYUc2IkYlRiU= LCgqKCIiIyIiIkkiYkc2IkYlSSJpR0YnRiVGJSooRiRGJUkiY0dGJ0YlSSJqR0YnRiVGJSooRiRGJUkiZEdGJ0YlSSJrR0YnRiVGJQ== q1:=a1+b1*i+c1*j+d1*k; q2:=a2+b2*i+c2*j+d2*k; q1+q2; collect(%,[i,j,k]); `&+`:=(q1,q2)->collect(q1+q2,[i,j,k]); q1&+q2; `&*`:=proc(q1,q2) local a1,a2,b1,b2,c1,c2,d1,d2; a1:=qRe(q1);a2:=qRe(q2);b1:=coeff(q1,i);b2:=coeff(q2,i); c1:=coeff(q1,j);c2:=coeff(q2,j);d1:=coeff(q1,k);d2:=coeff(q2,k); (a1*a2-b1*b2-c1*c2-d1*d2)+(a1*b2+a2*b1+c1*d2-d1*c2)*i+ (a1*c2+c1*a2+d1*b2-b1*d2)*j+(a1*d2+d1*a2+b1*c2-c1*b2)*k;end; q1&*q2; qconjugate(q1&+q2); qconjugate(q1)&+qconjugate(q2); qconjugate(q1&*q2);qconjugate(q2)&*qconjugate(q1); expand(%%-%); LCpJI2ExRzYiIiIiKiZJI2IxR0YkRiVJImlHRiRGJUYlKiZJI2MxR0YkRiVJImpHRiRGJUYlKiZJI2QxR0YkRiVJImtHRiRGJUYl LCpJI2EyRzYiIiIiKiZJI2IyR0YkRiVJImlHRiRGJUYlKiZJI2MyR0YkRiVJImpHRiRGJUYlKiZJI2QyR0YkRiVJImtHRiRGJUYl LDJJI2ExRzYiIiIiKiZJI2IxR0YkRiVJImlHRiRGJUYlKiZJI2MxR0YkRiVJImpHRiRGJUYlKiZJI2QxR0YkRiVJImtHRiRGJUYlSSNhMkdGJEYlKiZJI2IyR0YkRiVGKEYlRiUqJkkjYzJHRiRGJUYrRiVGJSomSSNkMkdGJEYlRi5GJUYl LCwqJiwmSSNiMUc2IiIiIkkjYjJHRiZGJ0YnSSJpR0YmRidGJyomLCZJI2MyR0YmRidJI2MxR0YmRidGJ0kiakdGJkYnRicqJiwmSSNkMkdGJkYnSSNkMUdGJkYnRidJImtHRiZGJ0YnSSNhMUdGJkYnSSNhMkdGJkYn Zio2JEkjcTFHNiJJI3EyR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJkYkIiIiRiZGMTclSSJpR0YlSSJqR0YlSSJrR0YlRiVGJUYl LCwqJiwmSSNiMUc2IiIiIkkjYjJHRiZGJ0YnSSJpR0YmRidGJyomLCZJI2MyR0YmRidJI2MxR0YmRidGJ0kiakdGJkYnRicqJiwmSSNkMkdGJkYnSSNkMUdGJkYnRidJImtHRiZGJ0YnSSNhMUdGJkYnSSNhMkdGJkYn 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 LDAqJkkjYTFHNiIiIiJJI2EyR0YlRiZGJiomSSNiMUdGJUYmSSNiMkdGJUYmISIiKiZJI2MxR0YlRiZJI2MyR0YlRiZGKyomSSNkMUdGJUYmSSNkMkdGJUYmRisqJiwqKiZGJEYmRipGJkYmKiZGJ0YmRilGJkYmKiZGLUYmRjFGJkYmKiZGMEYmRi5GJkYrRiZJImlHRiVGJkYmKiYsKiomRiRGJkYuRiZGJiomRi1GJkYnRiZGJiomRjBGJkYqRiZGJiomRilGJkYxRiZGK0YmSSJqR0YlRiZGJiomLCoqJkYkRiZGMUYmRiYqJkYwRiZGJ0YmRiYqJkYpRiZGLkYmRiYqJkYtRiZGKkYmRitGJkkia0dGJUYmRiY= LCxJI2ExRzYiIiIiSSNhMkdGJEYlKiYsJkkjYjFHRiRGJUkjYjJHRiRGJUYlSSJpR0YkRiUhIiIqJiwmSSNjMkdGJEYlSSNjMUdGJEYlRiVJImpHRiRGJUYsKiYsJkkjZDJHRiRGJUkjZDFHRiRGJUYlSSJrR0YkRiVGLA== LCwqJiwmSSNiMUc2IiEiIkkjYjJHRiZGJyIiIkkiaUdGJkYpRikqJiwmSSNjMkdGJkYnSSNjMUdGJkYnRilJImpHRiZGKUYpKiYsJkkjZDJHRiZGJ0kjZDFHRiZGJ0YpSSJrR0YmRilGKUkjYTFHRiZGKUkjYTJHRiZGKQ== LDAqJkkjYTFHNiIiIiJJI2EyR0YlRiZGJiomSSNiMUdGJUYmSSNiMkdGJUYmISIiKiZJI2MxR0YlRiZJI2MyR0YlRiZGKyomSSNkMUdGJUYmSSNkMkdGJUYmRisqJiwqKiZGJEYmRipGJkYmKiZGJ0YmRilGJkYmKiZGLUYmRjFGJkYmKiZGMEYmRi5GJkYrRiZJImlHRiVGJkYrKiYsKiomRiRGJkYuRiZGJiomRi1GJkYnRiZGJiomRjBGJkYqRiZGJiomRilGJkYxRiZGK0YmSSJqR0YlRiZGKyomLCoqJkYkRiZGMUYmRiYqJkYwRiZGJ0YmRiYqJkYpRiZGLkYmRiYqJkYtRiZGKkYmRitGJkkia0dGJUYmRis= LDAqJkkjYTFHNiIiIiJJI2EyR0YlRiZGJiomSSNiMUdGJUYmSSNiMkdGJUYmISIiKiZJI2MxR0YlRiZJI2MyR0YlRiZGKyomSSNkMUdGJUYmSSNkMkdGJUYmRisqJiwqKiZGJ0YmRilGJkYrKiZGJEYmRipGJkYrKiZGMEYmRi5GJkYmKiZGLUYmRjFGJkYrRiZJImlHRiVGJkYmKiYsKiomRi1GJkYnRiZGKyomRiRGJkYuRiZGKyomRilGJkYxRiZGJiomRjBGJkYqRiZGK0YmSSJqR0YlRiZGJiomLCoqJkYwRiZGJ0YmRisqJkYkRiZGMUYmRisqJkYtRiZGKkYmRiYqJkYpRiZGLkYmRitGJkkia0dGJUYmRiY= IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">3.4.29. Kvaterni\303\263k abszol\303\272t \303\251rt\303\251ke.</Font></Text-field> qabs:=q->sqrt(qRe(q)^2+coeff(q,i)^2+coeff(q,j)^2+coeff(q,k)^2); qabs(q); q&*qconjugate(q); Zio2I0kicUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCoqJCktSSRxUmVHRiVGIyIiIyIiIkY1KiQpLUkmY29lZmZHRiw2JEYkSSJpR0YlRjRGNUY1KiQpLUY5NiRGJEkiakdGJUY0RjVGNSokKS1GOTYkRiRJImtHRiVGNEY1RjVGJUYlRiU= KiQpLCoqJClJImFHNiIiIiMiIiJGKiokKUkiYkdGKEYpRipGKiokKUkiY0dGKEYpRipGKiokKUkiZEdGKEYpRipGKiNGKkYpRio= LCoqJClJImFHNiIiIiMiIiJGKCokKUkiYkdGJkYnRihGKCokKUkiY0dGJkYnRihGKCokKUkiZEdGJkYnRihGKA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->3.4.30. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*3.4.31. Vektori<Font encoding="UTF-8">\303\241</Font>lis szorz<Font encoding="UTF-8">\303\241</Font>s.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*3.4.32. Kvaterni\303\263k \303\251s a h\303\241romdimenzi\303\263s euklid\303\251szi t\303\251r.</Font></Text-field> q1:=b1*i+c1*j+d1*k; q2:=b2*i+c2*j+d2*k; q3:=b3*i+c3*j+d3*k; scalarprod:=(q1,q2)->-qRe(q1&*q2); scalarprod(q1,q2); vectorprod:=(q1,q2)->qIm(q1&*q2); vectorprod(q1,q2); mixedprod:=(q1,q2,q3)->scalarprod(q1,vectorprod(q2,q3)); mixedprod(q1,q2,q3); LCgqJkkjYjFHNiIiIiJJImlHRiVGJkYmKiZJI2MxR0YlRiZJImpHRiVGJkYmKiZJI2QxR0YlRiZJImtHRiVGJkYm LCgqJkkjYjJHNiIiIiJJImlHRiVGJkYmKiZJI2MyR0YlRiZJImpHRiVGJkYmKiZJI2QyR0YlRiZJImtHRiVGJkYm LCgqJkkjYjNHNiIiIiJJImlHRiVGJkYmKiZJI2MzR0YlRiZJImpHRiVGJkYmKiZJI2QzR0YlRiZJImtHRiVGJkYm Zio2JEkjcTFHNiJJI3EyR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJHFSZUdGJTYjLUkjJipHRiVGIyEiIkYlRiVGJQ== LCgqJkkjYjFHNiIiIiJJI2IyR0YlRiZGJiomSSNjMUdGJUYmSSNjMkdGJUYmRiYqJkkjZDFHRiVGJkkjZDJHRiVGJkYm Zio2JEkjcTFHNiJJI3EyR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSRxSW1HRiU2Iy1JIyYqR0YlRiNGJUYlRiU= LCgqJiwmKiZJI2MxRzYiIiIiSSNkMkdGJ0YoRigqJkkjZDFHRidGKEkjYzJHRidGKCEiIkYoSSJpR0YnRihGKComLCYqJkYrRihJI2IyR0YnRihGKComSSNiMUdGJ0YoRilGKEYtRihJImpHRidGKEYoKiYsJiomRjRGKEYsRihGKComRiZGKEYyRihGLUYoSSJrR0YnRihGKA== Zio2JUkjcTFHNiJJI3EyR0YlSSNxM0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkrc2NhbGFycHJvZEdGJTYkRiQtSSt2ZWN0b3Jwcm9kR0YlNiRGJkYnRiVGJUYl LCgqJkkjYjFHNiIiIiIsJiomSSNjMkdGJUYmSSNkM0dGJUYmRiYqJkkjZDJHRiVGJkkjYzNHRiVGJiEiIkYmRiYqJkkjYzFHRiVGJiwmKiZGLEYmSSNiM0dGJUYmRiYqJkkjYjJHRiVGJkYqRiZGLkYmRiYqJkkjZDFHRiVGJiwmKiZGNUYmRi1GJkYmKiZGKUYmRjNGJkYuRiZGJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*3.4.33. A szorz<Font encoding="UTF-8">\303\241</Font>sok geometriai jelent<Font encoding="UTF-8">\303\251</Font>se.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*3.4.34. Forgat<Font encoding="UTF-8">\303\241sok</Font>.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*3.4.35. A skal<Font encoding="UTF-8">\303\241</Font>ris es vektori<Font encoding="UTF-8">\303\241</Font>lis szorz<Font encoding="UTF-8">\303\241</Font>s geometriai alkalmaz<Font encoding="UTF-8">\303\241</Font>sai.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">* 3.4.36. Okt\303\241vok vagy Cayley-sz\303\241mok.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.37. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->3.4.38. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*3.4.39. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.40. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok megold<Font encoding="UTF-8">\303\241</Font>sokkal.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">3.4.41. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">4. V\303\251ges halmazok</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">5. V\303\251gtelen halmazok</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">6. Sz\303\241melm\303\251let</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">7. Gr\303\241felm\303\251let</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">8. Algebra</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">9. K\303\263dol\303\241s</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">10. Algoritmusok</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn