<Text-field style="Heading 1" layout="Heading 1">1. Halmazok</Text-field>

<Text-field style="Heading 1" layout="Heading 1">2. Sz\303\241mok</Text-field>

restart;

<Text-field style="Heading 2" layout="Heading 2">2.1. Val\303\263s sz\303\241mok</Text-field>

<Text-field style="Heading 3" layout="Heading 3">2.1.1. Test.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.2. P\303\251lda.</Text-field>

&+(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

IiIh

IiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.3. P\303\251ld\303\241k.</Text-field>

`&+`:=(x,y)->irem(x+y,5); `&*`:=(x,y)->irem(x*y,5); 3&+4; 3&*4;

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklaXJlbUclKnByb3RlY3RlZEc2JCwmRiQiIiJGJkYvIiImRiVGJUYl

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklaXJlbUclKnByb3RlY3RlZEc2JComRiQiIiJGJkYvIiImRiVGJUYl

IiIj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.4. Algebrai strukt\303\272r\303\241k.</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==

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==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.5. P\303\251ld\303\241k.</Text-field>

iff:=(x,y)->evalb((x implies y) and (y implies x)); X:={true,false}; isabeliangroup(X,(x,y)->iff(x,y));

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkmZXZhbGJHJSpwcm90ZWN0ZWRHNiMzaEYkRiZoRiZGJEYlRiVGJQ==

PCRJJmZhbHNlRyUqcHJvdGVjdGVkR0kldHJ1ZUdGJA==

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.6. P\303\251ld\303\241k.</Text-field>

X:={a,b,c};with(combinat,powerset);P:=powerset(X);isgroup(P,(x,y)->symmdiff(x,y));

PCVJImFHNiJJImJHRiRJImNHRiQ=

NyNJKXBvd2Vyc2V0RzYi

PCo8IjwlSSJhRzYiSSJiR0YmSSJjR0YmPCRGJ0YoPCNGKDwkRiVGKDwjRiU8I0YnPCRGJUYn

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.7. P\303\251ld\303\241k.</Text-field>

isring({0},(x,y)->0,(x,y)->0);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

isring(P,(x,y)->symmdiff(x,y),(x,y)->{});

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.8. P\303\251ld\303\241k.</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==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.9. P\303\251ld\303\241k.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.10. Rendezett test.</Text-field>

abs(7.4); abs(-3); abs(0); signum(7.4); signum(-3); signum(0);

JCIjdSEiIg==

IiIk

IiIh

IiIi

ISIi

IiIh

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.11. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.12. P\303\251ld\303\241k.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.13. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.14. Val\303\263s sz\303\241mok.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.15. Term\303\251szetes sz\303\241mok.</Text-field>

inc:=x->x+1;0;inc(%);inc(%);inc(%);inc(%); dec:=x->x-1;4;dec(%);dec(%);dec(%);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZGJCIiIkYqRipGJUYlRiU=

IiIh

IiIi

IiIj

IiIk

IiIl

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZGJCIiIkYqISIiRiVGJUYl

IiIl

IiIk

IiIj

IiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.16. Rekurzi\303\263t\303\251tel.</Text-field>

n:=16;twopower:=1;for i to n do twopower:=twopower*2; od;

IiM7

IiIi

IiIj

IiIl

IiIp

IiM7

IiNL

IiNr

IiRHIg==

IiRjIw==

IiQ3Jg==

IiVDNQ==

IiVbPw==

IiUnNCU=

IiUjPik=

IiYlUTs=

IiZvRiQ=

IiZPYic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.17. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.18. Sorozatok.</Text-field>

i:='i';j:='j';$3..9;i^2$i=2/3..10/3;x[i]$i=3..8;{j^i$j=i..8}$i=2..4;

SSJpRzYi

SSJqRzYi

NikiIiQiIiUiIiYiIiciIigiIikiIio=

NiUjIiIlIiIqIyIjREYlIyIja0Yl

NigmSSJ4RzYiNiMiIiQmRiQ2IyIiJSZGJDYjIiImJkYkNiMiIicmRiQ2IyIiKCZGJDYjIiIp

NiU8KSIiJSIiKiIjOyIjRCIjTyIjXCIjazwoIiNGRioiJEQiIiQ7IyIkViQiJDcmPCciJGMjIiREJyIlJ0giIiUsQyIlJzQl

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.19. P\303\251lda.</Text-field>

cat("ab","bcc");cat("bcc","ab");evalb(%=%%);"ab"||"bcc";

USZhYmJjYzYi

USZiY2NhYjYi

SSZmYWxzZUclKnByb3RlY3RlZEc=

USZhYmJjYzYi

with(StringTools,Generate):Generate(4,"abc");

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.20. Motiv\303\241ci\303\263: tov\303\241bbi rekurz\303\255v defin\303\255ci\303\263k.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.21. \303\201ltal\303\241nos rekurzi\303\263 t\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.22. Fibonacci-sz\303\241mok.</Text-field>

Fib:=proc(n::nonnegint) option remember; if n<2 then n else Fib(dec(n))+Fib(dec(dec(n))) fi end;interface(verboseproc=3): print(Fib); Fib(3);print(Fib); Fib(7);print(Fib);

Zio2IydJIm5HNiJJKm5vbm5lZ2ludEclKnByb3RlY3RlZEdGJjYjSSlyZW1lbWJlckdGJkYmQCUyRiUiIiNGJSwmLUkkRmliR0YmNiMtSSRkZWNHRiY2I0YlIiIiLUYwNiMtRjNGMUY1RiZGJkYm

IiIj

Zio2IydJIm5HNiJJKm5vbm5lZ2ludEclKnByb3RlY3RlZEdGJjYjSSlyZW1lbWJlckdGJkVccyUiIiFGLCIiIkYtIiIjRi0iIiRGLkAlMkYlRi5GJSwmLUkkRmliR0YmNiMtSSRkZWNHRiY2I0YlRi0tRjQ2Iy1GN0Y1Ri1GJkYmRiY=

IiM4

Zio2IydJIm5HNiJJKm5vbm5lZ2ludEclKnByb3RlY3RlZEdGJjYjSSlyZW1lbWJlckdGJkVccykiIiFGLCIiIkYtIiIjRi0iIiRGLiIiJkYwIiIlRi8iIigiIzgiIiciIilAJTJGJUYuRiUsJi1JJEZpYkdGJjYjLUkkZGVjR0YmNiNGJUYtLUY6NiMtRj1GO0YtRiZGJkYm

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.23. Szorzatok \303\251s \303\266sszegek.</Text-field>

prodfrom1:=proc(x,n) if n<1 then 1 elif n=1 then x[1] else prodfrom1(x,dec(n))*x[n] fi end; prodfrom1(y,5); product(y[i],i=0..7); sum(x[j],j=4.5..8.7);

Zio2JEkieEc2IkkibkdGJUYlRiVGJUAnMkYmIiIiRikvRiZGKSZGJDYjRikqJi1JKnByb2Rmcm9tMUdGJTYkRiQtSSRkZWNHRiU2I0YmRikmRiRGM0YpRiVGJUYl

KiwmSSJ5RzYiNiMiIiJGJyZGJDYjIiIjRicmRiQ2IyIiJEYnJkYkNiMiIiVGJyZGJDYjIiImRic=

KjImSSJ5RzYiNiMiIiEiIiImRiQ2I0YoRigmRiQ2IyIiI0YoJkYkNiMiIiRGKCZGJDYjIiIlRigmRiQ2IyIiJkYoJkYkNiMiIidGKCZGJDYjIiIoRig=

LComSSJ4RzYiNiMiIiYiIiImRiQ2IyIiJ0YoJkYkNiMiIihGKCZGJDYjIiIpRig=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.24. Az \303\241ltal\303\241nos disztributivit\303\241s t\303\251tele.</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">2.1.25. Faktori\303\241lis, binomi\303\241lis egy\303\274tthat\303\263.</Text-field>

0!; 1!; 2!; 3!; 4!; 5!; 6!;

IiIi

IiIj

IiIn

IiND

IiQ/Ig==

IiQ/KA==

binomial(6,0);binomial(6,1);binomial(6,2);binomial(6,3);binomial(6,4);

IiIi

IiIn

IiM6

IiM/

IiM6

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.26. Binomi\303\241lis t\303\251tel.</Text-field>

(x+y)^6;expand(%);

KiQpLCZJInhHNiIiIiJJInlHRiZGJyIiJ0Yn

LDAqJClJInhHNiIiIiciIiJGKCooRidGKClGJSIiJkYoSSJ5R0YmRihGKCooIiM6RigpRiUiIiVGKClGLCIiI0YoRigqKCIjP0YoKUYlIiIkRigpRixGNkYoRigqKEYuRigpRiVGMkYoKUYsRjBGKEYoKihGJ0YoRiVGKClGLEYrRihGKCokKUYsRidGKEYo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.27. K\303\266vetkezm\303\251ny.</Text-field>

sum(binomial(n,k),k=0..n);sum(binomial(n,k)*(-1)^k,k=0..n);

IiZPYic=

IiIh

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.28. Eg\303\251sz sz\303\241mok.</Text-field>

type(5,integer); type(-3,integer); type(0,integer); type(3.14,integer); type(5,posint); type(-3,negint); type(0,posint); type(0,nonnegint); type(0,nonposint);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

SSZmYWxzZUclKnByb3RlY3RlZEc=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

SSZmYWxzZUclKnByb3RlY3RlZEc=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.29. Hatv\303\241nyoz\303\241s eg\303\251sz kitev\305\221vel.</Text-field>

x^(-5); x^(n+m); expand(%); (x^4)^(-5); (x*y)^5;

KiQpSSJ4RzYiIiImISIi

KUkieEc2IiwmIiM7IiIiSSJtR0YkRic=

KiYpSSJ4RzYiIiM7IiIiKUYkSSJtR0YlRic=

KiQpSSJ4RzYiIiM/ISIi

KiYpSSJ4RzYiIiImIiIiKUkieUdGJUYmRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.30. Racion\303\241lis sz\303\241mok.</Text-field>

type(5/7,rational); type(0,rational);

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.31. Archim\303\251d\303\251szi tulajdons\303\241g.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.32. \303\201ll\303\255t\303\241s.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.33. Eg\303\251sz r\303\251sz, marad\303\251k.</Text-field>

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">2.1.34. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.35. T\303\251tel: gy\303\266kvon\303\241s.</Text-field>

root[2](2); evalf(%); sqrt(2); root[12](2); evalf(%);

KiQpIiIjIyIiIkYkRiY=

JCIraU5AOTkhIio=

KiQpIiIjIyIiIkYkRiY=

KiQpIiIjIyIiIiIjN0Ym

JCIrJTRqJWY1ISIq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.36. K\303\266vetkezm\303\251ny.</Text-field>

<Text-field style="Heading 3" layout="Heading 3">* 2.1.37. \303\201ll\303\255t\303\241s.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.1.38. \303\201ll\303\255t\303\241s.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.1.39. B\305\221v\303\255tett val\303\263s sz\303\241mok.</Text-field>

infinity; -infinity; evalb(5<infinity); evalb(5<-infinity);

SSlpbmZpbml0eUclKnByb3RlY3RlZEc=

LCRJKWluZmluaXR5RyUqcHJvdGVjdGVkRyEiIg==

SSV0cnVlRyUqcHJvdGVjdGVkRw==

SSZmYWxzZUclKnByb3RlY3RlZEc=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 2" layout="Heading 2">2.2. Megsz\303\241ml\303\241lhat\303\263 halmazok.</Text-field>

restart;

<Text-field style="Heading 3" layout="Heading 3">2.2.1. Halmazok ekvivalenci\303\241ja.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.2. \303\201ll\303\255t\303\241s.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.3. Megjegyz\303\251s.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.4. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.5. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.6. V\303\251ges \303\251s v\303\251gtelen halmazok.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.7. Karakterisztikus f\303\274ggv\303\251nyek.</Text-field>

X:={a,b,c,d,e}; Y:={a,c,e}; chi:=x-> if x in Y then 1 elif x in X then 0 else FAIL fi;chi(a);chi(b);chi(1);

PCdJImFHNiJJImJHRiRJImNHRiRJImVHRiRJImRHRiQ=

PCVJImFHNiJJImNHRiRJImVHRiQ=

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlQCctSSNpbkclKnByb3RlY3RlZEc2JEYkSSJZR0YlIiIiLUYrNiRGJEkiWEdGJSIiIUklRkFJTEdGLEYlRiVGJQ==

IiIi

IiIh

SSVGQUlMRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.8. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.9 Skatulya elv.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.10. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.11. Megsz\303\241ml\303\241lhat\303\263 halmazok.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.12. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.13. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.14. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.15. T\303\251tel.</Text-field>

for k from 0 do for m from 0 to k do n:=k-m: T:=time(); while time()<T+1 do od; print([m,n],(k*(k+1)/2)+m); od; od;

NiQ3JCIiIUYkRiQ=

NiQ3JCIiISIiIkYl

NiQ3JCIiIiIiISIiIw==

NiQ3JCIiISIiIyIiJA==

NiQ3JCIiIkYkIiIl

NiQ3JCIiIyIiISIiJg==

NiQ3JCIiISIiJCIiJw==

NiQ3JCIiIiIiIyIiKA==

NiQ3JCIiIyIiIiIiKQ==

NiQ3JCIiJCIiISIiKg==

NiQ3JCIiISIiJSIjNQ==

NiQ3JCIiIiIiJCIjNg==

NiQ3JCIiI0YkIiM3

NiQ3JCIiJCIiIiIjOA==

Warning, computation interrupted

<Text-field style="Heading 3" layout="Heading 3">2.2.16. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.17. K\303\266vetkezm\303\251ny.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.18. T\303\251tel.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.19. K\303\266vetkezm\303\251ny.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.2.20. Cantor t\303\251tele.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 2" layout="Heading 2">2.3. Komplex sz\303\241mok</Text-field>

restart;

<Text-field style="Heading 3" layout="Heading 3">2.3.1. Komplex sz\303\241mok.</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=

LCYiIiMhIiIqJiIiJyIiIl4jRidGJ0Yk

LCYiI0MiIiIqJiIjR0YkXiNGJEYkISIi

IiIk

IiIm

LCYiIiQiIiIqJiIiJkYkXiNGJEYkISIi

z:='z';w:='w'; conjugate(z); conjugate(conjugate(z));conjugate(z+w);conjugate(1/z);

SSJ6RzYi

SSJ3RzYi

LUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjSSJ6RzYi

SSJ6RzYi

LUkqY29uanVnYXRlRyUqcHJvdGVjdGVkRzYjLCZJInpHNiIiIiJJIndHRihGKQ==

KiQtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJInpHNiIhIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.3.2. P\303\251lda.</Text-field>

64/(3^(1/2)+I); evalc(%);

LCQqJiIjayIiIiwmKiQpIiIkI0YlIiIjRiVGJV4jRiVGJSEiIkYl

LCYqJiIjOyIiIikiIiQjRiUiIiNGJUYlKiZGJEYlXiNGJUYlISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.3.3. Komplex sz\303\241mok abszol\303\272t \303\251rt\303\251ke.</Text-field>

z:=x+I*y;abs(z);evalc(%);evalc(1/(x+I*y));evalc(conjugate(z)/abs(z)^2);

LCZJInhHNiIiIiIqJl4jRiVGJUkieUdGJEYlRiU=

LUkkYWJzRyUqcHJvdGVjdGVkRzYjLCZJInhHNiIiIiIqJl4jRilGKUkieUdGKEYpRik=

KiQpLCYqJClJInhHNiIiIiMiIiJGKiokKUkieUdGKEYpRipGKiNGKkYpRio=

LCYqJkkieEc2IiIiIiwmKiQpRiQiIiNGJkYmKiQpSSJ5R0YlRipGJkYmISIiRiYqKCwkXiNGJkYmRiZGLUYmRidGLkYu

signum(3+4*I); signum(-5); signum(0);

LCYjIiIkIiImIiIiKiYjIiIlRiVGJl4jRiZGJkYm

ISIi

IiIh

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.3.4. Komplex sz\303\241mok argumentuma \303\251s trigonometrikus alakja.</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">2.3.5. P\303\251lda.</Text-field>

z:=16*sqrt(3)-I*16; polar(z);

LCYqJiIjOyIiIikiIiQjRiUiIiNGJUYlKiZGJEYlXiNGJUYlISIi

LUkmcG9sYXJHSShfc3lzbGliRzYiNiQiI0ssJComIyIiIiIiJ0YrSSNQaUclKnByb3RlY3RlZEdGKyEiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.3.6. Gy\303\266kvon\303\241s komplex sz\303\241mb\303\263l.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.3.7. P\303\251lda.</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">2.3.8. B\305\221v\303\255tett komplex sz\303\241mok.</Text-field>

z:=infinity+I*infinity; w:=infinity-I*infinity; evalb(z=w);

LCZJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiIiomRiNGJV4jRiVGJUYl

LCZJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiIiomLCRGI0YlRiVeI0YlRiUhIiI=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.3.9. Kvaterni\303\263k.</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

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

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

NyQsJiomSSJ6RzYiIiIiLCZJI3oxR0YmRidJI3oyR0YmRidGJ0YnKiYtSSpjb25qdWdhdGVHJSpwcm90ZWN0ZWRHNiNJIndHRiZGJywmSSN3MUdGJkYnSSN3MkdGJkYnRichIiIsJiomRihGJ0YwRidGJyomRjFGJy1GLTYjRiVGJ0Yn

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==

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">2.3.10. Kvaterni\303\263k \303\251s a h\303\241rom dimenzi\303\263s euklid\303\251szi t\303\251r.</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">2.3.11. Kvaterni\303\263k abszol\303\272t \303\251rt\303\251ke.</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">2.3.12. A skal\303\241r- vektor- \303\251s vegyes szorzat geometriai jelent\303\251se.</Text-field>

qabs(q1&+q2)^2-qabs(q1-q2)^2;

LCgqKCIiJSIiIkkjYjFHNiJGJUkjYjJHRidGJUYlKihGJEYlSSNjMUdGJ0YlSSNjMkdGJ0YlRiUqKEYkRiVJI2QxR0YnRiVJI2QyR0YnRiVGJQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 2" layout="Heading 2">2.4. Polinomok</Text-field>

restart;

<Text-field style="Heading 3" layout="Heading 3">2.4.1. Jel\303\266l\303\251s.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.2. Polinomok \303\251s racion\303\241lis t\303\266rtf\303\274ggv\303\251nyek.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.3. A marad\303\251kos oszt\303\241s t\303\251tele polinomokra.</Text-field>

f:=3*x^3+6*x^2+7*x-8; g:=x^2+8*x-2; q:=quo(f,g,x); r:=rem(f,g,x); g*q+r; expand(%);

LCoqJiIiJCIiIilJInhHNiJGJEYlRiUqJiIiJ0YlKUYnIiIjRiVGJSomIiIoRiVGJ0YlRiUiIikhIiI=

LCgqJClJInhHNiIiIiMiIiJGKComIiIpRihGJUYoRihGJyEiIg==

LCYqJiIiJCIiIkkieEc2IkYlRiUiIz0hIiI=

LCYiI1chIiIqJiIkZCIiIiJJInhHNiJGJ0Yn

LCgqJiwoKiQpSSJ4RzYiIiIjIiIiRioqJiIiKUYqRidGKkYqRikhIiJGKiwmKiYiIiRGKkYnRipGKiIjPUYtRipGKiIjV0YtKiYiJGQiRipGJ0YqRio=

LCoqJiIiJCIiIilJInhHNiJGJEYlRiUqJiIiJ0YlKUYnIiIjRiVGJSomIiIoRiVGJ0YlRiUiIikhIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.4. K\303\266vetkezm\303\251ny: Horner-elrendez\303\251s.</Text-field>

rem(f,x-5,x); subs(x=5,f);

IiRfJg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.5. K\303\266vetkezm\303\251ny: gy\303\266kt\303\251nyez\305\221 lev\303\241laszt\303\241sa.</Text-field>

f:=(x-1)^2*(x-2); f:=expand(f); quo(f,x-2,x);

KiYpLCZJInhHNiIiIiJGJyEiIiIiI0YnLCZGJUYnRilGKEYn

LCoqJClJInhHNiIiIiQiIiJGKComIiIlRigpRiUiIiNGKCEiIiomIiImRihGJUYoRihGLEYt

LCgqJClJInhHNiIiIiMiIiJGKComRidGKEYlRighIiJGKEYo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.6. K\303\266vetkezm\303\251ny.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.7. K\303\266vetkezm\303\251ny.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.8. Polinomok egy\303\251rtelm\305\261 fel\303\255r\303\241sa.</Text-field>

coeff(f,x,0); coeff(f,x,1); coeff(f,x,2); coeff(f,x,3); coeff(f,x,4); lcoeff(f,x); lcoeff(0,x);

ISIj

IiIm

ISIl

IiIi

IiIh

IiIi

IiIh

degree(f); degree(0);

IiIk

LCRJKWluZmluaXR5RyUqcHJvdGVjdGVkRyEiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.9. A marad\303\251kos oszt\303\241s egy\303\251rtelm\305\261s\303\251ge.</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.10. T\303\266bbsz\303\266r\303\266s gy\303\266k\303\266k.</Text-field>

subs(x=1,f); quo(f,x-1,x); subs(x=1,f); quo(f,(x-1)^2,x);

IiIh

LCgqJClJInhHNiIiIiMiIiJGKComIiIkRihGJUYoISIiRidGKA==

IiIh

LCZJInhHNiIiIiIiIiMhIiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.11. Az algebra alapt\303\251tele.</Text-field>

solve(f,x); solve(x^3=1,x); r:=[%];

NiUiIiMiIiJGJA==

NiUiIiIsJiNGIyIiIyEiIiomKiZGJUYjXiNGI0YjRiMpIiIkRiVGI0YjLCZGJUYnKiYsJEYpRiNGI0YrRiNGJw==

NyUiIiIsJiNGIyIiIyEiIiomKiZGJUYjXiNGI0YjRiMpIiIkRiVGI0YjLCZGJUYnKiYsJEYpRiNGI0YrRiNGJw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.12. Gy\303\266kt\303\251nyez\305\221s el\305\221\303\241ll\303\255t\303\241s.</Text-field>

map(y->x-y,r); convert(%,`*`); evalc(%);

NyUsJkkieEc2IiIiIkYmISIiLChGJEYmI0YmIiIjRiYqJiwkKiZGKUYmXiNGJkYmRiZGJikiIiRGKUYmRicsKEYkRiZGKUYmKiZGLUYmRi9GJkYm

KigsJkkieEc2IiIiIkYmISIiRiYsKEYkRiYjRiYiIiNGJiomLCQqJkYpRiZeI0YmRiZGJkYmKSIiJEYpRiZGJ0YmLChGJEYmRilGJiomRi1GJkYvRiZGJkYm

LCYqJClJInhHNiIiIiQiIiJGKEYoISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">* 2.4.13. Lagrange-interpol\303\241ci\303\263.</Text-field>

CurveFitting[PolynomialInterpolation]([0,1,2,3],[0,3,1,3],x); CurveFitting[PolynomialInterpolation]([0,1,2,3],[0,3,1,3],x,form=Lagrange);

LCgqJiMiIiQiIiMiIiIpSSJ4RzYiRiVGJ0YnKiYiIihGJylGKUYmRichIiIqJiMiIzxGJkYnRilGJ0Yn

LCgqKiMiIiQiIiMiIiJJInhHNiJGJywmRihGJ0YmISIiRicsJkYoRidGJUYrRidGJyoqI0YnRiZGJ0YoRicsJkYoRidGJ0YrRidGLEYnRisqKkYuRidGKEYnRi9GJ0YqRidGJw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" layout="Heading 3">2.4.14. T\303\266bbv\303\241ltoz\303\263s polinomok \303\251s racion\303\241lis t\303\266rtf\303\274ggv\303\251nyek.</Text-field>

f:=5*x^2*y; type(f,monomial); type(f,polynom); degree(f); f:=f+x*y^3; type(f,monomial); type(f,polynom); degree(f);

LCQqKCIiJiIiIilJInhHNiIiIiNGJUkieUdGKEYlRiU=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IiIk

LCYqKCIiJiIiIilJInhHNiIiIiNGJUkieUdGKEYlRiUqJkYnRiUpRioiIiRGJUYl

SSZmYWxzZUclKnByb3RlY3RlZEc=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

IiIl

f/(2*x*y-5*x^2*y^2); type(f,ratpoly);

KiYsJiooIiImIiIiKUkieEc2IiIiI0YmSSJ5R0YpRiZGJiomRihGJilGKyIiJEYmRiZGJiwmKihGKkYmRihGJkYrRiZGJiooRiVGJkYnRiYpRitGKkYmISIiRjM=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn