function RAT(t,b){  // if infinite, sets  t=1 and b=0
  if(!isZ(t))halt("Cant make Q from num:"+t);
  if(!isZ(b))halt("cant make Q from den:"+b);
  if(b==0){t=(t>0?1:-1)}
  else{var g=gcd(t,b);if(g>1){t=Math.round(t/g);b=Math.round(b/g)}};
  if(b<0){t=-t;b=-b}
  this.top=NB(t);
  this.bot=NB(b);
  this.isfinite=b!=0;
  this.toString=function(){
      return (this.isfinite?(this.bot==1?this.top.toString()
                                        :this.top.toString()+"/"+this.bot.toString()+"")
                           :(this.top<0?"-":"")+"Infinity")};
  this.val=NB(t/b);
  this.sgnum=(this.top<0?-1:this.top==0?0:1);
};
function RATmul(a,b){if(typeof a=="number")a=new RAT(a,1);if(typeof b =="number")b=new RAT(b,1);
   return new RAT(a.top*b.top,a.bot*b.bot)};
function RATdiv(a,b){if(typeof a=="number")a=new RAT(a,1);if(typeof b =="number")b=new RAT(b,1);
   return new RAT(a.top*b.bot,a.bot*b.top)};
function RATadd(a,b){if(typeof a=="number")a=new RAT(a,1);if(typeof b =="number")b=new RAT(b,1);
   return new RAT(a.top*b.bot+b.top*a.bot,a.bot*b.bot)};
function RATsub(a,b){if(typeof a=="number")a=new RAT(a,1);if(typeof b=="number")b=new RAT(b,1);
   return RATadd(a,RATneg(b))};
function RATneg(a){return (typeof a=="number"?new RAT(-a,1):new RAT(-a.top,a.bot))};
function RATis(a,op,b){if(typeof a=="number")a=new RAT(a,1);if(typeof b=="number")b=new RAT(b,1);
     var L=a.top*b.bot,R=a.bot*b.top;
     return eval(isFinite(L)&&isFinite(R)?L+op+R:a.val+op+b.val)};


var toosmall=0.0000000001,toolarge=100000;
function NB(x){if(Math.abs(Math.round(x)-x)<=toosmall)return Math.round(x)
    else if(x>toolarge)return Number.POSITIVE_INFINITY
    else if(x<-toolarge)return Number.NEGATIVE_INFINITY
    else return x};
function norm(x){
   //if(isPOLY(x)&&x.deg==0){x=POLYcoeff(x,0)};
   //if(isC(x)&&NB(Cim(x))==0){x=Cre(x)};
   if(isQ(x)&&x.bot==1){x=x.top};
   if(isR(x)&&Math.abs(Math.round(x)-x)<toosmall)x=Math.round(x);
   return x
};
function Z2Q(n){return new RAT(n,1)};
//function Q2C(r){return new COMPLX(r,0)};
function Q2R(x){return x.val};
//function R2C(x){return new COMPLX(x,0)};
function isZ(x){return !isNaN(x) && isFinite(x) &&  x.toString().indexOf(".")==-1};
function isR(x){return !isNaN(x) && isFinite(x) && x.toString().indexOf(".")!=-1};

function isNb(x){return isZ(x)||isR(x)};
function isQ(x){return typeof x=="object" && x.top};   //x instanceof RAT }  
//function isC(x){return typeof x=="object" && x.im};   //x instanceof COMPLX};
function isfinite(n){return isQ(n)? n.isfinite:isFinite(n)};
function isInfinite(x){return isQ(x)&&!x.isfinite //|| !isC(x) &&!isfinite(x)
};
var notyp="?no-type?"
function typ(x){return isZ(x)?"Z":isR(x)?"R":isQ(x)?"Q"//:isC(x)?"C":isPOLY(x)?"P"
          :notyp};
function isString(s){return typeof s == "string"};
function toR(x){return //isC(x)?x:
      isQ(x)?x.val:isR(x)||isZ(x)?x:halt("cannot convert "+x+" to R")};
//function toC(x){if(typ(x)=="C"){return x }else{ return [x,0]}};
function toQ(x){switch(typ(x)){
   case "Q":return x;
  // case "C":halt("Cant convert CtoQ "+C);
   case "R":halt("Cant convert RtoQ "+x);
   case "Z":return Z2Q(x);}};
function toR(x){
     if(isQ(x))return x.val;
     //if(isC(x)){if(x[1]==0){return x[0]}else{halt("Cant convert CtoR:"+x)}};
     if(isNb(x))return x;
     };


function add(x,y){//document.writeln(">>> ",typeof x,"=",x,"  ",typeof y,"=",y);
        if(isInfinite(x) && isInfinite(y)) return (cmp(x,">",0)==cmp(y,">",0)?x:Number.NaN)
   else if(isInfinite(x))return x
   else if(isInfinite(y))return y
   else if(isR(x)||isR(y))return norm(toR(x)+toR(y))
   else if(isQ(x)||isQ(y)){x=toQ(x);y=toQ(y);return norm(RATadd(x,y))};
   return norm(NB(eval(x)+eval(y)));
 };
function sub(x,y){return add(x,neg(y))};
function neg(x){ if(isNb(x))return -x;
   if(isInfinite(x))return -x;
   if(isQ(x))return RATneg(x);
   halt("??? negate "+x)};
function mul(a,b){
        if(isInfinite(a))return (b==0?Number.NaN:cmp(b,">",0)?a:-a)
   else if(isInfinite(b))return (a==0?Number.NaN:cmp(a,">",0)?b:-b)
 //  else if(isC(a)||isC(b)){a=toC(a);b=toC(b);return norm(Cmul(a,b))}
   else if(isR(a)||isR(b))return norm(toR(a)*toR(b));
   else if(isQ(a)||isQ(b)){a=toQ(a);b=toQ(b);return norm(RATmul(a,b))}
   return norm(a*b)};
function mod(a,b){
  if(isInfinite(b))return (!isInfinite(a)?a:Number.NaN);
  if(isZ(a)&&isZ(b))return a%b
  return Number.NaN};
function div(n,d){//alert("div "+typ(n)+" "+typ(d));
   if(isR(n)||isR(d)){n=toR(n);d=toR(d);return norm(NB(n/d))};
   //alert(n+" mod "+d+" = "+mod(n,d)+" flr="+Math.floor(n/d));
   if(isZ(n)&&isZ(d))
       if(mod(n,d)==0)return norm(Math.round(n/d))
       else return norm(new RAT(n,d));
   n=toQ(n);d=toQ(d);return norm(RATdiv(n,d))
};
function pow(a,b){//alert("pow "+typ(a)+"^"+typ(b));
  if(isQ(b))b=b.val;
  if(isQ(a)){
             if(isZ(b)&&b<0)return new RAT(Math.pow(a.bot,-b),Math.pow(a.top,-b))
             else           return new RAT(Math.pow(a.top,b),Math.pow(a.bot,b))};
  a=toR(a);return norm(NB(Math.pow(a,b)))
};
function cmp(a,op,b){
   if(op=="=")op="==";
  if(isQ(a)||isQ(b)){a=toQ(a);b=toQ(b);return RATis(a,op,b)};
   return eval(a+op+b)};
function abs(x){if(cmp(x,"<",0))x=neg(x);return x};
function num(n){ return isQ(n)?n.top:n };
function den(n){return isQ(n)?n.bot:1};