@nm Of course, from the standpoint of common sense, there is no difference. But Maple is a system of computer algebra and everything that is obvious to a person must be properly programmed. Clearly, developers have a lot more work to do on this.

For example, Maple does not know how to find  RealRange(0,2) union RealRange(1,3) :

R1:=RealRange(0,2);
R2:=RealRange(1,3);
R1 union R2;
simplify(%);
is(%=RealDomain(0,3));
                     

@Carl Love  We have

solve(evalc(Im(x*ln(y))), {x,y}); 
                                        {x = 0, y = y}, {x = x, 0 <= y}

But  ln(0)  does not exist.

Unfortunately, Maple makes mistakes in the simplest examples, for example:

solve(x*ln(y)<0);
RealDomain:-solve(x*ln(y)<0);
                                         {y = y, x < 0}, {y = y, 0 < x}
                                         {y = y, x < 0}, {y = y, 0 < x}
 

@Rouben Rostamian  Here is another animation, in which the quadrilateral takes various forms, in particular, becomes non-convex and even degenerate. For  t = 0  this coincides with my picture above.

restart;
VanAubel:=proc(t)
local P, Q, A, B, Squares, Pol, L1, L2, T1, T2;
uses plottools, plots;
P[1], P[2], P[3], P[4]:=<0,0>,<7*cos(t),7*sin(t)>,<sqrt(52)*cos(2*t+arctan(2/3)),sqrt(52)*sin(2*t+arctan(2/3))>,<sqrt(5)*cos(3*t+arctan(2)),sqrt(5)*sin(3*t+arctan(2))>:
P[5]:=P[1]:
assign(seq(Q[i]=(P[i]+P[i+1])/2+<0,1; -1,0>.(P[i+1]-P[i])/2, i=1..4)):
assign(seq(A[i]=Q[i]+(Q[i]-P[i]), i=1..4)):
assign(seq(B[i]=Q[i]+(Q[i]-P[i+1]), i=1..4)):
Pol:=polygon([seq(convert(P[i],list),i=1..4)],style=line, color=blue, thickness=2):
Squares:=seq(polygon(convert~([P[i],P[i+1],A[i],B[i]],list),style=line, color=green, thickness=2), i=1..4):
L1:=line(convert~([Q[1],Q[3]],list)[], color=red, thickness=2):
L2:=line(convert~([Q[2],Q[4]],list)[], color=red, thickness=2):
T1:=seq(textplot([convert(P[i],list)[],convert(cat(P,i),string)], align=left), i=1..4):
T2:=seq(textplot([convert(Q[i],list)[],convert(cat(Q,i),string)], align={above,left}), i=1..4):
display(Pol,Squares, L1, L2, T1, T2, scaling=constrained, axes=none, size=[400,500]);
end proc:

plots:-animate(VanAubel,[t], t=0..2*Pi, frames=180, paraminfo=false);

@tomleslie If we add the multiplier  signum(x) , then your example will work in all cases (of course in the real domain):

restart;
A:=sqrt(x^2+y^2)/x;
B:=signum(x)*sqrt(collect(A^2,x));
eval(A,{y=1,x=-2});
eval(B,{y=1,x=-2});
 

@vv  A wonderful geometric application of complex numbers.

@Carl Love  In this context, assuming real  is needed not for  solve, but for the correct operation of  Re  and  Im  commands. Therefore, your criticism seems unfounded to me.

Compare:

Re(x+I*y);
Re(x+I*y) assuming real;
 

@Carl Love  Here we do not get anything new. This will not be a solution for any  k , but only for  k=1  or  k=-1 . See:

restart;
sys:={-6*c+(3/2)*c^2-2*b-3*b*c+(3/2)*b^2-3*a*c+(k^2)*b-b+(3/2)*(a^2)+(k^2)*a-a=0,
-2*b-3*b*c+3*(b^2)-6*a*c+2*(k^2)*c-2*c-9*a*b+3*(k^2)*b-3*b+6*(a^2)+4*(k^2)*a-4*a=0,
(3/2)*(b^2)-3*a*c+(k^2)*c-c-9*a*b+3*(k^2)*b-3*b+9*(a^2)+6*(k^2)*a-a=0, 
-3*a*b+(k^2)*b-b+6*(a^2)+4*(k^2)*a-4*a=0, 
(3/2)*(a^2)+(k^2)*a-a=0}:
simplify(eval(sys, {a = (k^2-1)/3, b= 0, c= 0}));
solve(%);

@Sabrina Kamal  Just add this command in the end of your code.

Please upload the worksheet in which this error occurs using the thick green up-arrow in the MaplePrimes's editor.

@Carl Love  Thank you for indicating this inaccuracy.
At first I did not notice the third example at all, but when I noticed, I forgot about this phrase.
By the way, if we are very scrupulous, it is worth noting that, for example,  A = 1 is not always, but if  s<>Pi*k  (k  is an integer) because

eval(A, s=Pi*k) assuming k::integer;
Error, (in assuming) when calling '`one of {cos, eval, sin}`'. Received: 'numeric exception: division by zero'

 

@farah adanan Probably you are right, since in Maple 2018 everything works properly. I do not have Maple 18.

Addition - try to remove this line  local gamma;  as Preben indicated.

@farah adanan Probably you have not fixed everything. Save your worksheet in which the error occurs and load a link to it here using the bold green up-arrow in the mapleprime's editor. I'll check.

@Markiyan Hirnyk  Obviously the reason is in  rational  option. Examples show that in a solidly filled region the function takes complex values.
Look at these simple examples:

plots:-implicitplot(sqrt(-x^2-y^2), x=0..1,y=0..1);
plots:-implicitplot(sqrt(-x^2-y^2), x=0..1,y=0..1, rational);
plots:-implicitplot(sqrt(-x^2-y^2), x=0..1,y=0..1, gridrefine=2, rational);
 

@nm  The reason is probably in this:

whattype(1/sqrt(x));
op(1/sqrt(x));
                                                

A workaround:

expr:=1/sqrt(x)+sqrt(x);    
subs([x^(1/2)=abs(x)^(1/2), x^(-1/2)=(abs(x))^(-1/2)], expr);
                                           

 

@digerdiga  Yes. Here is an example:

simplify(sqrt((R-sqrt(R^2+z^2))^2));
simplify(sqrt((R-sqrt(R^2+z^2))^2))  assuming R-sqrt(R^2+z^2)<=0;