Computations with irrational numbers
1/(2^(1/2) + 3^(1/2));
1 (%o15) ----------------- sqrt(3) + sqrt(2)
This is a number in the field Q[sqrt(2), sqrt(3)]. It has a representation in the form
a + b*sqrt(2) + c*sqrt(3) + d*sqrt(2)*sqrt(3)
ev (ratdisrep(rat(%)), algebraic);
And we get this result:
sqrt(3) - sqrt(2)
Here is a more complicated example that computes in Q[sqrt(2), 3^(1/3)]:
ir2: 2^(1/2) + 3^(1/3);
1/3 (%o20) 3 + sqrt(2)
1/ir2;
1 (%o26) -------------- 1/3 3 + sqrt(2)
ev (ratdisrep(rat(%)), algebraic);
2/3 1/3 (%o27) (3 - 2 sqrt(2)) 3 + (4 - 3 sqrt(2)) 3 - 4 sqrt(2) + 6
%*ir2;
1/3 2/3 1/3 (%o28) (3 + sqrt(2)) ((3 - 2 sqrt(2)) 3 + (4 - 3 sqrt(2)) 3 - 4 sqrt(2) + 6)
expand(%);
1
The third example is even more complicated. We use a nested root to extend the field Q:
r: 33^(1/2)*3;
3 sqrt(33)
With this root, we define two third roots:
r1: (17 - r)^(1/3);
1/3 (%o2) (17 - 3 sqrt(33))
r2: (17 + r)^(1/3);
1/3 (%o3) (3 sqrt(33) + 17)
With these roots we can now write the number that we want to invert:
rn: r1+r2;
1/3 1/3 (%o8) (3 sqrt(33) + 17) + (17 - 3 sqrt(33))
The reciprocal is:
1/rn;
1 (%o9) ------------------------------------------- 1/3 1/3 (3 sqrt(33) + 17) + (17 - 3 sqrt(33))
ev (ratdisrep(rat(%)), algebraic);
We obtain:
2/3 1/3 1/3 (%o10)((3 sqrt(33) + 17) - (17 - 3 sqrt(33)) (3 sqrt(33) + 17) 2/3 + (17 - 3 sqrt(33)) )/34
This product is simplified with rootscontract:
rootscontract(%);
2/3 2/3 (3 sqrt(33) + 17) + (17 - 3 sqrt(33)) + 2 (%o11) ----------------------------------------------- 34
This is the searched reciprocal in the desired representation.
Multiplication with rn gives:
%*rn;
1/3 1/3 2/3 2/3 ((3 sqrt(33) + 17) + (17 - 3 sqrt(33)) ) ((3 sqrt(33) + 17) + (17 - 3 sqrt(33)) + 2) (%o12) ----------------------------------------------------------------------------------------------- 34
To continue our computation, we expand this:
expand(%);
1/3 2/3 2/3 1/3 1/3
(17 - 3 sqrt(33)) (3 sqrt(33) + 17) (17 - 3 sqrt(33)) (3 sqrt(33) + 17) (3 sqrt(33) + 17)
(%o22)----------------------------------------- + ----------------------------------------- + --------------------
34 34 17
1/3
(17 - 3 sqrt(33))
+ -------------------- + 1
17
The product of the roots is removed with rootscontract:
rootscontract(%);
1/3 1/3 1/3 1/3 (24 sqrt(33) - 136) (3 sqrt(33) + 17) (17 - 3 sqrt(33)) (- 24 sqrt(33) - 136) (%o24) ---------------------- + -------------------- + -------------------- + ------------------------ + 1 34 17 17 34 17
To simplify this, we canonicalize this expression with the option radexpand:all. This option transforms (24*SQRT(33) - 136) into 2*(3*SQRT(33) - 17).
radcan(%), radexpand:all;
1
It is convenient to define a function that computes the reciprocal and performs all desired simplifications:
inv(x) := block([r:1/x], rootscontract(ev (ratdisrep(rat(r)), algebraic)));
We can now write:
inv(r1 + r2);
to obtain:
2/3 2/3 (3 sqrt(33) + 17) + (17 - 3 sqrt(33)) + 2 (%o5) ----------------------------------------------- 34