Let's say I want to multiply (−1−i√32)(−1+i√32). Doing it by hand, we easily see that the answer is 1.
However, when I type
n(((-1 + sqrt(-3))/2)*((-1-sqrt(-3))/2))
in sage, I get
1 - 5.55111512312578 x 10^(-17)i
which is (of course) very close to 1...but its not exactly 1. Why is that? How do I fix this??
Default complex numbers use double precision floating points and their rounding errors (which can't be avoided).
Use variable from the right field and arithmetic specific to complex numbers
var('t,v,a,b', domain="complex")
v=sqrt(-3);
a=(-1+v)/2;
b=conjugate(a);
t = a*b;
v;
a;
b;
t.real();
t.imag();displays
sqrt(-3)
1/2*sqrt(-3) - 1/2
1/2*conjugate(sqrt(-3)) - 1/2
1
0Sage complex numbers are based on floating points, 1+2i is implemented as 1.00000000 + 2.00000000000 * i.
OOOPs it doesn't work
i=CC(-3);
j=CC(-1);
k=CC(2);
t = ((j + sqrt(i))/k)*((j - sqrt(i))/k);
t.real();
t.imag();The error moved into he imaginary part (approx 0.5 E-16).
I must edit the answer .....
Can be written this way too
var('t,v,a,b', domain="complex");
v=sqrt(-3); a=(-1+v)/2; b=(-1-v)/2;
t=a*b; t.real(); t.imag();or this way
var('t,v', domain="complex");
v=sqrt(-3);
t=((-1+v)/2)*((-1-v)/2);
t.expand().factor();There is a similar question in this thread
You can directly try to simplify the symbolic expresssion, instead of looking for its numerical approximation (which is the of role the n() function):
sage: a = ((-1 + sqrt(-3))/2)*((-1-sqrt(-3))/2)
sage: a
-1/4*(sqrt(-3) + 1)*(sqrt(-3) - 1)
sage: a.parent()
Symbolic Ring
sage: a.full_simplify()
1Now, if you follow the link provided by @Pierre last comment, you will see that Sage is not very reliable concerning Symbolic expressions (there are also undecidability results behind this). An alternative is to use the Algebraic Field which is much more reliable:
sage: a = QQbar((-1 + sqrt(-3))/2*(-1-sqrt(-3))/2)
sage: a
1.000000000000000? + 0.?e-19*I
sage: a.parent()
Algebraic Field
sage: a.simplify()
sage: a
1Asked: 10 years ago
Seen: 814 times
Last updated: Apr 11 '15
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.
Note that if you use a slightly different expression (with simply remove parentheses), you get the correct answer. In the following example, I just removed the (useless) parentheses around the fractions, and you get the same answer if you replace the two divisions by 2 by a division by 4.