Revision history [back]
 | 1 | initial version |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar((-1)^(1/3))
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number, the other cubic roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar((-1)^(1/3))^2
sage: complex_cubic_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cubic_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 2 | No.2 Revision |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar((-1)^(1/3))
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a a is a cubic root of some complex number, number (in your case -1), the other cubic roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar((-1)^(1/3))^2
sage: complex_cubic_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cubic_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 3 | No.3 Revision |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar((-1)^(1/3))
QQbar(-1)^(1/3)
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number (in your case -1), the other cubic roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar((-1)^(1/3))^2
sage: complex_cubic_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cubic_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 4 | No.4 Revision |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar(-1)^(1/3)
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number (in your case -1), the other cubic roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar((-1)^(1/3))^2
QQbar(-1)^(2/3)
sage: complex_cubic_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cubic_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 5 | cubic -> cube |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar(-1)^(1/3)
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number (in your case -1), the other cubic cube roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar(-1)^(2/3)
sage: complex_cubic_roots complex_cube_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cubic_roots:
complex_cube_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 6 | No.6 Revision |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar(-1)^(1/3)
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number (in your case -1), the other cube roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar(-1)^(2/3)
sage: j == QQbar(e^(2*I*pi/3))
True
sage: complex_cube_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cube_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 7 | No.7 Revision |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar(-1)^(1/3)
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number (in your case -1), the other cube roots are a*j and a*j^2, where j=exp(2*I*pi/3). Hence you can define
sage: j = QQbar(-1)^(2/3)
sage: j == QQbar(e^(2*I*pi/3))
True
sage: 1+j+j^2 == 0
True
sage: complex_cube_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cube_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
| | 8 | No.8 Revision |
Fist, when you write
sage: a = (-1)^(1/3)
you define an element of the Symbolic Ring, which is not a safe place:
sage: a.parent()
Symbolic Ring
sage: a^2
1
So it is better to work on QQbar, the set of algebraic complex numbers.
sage: a = QQbar(-1)^(1/3)
sage: a.parent()
Algebraic Field
sage: a^2
-0.500000000000000? + 0.866025403784439?*I
Now, if a is a cubic root of some complex number (in your case -1), the other cube roots are a*j and a*j^2, where j=exp(2*I*pi/3)j=exp(2*I*pi/3)=(-1+I*sqrt(3))/2
sage: j = QQbar(-1)^(2/3)
sage: j == QQbar(e^(2*I*pi/3))
True
sage: j == QQbar(-1+I*sqrt(3))/2
True
sage: 1+j+j^2 == 0
True
sage: complex_cube_roots = [a, a*j, a*j^2]
and check
sage: for i in complex_cube_roots:
....: print i, 'whose cube is', i^3
0.500000000000000? + 0.866025403784439?*I whose cube is -1
-1 whose cube is -1
0.500000000000000? - 0.866025403784439?*I whose cube is -1
