defined (up to a sign) in two dimensions and all bivectors are multiples of e12. As the highest grade element of the algebra e12 is also the pseudoscalar which is given the symbol i. Complex numbers With the properties of negative square and unit magnitude, the unit bivector can be identified with the imaginary unit from complex numbers. The bivectors and scalars together form the even subalgebra of the geometric algebra, which is isomorphic to the complex numbers C. The even subalgebra has basis (1, e12), the whole algebra has basis (1, e1, e2, e12). The complex numbers are usually identified with the coordinate axes and two-dimensional vectors, which would mean associating them with the vector elements of the geometric algebra. There is no contradiction in this, as to get from a general vector to a complex number an axis needs to be identified as the real axis, e1 say. This multiplies by all vectors to generate the elements of even subalgebra. All the properties of complex numbers can be derived from bivectors, but two are of particular interest. First as with complex numbers products of bivectors and so the even subalgebra are commutative. This is only true in two dimensions, so properties of the bivector in two dimensions that depend on commutativity do not usually generalise to higher dimensions. Second a general bivector can be written where θ is a real number. Putting this into the Taylor series for the exponential map and using the property e122 = −1 results in a bivector version of Euler's formula, which when multiplied by any vector rotates it through an angle θ about the origin: The product of a vector with a bivector in two dimensions is anticommutative, so the following products all generate the same rotation Of these the last product is the one that generalises into higher dimensions. The quantity needed is called a rotor and is given the symbol R, so in two dimensions a rotor that rotates through angle θ can be written and the rotation it generates is Three dimensions In three dimensions the geometric product of two vectors is This can be split into the symmetric, scalar-valued, scalar product and the antisymmetric, bivector-valued, exterior product: In three dimensions all bivectors are simple and so the result of an exterior product. The unit bivectors e23, e31 and e12 form a basis for the space of bivectors ⋀2R3,