Editing Imaginary unit (section)

=== Geometric algebra ===

In the [[geometric algebra]] of the [[Euclidean plane]], the geometric product or quotient of two arbitrary [[Euclidean vector|vectors]] is a sum of a scalar (real number) part and a [[bivector]] part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector is a negative scalar.

The quotient of a vector with itself is the scalar {{math|1=1 = ''u''/''u''}}, and when multiplied by any vector leaves it unchanged (the [[Identity function|identity transformation]]). The quotient of any two perpendicular vectors of the same magnitude, {{math|1=''J'' = ''u''/''v''}}, which when multiplied rotates the divisor a quarter turn into the dividend, {{math|1=''Jv'' = ''u''}}, is a unit bivector which squares to {{math|&minus;1}}, and can thus be taken as a representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is [[isomorphic]] to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.<ref>The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed by [[Hermann Grassmann]] in the foreword to his ''Ausdehnungslehre'' of 1844; later [[William Kingdon Clifford|William Clifford]] realized that this ratio could be interpreted as a bivector. {{pb}} {{cite book |last=Hestenes |first=David |author-link=David Hestenes |year=1996 |chapter=Grassmann’s Vision |editor-last=Schubring |editor-first=G. |title=Hermann Günther Graßmann (1809–1877) |series=Boston Studies in the Philosophy of Science |volume=187 |pages=243–254 |publisher=Springer |doi=10.1007/978-94-015-8753-2_20 |isbn=978-90-481-4758-8 |chapter-url=https://davidhestenes.net/geocalc/pdf/GrassmannsVision.pdf }}</ref>

More generally, in the geometric algebra of any higher-dimensional [[Euclidean space]], a unit bivector of any arbitrary planar orientation squares to {{math|−1}}, so can be taken to represent the imaginary unit {{mvar|i}}.