Teaching Material on Complex Numbers : From Geometric Viewpoint

Abstract

An extension of real numbers to complex numbers has a certain gap, since the latter is the 2-dimensional vector space consists of real part and imaginary part. In this paper, we compare the geometric meanning of two notations a+bi and (a, b) for a complex number. We may consider the former as a geometric vector of the 2-dimensional vector space spanned by orthonormal vectors 1 and i, where 1 is the unit element and ii=-1, the latter as a number vector of the 2-dimensional vector space spanned by (1, 0) and (0, 1). Therefore, we can easily see that two definitions are the same essentially. The multiplication by i for number w in the complex number plane is a rotation by π/2 around the origin, however this multiplication is not a homomorphism with respect to the multiplication of complex numbers. The multiplication by i is the homomorphism for a ternary compositions u-v+w and uv^<-l>w (v&bne;0), which due to E. Cartan (1927). The complex number relates a matrix of left multiplication and also cosine, sine function. Therefore, for understanding of complex number, it is useful to consider it together with matrices and trigonometric functions.