M  A  T  H 
2  1  B 
Complex numbersA rotation dilation matrixA =  a b   b a can be associated with a complex number z = a + i b. The matrix J =  0 1   1 0 plays the role of the imaginary number i, because it satisfies i^{2} = 1. If we take two complex numbers z = a+ib, w=u+iv, then their product is z w = (aubv) + i (av+bu). The product of the matrices A B =  a b   u v  =  aubv av+bu   b a  v u   (av+bu) aubv indeed is again a rotation dilation matrix which is associated to the complex number z w. This explanation takes a bit the mystery of the question, what is the imaginary number I because matrices allow complx numbers to be realized as a concrete object. The connection also illustrates again what the multiplication of complex numbers does: multiplying with z is a rotation and a scaling by z. 
Dual numbersThe set of 2x2 matrices of the form a b   0 a forms a set called dual numbers. One can add and multiply dual numbers and again gets a dual number. Similarly as in the case of complex numbers, one can write any dual number as z = a + e b. But now, instead of i^{2} = 1, we have the rule e^{2} = 0. Problem: verify that the analogue for the fundamental identity exp(i t) = cos(t) + i sin(t)for complex numbers is exp(e t) = 1+ e t.The use of dual numbers is motivated by infinitesimal calculus. If we think of e as an infinitesimal number, then e^{2} = 0. Dual numbers are also motivated from physics: if e denotes a Fermionic direction, then e^{2}=0 is a model for the Pauli exclusion principle. 
Split complex numbersThe set of 2x2 matrices of the form a b   b a forms a set called split complex numbers. One can add and multiply split complex numbers and again get a split complex number. Similarly as in the case of complex numbers, one can write any split complex number as z = a + j b. But now, instead of i^{2} = 1, one has to use the rule j^{2} = 1. Problem: verify that the analogue for the fundamental identities for complex and dual numbers exp(i t) = cos(t) + i sin(t) exp(e t) = 1+ e t.is the identity exp(j t) = cosh(t) + j sinh(t) .The use of split complex numbers is motivated by special relativity. The correponding trnasformation is a Lorentz boost, a special Lorentz transformation. 
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Math21b  Oliver Knill  Spring 2008 
Department of Mathematics 
Faculty of Art and Sciences 
Harvard University
