  Dirac talked about q-numbers versus c-numbers. Quantum observables are
q-number variables that generally do not commute among themselves. He was
proposing to have a generalized form of numbers as elements of a noncommutative
algebra. That was Dirac's appreciation of the mathematical properties of the
physical quantities as presented in Heisenberg's new quantum theory. After all,
the familiar real, or complex, number system only came into existence through
the history of mathematics. Values of physical quantities having a commutative
product is an assumption that is not compatible with quantum physics. The
revolutionary idea of Heisenberg and Dirac was pulled back to a much more
conservative setting by the work of Schr\"odinger, followed by Born and Bohr.
What Bohr missed is that the real number values we obtained from our
measurements are only a consequence of the design of the kind of experiments
and our using real numbers to calibrate the output scales of our apparatus. It
is only our modeling of the information obtained about the physical quantities
rather than what Nature dictates. We have proposed an explicit notion of
definite noncommutative values of observables that gives a picture of quantum
mechanics as realistic as the classical theory. In this article, we illustrate
how matrices can be taken as noncommutative (q-)numbers serving as the values
of physical quantities, each to be seen as a piece of quantum information. Our
main task is to clarify the subtle issues involved in setting up a conventional
scheme assigning matrices as values to the physical quantities.
