Complex numbers in today's computers are represented and manipulated by splitting them into their real and imaginary components and then each pair of real parts and imaginary parts undergoes arithmetic operations independent of one another. Thus, a simple addition/subtraction of two complex numbers (a+jb) and (c+jd) involves two individual additions/subtractions, (a±c) and (b±d). This gets more complicated for multiplication and division operations of complex numbers. Complex Binary Number System (CBNS) provides a novel way of representing each complex number as a single entity in the binary format. In this paper, a tutorial on CBNS and its arithmetic algorithms is presented with brief details about the hardware circuits implementing this number system. An associative dataflow processor, incorporating CBNS, which has been granted Innovation Patent Number: 2010100706 by the Australian Patent Office (IP Australia) is also described.