Imaginary numbers for combining linear equation models via Dempsterʼs rule

•Proposed a new concept of imaginary extreme numbers.•Resolved the division-by-zero problem in sweeping and Gaussian elimination operations.•Applied imaginary numbers to representing linear models in knowledge-based systems.•Established a new method for solving linear equations via combining belief functions.•Reduced exponential complexity into polynomial-time for linear models-based inferences.

This paper proposes the concept of imaginary extreme numbers, which are like traditional imaginary number a+bia+bi with i=−1 being replaced by e=1/0e=1/0, along with the usual operations on these numbers including addition, subtraction, and division. It then applies the concept to representing linear equations in knowledge-based systems. It proves that the combination of linear equations via Dempsterʼs rule is equivalent to solving a system of simultaneous equations or finding a least-squares estimate when they are overdetermined.