Mathematics has always seemed like a mystery to many people, especially to children of basic school age. When children at this stage of learning encounter mathematical concepts, they tend to be so perplexed that many of them end up hating the subject irrespective of its relevance to life. The reasons for this perplexity are not far-fetched; teachers of the subject have serious difficulty demystifying the underlying of its most rudimentary concepts. Children progress from class to class with these unresolved challenges, which actually accumulate into a bigger challenges leading to hatred and phobia for the subject. Recall that mathematics developed several millennia ago from its most rudimentary form; number sense into arithmetic advancing gradually to its current climax; advanced mathematics. For countless ages now, mathematicians have sort to advance mathematics from its most rudimentary concepts to higher concepts, seemingly oblivious of the converse situation. In other words it seems quite likely that interest in the utilization of advanced concepts of mathematics to improve understanding of elementary concepts is either lacking or inadequate. A major question that can be impressed upon the mind is therefore; “How can we develop modified arithmetical approaches that will obliterate or reduce to the barest minimum the perplexity in arithmetic thereby making learning of mathematics easily accessible to all”. This study therefore, seeks to develop new elementary semiotic approaches to number sense and place value with special recourse to arithmetical operations such as addition (+), subtraction (-), multiplication (x), division (÷ ), etc. The study attempts to provide alternative semiotic approaches to elementary arithmetic by synthesizing concepts from advanced mathematics such as linear algebra and number theory. Hence the concept of diagonal arithmetic.