"how numbers are stored and used in computers"

Fangcheng

In China, from about the first century C.E. through the seventeenth century, anonymous adepts practiced an art known as fāngchéng (方程), often translated into English as "rectangular arrays". This art provided procedures for manipulating counting rods on a counting board, enabling practitioners to produce answers to seemingly insoluble riddles.

The ancient calculations were made with implements called counting rods (the abacus being a comparatively recent invention). The rods were short sticks that could be arranged to represent digits in a decimal notation for positive and negative integers. Elaborate calculations were made by placing the rods inside squares on a counting “board” or “table.” No ancient tables survive, but by one surmise, they were any flat surface, perhaps covered by a sheet of cloth ruled into squares. The counting table was literally an ancient “spreadsheet” for manual computing in which numbers could be entered and changed as a calculation progressed.

Joseph F. Grcar, The Chinese Roots of Linear Algebra

Example fāngchéng problem

3 bundles of high-quality rice straws, 2 bundles of mid-quality rice straws and 1 bundle of low-quality rice straw produce 39 units of rice
2 bundles of high-quality rice straws, 3 bundles of mid-quality rice straws and 1 bundle of low-quality rice straw produce 34 units of rice
1 bundles of high-quality rice straw, 2 bundles of mid-quality rice straws and 3 bundle of low-quality rice straws produce 26 units of rice

How many units of rice can high, mid and low quality rice straw produce respectively?

Try changing the system of equations below to see the result of the elimination process.

x+y+z=x+y+z=x+y+z=

Over a period of around sixteen centuries, bibliographies of imperial libraries record the titles of hundreds of treatises on the mathematical arts (suàn fǎ, 算法). Many of these are still extant, and many include fāngchéng problems.

Remarkable features

Fāngchéng is remarkable for several reasons:

it is essentially equivalent to the solution of systems of equations in unknowns in modern linear algebra;
the earliest recorded fāngchéng procedure is in many ways quite similar to what we now call Gaussian elimination;
there is a reasonable possibility that these procedures were transmitted to Europe and Japan, serving perhaps as precursors of modern matrices, Gaussian elimination, and determinants.