The History of Mathematical Proof in Ancient Traditions

478 karine chemla

In each of the cases when a dividend does not fi ll up a divisor, they hence have the

names of denominator and numerator.

In other words, one may choose to read an array of two lines on the

surface for computing, as in Figure 13.10 , in two ways.

On the one hand, the array is the layout of an operation of division,

which we shall represent as a : b. On the other hand, when a is smaller than

b , which is precisely the case ‘when a dividend does not fi ll up a divisor’, it

can be read as the quantity resulting from carrying out the operation, that

is, the fraction a / b.

Th ese dual points of view allow Liu Hui to link the fraction and the

numerator operationally. Placing himself at the most general level, as we

have seen him oft en do in proofs, he writes:

If there are parts (i.e. fractions), and if, when expanding the corresponding divi-

dend by multiplication, then, correlatively, it (the dividend produced by the multi-

plication) 75 fi lls up the divisor, the (division) hence only yields an integer.

Th e application of this remark that appears relevant in the context in

which it is formulated is that the sequence of a multiplication and a divi-

sion like ( b · a ) : b yields a. Seen from the other point of view, this remark

leads to stating that the multiplication b · a / b yields a as its result. Th e

numerator can thereby be seen as a quantity that is b times larger than the

fraction.

If, furthermore, one multiplies something by the numerator, the denominator must

consequently divide (the product) in return (baochu). Dividing in return is ‘divid-

ing the dividend by the divisor’.

Th is is the point where Liu Hui introduces the operation of ‘dividing in

return’, which we already mentioned above and which occurs only later in

the text of Th e Nine Chapters. In terms of operations, ‘dividing in return’ is

a simple division. However, the expression by which it is prescribed indi-

Figure 13.10 Th e layout of a division or a fraction on the surface for computing.

a

b

75 Th e name of ‘dividend’ designates what is in the position of the ‘dividend’ on the surface for

computing, at the moment when it is used. Th is is the assignment of variables typical of the

description of algorithms in Th e Nine Chapters.