The number 38057, for example, would be represented thus:
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The number 1267, represented by the sangi without the ruled board. Is shown in Fig. 7.
![The number 1297 represented by sangi.](http://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/A_History_of_Japanese_Mathematics_-_Fig._7.png/400px-A_History_of_Japanese_Mathematics_-_Fig._7.png)
Fig. 7. The number 1297 represented by sangi.
From representing the numbers by the sangi on a ruled board came a much later method of transferring the lines to paper, and using a circle to represent the vacant square. This could only have occurred after the zero had reached China and had been passed on to Japan, but the date is only a matter of conjecture. By this method, instead of having 38057 represented as shown above, we should have it written thus:
In laying down the rods a red piece indicated a positive number and a black one a negative. In writing, however, a marked placed obliquely across a number indicated subtraction. Thus, (symbol characters) meant (
symbol characters) 3, and (
symbol characters) meant (
symbol characters) 6.
The use of sangi in the fundamental operations may be illustrated by the following example in which we are required