NEW USES OF THE ABSTRACT—BORHM 319 
through rounding off numbers. Computers are particularly liable 
to commit such errors, for there is a limit to the size of the numbers 
they can manipulate. If a machine gets a very long number, it has 
to drop the digits at the end and work with an approximation. While 
the approximation may be extremely close, the error may grow to be 
enormous if the number is multiplied by a large factor at a later stage 
of the problem. It is generally safe to assume that rounding off tends 
to even out in long arithmetic examples. In adding a long column of 
figures, for instance, you probably won’t go far wrong if you con- 
sider 44.23 simply as 44, and 517.61 as 518. But it is sheer super- 
stition to suppose that rounding off cannot possibly build up a serious 
accumulation of errors. (It obviously would if all the numbers hap- 
pened to end in .499.) 
There are subtler pitfalls in certain more elaborate kinds of com- 
putation. In some typical computer problems involving matrices 
that are used to solve simultaneous equations, John Todd of the Cali- 
Figure 5.—Solution to problem in figure 4. The key to the solution is to imagine that 
adjacent squares have different colors, as on a chessboard. Then it becomes obvious 
that each rectangle has to cover precisely one black square and one white square. Since 
the large figure contains unequal numbers of black and white squares, there can be no way 
to cover it with rectangles. ‘The solution represents a conclusive negative proof, a logical 
feat that is peculiar to mathematics; in other sciences negative conclusions are invariably 
risky. The postulation of color isa relatively easy abstraction, but it is characteristic of 
a of the more complex abstractions that mathematicians use to simplify problems and 
theorems. 
536608—60——22 
