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We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.

The theory may be extended to the cases of p= i and p = o; so that a 3 means a.a.a.1, a 2 means a.a.i, a 1 means a.i, and a° means I (there being then none of the multipliers a).

The principle is that 162.4 27 =100.4 2 7+ 60.4 2 7+ 2.4 2 7 = 1.42700+6.4270+2.427; but, instead of 427 writing down the separate products, we 6 427 (in effect) write 42700, 4270 and 427 in 2 4?7 separate rows, with the multipliers 1, 6, 2 69 174 in the margin, and then multiply each number in each column by the corresponding multiplier in the margin, making allowance for any figures to be " carried."