Elementary arithmetic, with brief notices of its history... by Robert Potts.

12

symbol of addition or subtraction are included in a parenthesis or a
brace, thus (5 + 2) and (5- 2), or (5 + 2 and 5 - 2), and sometimes
by a line placed over them in this manner, 5 + 2 and 5 -2.
One number may be considered the complement of another when
the sum of the two numbers make up any given number.
7. DEE.-The arithmetic complement of a number is defined to be
the difference between any given number and the unit of the next
Superior order; as 6 is the arithmetic complement of 4, 47 of 53, 845
of 155, and so on, being the differences respectively of 4, 53, 155, and
10, 100, 1,000, the next superior units to these numbers. Conversely,
also, 4, 53, 155 are the arithmetic complements of 6, 47, 845
respectively.'
The arithmetic complement of a number may always be found by
subtracting the figure in the unit's place from 10 and the rest of the
figures of the number from 9.
Since from the definition, the arithmetic complement of 155 is 845
=1000-155, whence it follows that 155+845 = 1000, or that the sum
of any number and ifs arithmetic complement will always be equal to
the unit of the next superior order.
Again, since 155 = 1000 -845; if 155 or its equivalent in terms of
the arithmetic complement be subtractive, it may be written 1845, by
placing the subtractive unit before the left digit of the arithmetic
complement, as is clone in the characteristics of logarithms, when they
are subtractive.
8. PioB. —If thle arithmetic comiplement be added to any other mnuber of
the same number of figures, the sum wvill exceed the difference of the t'wo
inumbers by an unit of the next superior order.
If 155 be subtracted from 768, the remainder is 613, the difference
between them.
But if the arithmetic complement 845 of 155, the less number, be
added to 768, the greater, the sum will be 1613, one unit (1000 in this
case) of the next superior order greater than the difference of the two
numbers. By removing this unit, the number left will be equal to
1 The arithmetic complement can be employed to find the difference of two
num]bers, as also the aggregate of several numbers when some of them are additive
and some subtractive. It is employed with the greatest advantage in the arithmetic
of logarithms, in cases where some logarithms are to be added and some to be
subtracted in the same computation. Instead of finding the two sums and sub.
tracting one from the other, the sum of the logarithms to be added and the
arithmetic complements of the logarithms to be subtracted will give the correct
difference of the sums of the additive and subtractive logarithms. The mode of
'writing the arithmetic complement is so simple that the arithmetic complement of
5a logarithm can be as readily written as the logarithm itself (an be copied from
the tables.