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

13

hle difference of 768 and 155; so that the difference of two numbers
acan be found by addition. The formal removal may be avoided by
writing the arithmetic complement thus, 1845, with the subtractive
unit on the left, which when added to 768, the sum will be 613, the
-additive and subtractive units being equal to zero.'
9. As numbers may be equal or unequal to one another, one number,,or an aggregate of numbers, is said to be equal to, or equivalent to
another number, when both are composed of the same number of
units. And one number is said to be greater than another when there
are more units in the former than in the latter; and the latter is said
to be less than the former. The following axioms will be found to be
useful in arithmetical reasonings.
1. Numbers are equal to one another which consist of the same
number of units.
2. A whole number or integer is equal to the sum of all the parts
of it.
3. Any whole number is greater than any part of it; and any part
of a number is less than the number itself.
4. Numbers or aggregates of numbers which are equal to the same
number are equal to one another.
5. If equal numbers be added to equal numbers, the sums are equal.
6. If equal numbers be added to unequal numbers: or if unequal
numbers be added to equal numbers, the sums are unequal.
7. If equal numbers be taken from, or subtracted from equal
numbers, the remainders are equal.
8. If equal numbers be subtracted from unequal numbers, or
if unequal numbers be subtracted from equal numbers, the remainders are unequal.
9. If the same number be added to, and subtracted from another,
the value of the latter is unaltered.2
1 Ex. 1. Find the difference between 328011 and 23412 by means of the arithmetic complement.
328011
A. C. of 23412 is 176588
By addition 304599 is the difference of the two numbers.
Ex. 2. Find the difference between the sum of 7970597, 1038037, and 3406424,
and the sum of 3279120 and 8441042 by the arithmetic complement.
7970597
1038037
3406424
A. C. of 3272190 is 16727810
A. C. of 8441042 is 11558958
1701826 the difference required.
As for example, 12 = 12+5- 5 = 17 - 5 =7 5.