Addition is also used for the additament or the thing added itself. In new editions of books, authors often make additions; they frequently make needless additions, in lieu of retrenching superfluities and impertinences.
'Tis an axiom that if you add unequal quantities to equal quantities, the excess of the wholes will be the same as the excess of the additional parts.
Addition, in Arithmetic, is the first of the four fundamental rules or operations of that art. See RULE and ARITHMETIC.
Addition consists of finding the amount of several numbers or quantities separately added to one another. Or, Addition is the invention of a number, from two or more homogeneous ones given, which is equal to the given numbers taken jointly or together. See NUMBER.
This number, thus found, is called the sum or aggregate of the numbers given. See SUM.
The character of addition is +, which we usually express by Plus. Thus 3 + 4 denotes the sum of 3 and 4 and is read 3 plus 4. See CHARACTER.
To add any given numbers together:The addition of simple numbers is easy. Thus it is readily perceived that 7 and 9, or 7 + 9, make 16, and 11 + 15 make 26. In longer or compounded numbers, the business is performed by writing the given numbers in a row, downwards; homogeneous under homogeneous, i.e., units under units, tens under tens, etc., and singly collecting the sums of the respective columns.
To do this, we begin at the bottom of the outermost row or column, to the right; and if the amount of this column does not exceed 9, we write it down at the foot of the same column: If it does exceed 9, the excess is only to be written down, and the rest reserved to be carried to the next row and added thereto, as being of the same kind or denomination.
Suppose, e.g., the numbers 1357 and 172 were given to be added; write either of them, e.g., 172, under the other, 1357, so that the units of the one, viz. 2, stand under the units of the other, viz. 7, and the other numbers of the one, under the corresponding ones of the other, viz. the place of tens under tens, as 7 under 5; and that of hundreds, viz. 1, under the place of hundreds of the other, 3. Then, beginning, say, 2 and 7 make 9, which write underneath; also 7 and 5 make 12; the last of which two numbers, like:
(1)
1357
+ 172
______
29
viz. 2, is to be written, and the other 1, reserved in your mind to be added to the next row, 1 and 3: then say, 1 and 1 make 2, which added to 3 make 5; this write underneath, and there will remain only 1, the first figure of the upper row of numbers, which also must be written underneath; and thus you have the whole sum, viz. 1529:
1357
+ 172
______
1529
87899
+ 13403
+ 885
+ 1920
_______
104107
Addition of numbers of different denominations, for instance, of Pounds, Shillings, and Pence, is performed by adding or summing up each denomination by itself, always beginning with the lowest; and if after the addition, there be enough to make one of the next higher denomination, for instance, Pence enough to make one or more Shillings; they must be added to the figures of that denomination, that is, to the Shillings; only reserving the odd remaining Pence to be put down in the place of Pence. And the same rule is to be observed in Shillings, with regard to Pounds.
For an instance, 5 Pence and 9 Pence make 14 Pence; now in 14 there is once 12, or a Shilling, and two remaining Pence; put the Pence down and reserve 1 Shilling to be added to the next column, which consists of Shillings. Then 1 and 8 and 2 make 11, and 11 and 5 make 16: put the 6 down and carry the 1 to the column of Tens; 1 and 1 and 1 make three Tens of Shillings, or 30 Shillings; in 30 Shillings there is once 20 Shillings, or a Pound, and 10 over: write one in the column of Tens of Shillings, and carry 1 to the column of Pounds; and continue the addition of Pounds, according to the former rules.
l. s. d.
120 15 9
65 12 5
9 8 0
__________
195 16 2
Addition of Decimals is performed in the same manner as that of Whole Numbers; as may be seen in the following example. See also Decimal.
630.953
51.0807
305.27
________
987.3037
Addition of Vulgar Fractions, see under the article FRACTION.
Addition, in Algebra, or the Addition of Species, is performed by connecting the quantities to be added by their proper signs; and also by uniting into one sum those that can be so united. See QUANTITY, SPECIES, etc.
Thus, a and b make a+b; a and -b make a-b; -a and -b make -a-b; 7a and 9a make 7a+9a; -a√ac and b√ac make -a√ac + b√ac, or b√ac - a√ac; for it is all one in whatever order they be written.
But, particularly, 1°, Affirmative quantities of the same species or kind are united by adding the prefixed numbers whereby the species are multiplied. See POSITIVE.
Thus, 7a + 9a make 16a. And 11bc + 15bc make 26bc. Also 3a/c + 5a/c make 8a/c and 2√ac + 7√ac make 9√ac; 6√(ab-xx) + 7√(ab-xx) make 13√(ab-xx).
And in like manner 6√3 + 7√3 make 13√3. Again, ay/√ac + by/√ac make (a+b)y/√ac, by adding together a and b, as numbers multiplying ac. And so 2a+3cy/3a√ex - x² + 3y/3a√ex - a² make 5a+3cy/3a√ex - x² since 2a + 3c and 3a make 5a + 3c.
2°, Affirmative fractions, which have the same denominator, are added together by adding their numerators.
Thus, 2/4 + 3/4 make 5/4, and a/b + c/b make (a+c)/b. And thus 5a/√cx + 7a/√cx make 12a/√cx. See FRACTION.
3°, Negative quantities are added after the same manner as affirmative. See NEGATIVE.
Thus, -2 and -3 make -5; -7z and -z make -8z; -ay/√ac and -by/√ac make -a-by/√ac, etc.
When a negative quantity is to be added to an affirmative one, the affirmative must be diminished by a negative one.
Thus, 3 and -2 make 1; 14x² and -9x² make 5x²; -ax/√ac and bx/√ac make b-ax/√ac.
And note that when the negative quantity is greater than the affirmative, the aggregate or sum will be negative. Thus, 2 and -3 make -1; and 2a and -7a make -5a; 2y/√ac and -7y/√ac make -5y/√ac.
Addition, in Algebra, or the Addition of Species, is performed by connecting the quantities to be added by their proper signs; and also by uniting into one sum those that can be so united. See QUANTITY, SPECIES, etc.
Thus, a and b make a+b; a and -b make a-b; -a and -b make -a-b; 7a and 9a make 16a; -a√ac and b√ac make b√ac - a√ac; for it is all one in whatever order they be written.
But, particularly, 1°, Affirmative quantities of the same species or kind are united by adding the prefixed numbers whereby the species are multiplied. See POSITIVE.
Thus, 7a + 9a make 16a. And 11bc + 15bc make 26bc. Also, 3a/c + 5a/c make 8a/c and 2√ac + 7√ac make 9√ac; 6√(ab-xx) + 7√(ab-xx) make 13√(ab-xx).
And in like manner, 6√3 + 7√3 make 13√3. Again, ay/√ac + by/√ac make (a+b)y/√ac, by adding together a and b, as numbers multiplying y. And so, (2a + 3cy)/(3a√ex - x²) + 3y/(3a√ex - a²) yields (5a + 3cy)/(3a√ex - x²).
2°, Affirmative fractions, which have the same denominator, are added together by adding their numerators.
Thus, 2/4 + 3/4 make 5/4, and a/b + c/b make (a+c)/b. And thus, 5a/√cx + 7a/√cx make 12a/√cx. See FRACTION.
3°, Negative quantities are added after the same manner as affirmative. See NEGATIVE.
Thus, -2 and -3 make -5; -7z and -z make -8z; -ay/√ac and -by/√ac make (-a-b)y/√ac.
When a negative quantity is to be added to an affirmative one, the affirmative must be diminished by the negative one.
Thus, 3 and -2 make 1; 14x² and -9x² make 5x²; -ax/√ac and bx/√ac yield (b-a)x/√ac.
And note that when the negative quantity is greater than the affirmative, the aggregate or sum will be negative. Thus, 2 and -3 make -1; 2a and -7a yield -5a; 2y/√ac and -7y/√ac result in -5y/√ac.