ADDITION

, the uniting or joining of two or more things together; or the finding of one quantity equal to two or more others taken together.

, is the first of the four fundamental rules or operations of that science; and it consists in finding a number equal to several others taken together, or in finding the most simple expression of a number according to the established notation. The quantity so found equal to several others taken together, is named their sum.

The sign or character of addition is +, and is called plus. This character is set between the quantities to be added, to denote their sum: thus, , that is, 3 plus 6 are equal to 9; and , that is, 2 plus 4 plus 6 are equal to 12.

Simple numbers are either added as above; or else by placing them under one another, as in the margin, and adding them together, one after another, beginning at the bottom: thus 2 and 4 make 6, and 6 make 12.

Compound numbers, or numbers consisting of more figures than one, are added, by first ranging the numbers in columns under each other, placing always the numbers of the same denomination under each other, that is, units under units, tens under tens, and so on; and then adding up each column separately, beginning at the right hand, setting down the sum of each column below it, unless it amount to ten or some number of tens, and in that case setting down only the overplus, and carrying one for each ten to the next column. Thus, to add 451 and 326,

Also to add the numbers ; set them down as in the margin, and beginning at the lowest number on the right hand, say 8 and 7 make 15, and 2 make 17, and 9 make 26; set down 6, and carry 2 to the next column, saying 2 and 4 make 6, and 4 make 10, and 6 make 16, and 2 make 18; set down 8, and carry 1, saying 1 and 3 make 4, and 5 make 9, and 3 make 12; set down 2, and carry 1, saying 1 and 2 make 3, and 1 make 4, which set down; then 1 and 2 make 3; and 7 is 7 to set down: so the sum of all together is 734286. Or it is the same as the sums of the columns set under one another, as in the margin, and then these added up in the same manner.

When a great number of separate sums or numbers are to be added, as in long accounts, it is easier to break or separate them into two or more parcels, which are added up severally, and then their sums added together for the total sum. And thus also the truth of the addition may be proved, by dividing the numbers into parcels different ways, as the totals must be the same in both cases when the operation is right.

Another method of proving addition was given by Dr. Wallis, in his Arithmetic, published 1657, by casting out the nines, which method of proof extends also to the other rules of arithmetic. The method is this: add the figures of each line of numbers together severally, casting out always 9 from the sums as they arise in so adding, adding the overplus to the next figure, and setting at the end of each line what is over the nine or nines; then do the same by the sum-total, as also by the former excesses of 9, so shall the last excesses be equal when the work is right. So the former example will be proved as below:

When the numbers are of different denominations; as pounds, shillings, and pence; or yards, seet, and inches; place the numbers of the same kind under one another, as pence under pence, shillings under shillings, &c; then add each column separately, and carry the overplus as before, from one column to another. As in the following examples:

Addition of Decimals, is performed in the same manner as that of whole numbers, placing the numbers of the same denomination under each other, in which case the decimal separating points will range straight in one column; as in this example, to add together these numbers .

Addition of Vulgar Fractions, is performed by bringing all the proposed fractions to a common denominator, if they have different ones, which is an indispensable preparation; then adding all the numerators | together, and placing their sum over the common denominator for the sum total required.

So .

ADDITION in Algebra, or the addition of indeterminate quantities, denoted by letters of the alphabet, is performed by connecting the quantities together by their proper signs, and uniting or reducing such as are susceptible of it, namely similar quantities, by adding their co-efficients together if the signs are the same, but subtracting them when different. Thus the quantity a added to the quantity b, makes a + b; and a joined with-b, makes a-b; also-a and-b make-a-b; and 3a and 5a make 3a + 5a or 8a, by uniting the similar numbers 3 and 5 to make 8. Thus also .

In the addition of surd or irrational quantities, they must be reduced to the same denomination, or to the same radical, if that can be done; then add or unite the rational parts, and subjoin the common surd. Otherwise connect them with their own signs.

So ; but of √5 and √6 the sum is set down √5 + √6, because the terms are incommensurable, and not reducible to a common surd.

Addition of Logarithms. See Logarithms.

Addition of Ratios, the same as composition of ratios; which see.