may be written 4a3b-2ab+7a3c-3a°c + 9ac3 - 662 (Art. 28). Now 4a3b - 2ab is the same thing as 2ab, and 7a2c-3a2c is the same thing as 4a3c. Thus the expression may be put in the simpler form 2ab+4a3c + 9ac2 - 6b2.

ADDITION.

31. The addition of algebraical expressions is performed by writing the terms in succession each preceded by its proper sign.

this is the

Now if we

For suppose we have to add c-d+e to a-b; same thing as adding c+ed to a−b (Art. 28). add ce to a-b we obtain a−b+c+e; we have however thus added d too much, and must consequently subtract d. Hence we obtain a-b+c+e-d, which is the same as aɩ − b + c −d +e; thus the result agrees with the rule above given. The result is called the sum.

We may write our result thus:

a-b+(c-d+e)=a-b+c-d+e.

32. When the terms of the expressions which are to be dded are all unlike, the sum obtained by the rule does not dmit of simplification. But when like terms occur in the exressions, we may simplify as in Art. 30. Hence we have the ollowing rules:

When like terms have the same sign their sum is found by king the sum of the coefficients with that sign and annexing the mmon letters.

Example; add 5a-36 and 4a-7b; the sum is 9a-106. or the 5a and the 4a together make 9a, and the 36 and 76 gether make 106.

Again; add 4a2c - 10bde, 6a2c-9bde and 11a c-3bde. The m is 21a c-22bde.

When like terms occur with different signs their sum is found

sign - with those preceded by the sign +. Thus, for example, if a represent 10, b represent 6, and c represent 5, then

If however a represent 2, 6 represent 6, and c represent 5, then the expression a-c+b presents a difficulty because we are thus apparently required to take a greater number from a less, namely 5 from 2. It will be convenient to agree that such an expression as a c+b when c is greater than a shall be understood to mean the same thing as a + b − c. At present we shall never use such an expression except when c is less than a + b, so that a + b − c presents no difficulty. Similarly we shall consider − b + a to mean the same thing as a -b. We shall recur to this point hereafter in Chapter V.

28. Thus the numerical value of an expression remains the same whatever may be the order of the terms which compose it. This, as we have seen, follows, partly from our notions of addition and subtraction, and partly from an agreement as to the meaning we ascribe to an expression when our ordinary arithmetical notions are not strictly applicable. Such an agreement is called in Algebra a convention, and conventional is the corresponding adjective.

29. We shall frequently, as in Article 26, have to distinguish the terms of an expression which are preceded by the sign + from the terms which are preceded by the sign —, and thus the following definition is adopted. The terms in an algebraical expression which are preceded by no sign or which are preceded by the sign + are called positive terms; the terms which are preceded by the sign are called negative terms. This definition is introduced merely for the sake of brevity, and no meaning is to be given to the words positive and negative beyond what is expressed in the definition. The student will notice that terms preceded by no sign are treated as if they were preceded by the sign +.

30. Sometimes an expression includes several like terms; in this case the expression admits of simplification. For example,

consider the expression 4a2b-3a2c+9ac2 - 2ab+7a3c - 662; this may be written 4a3b-2ab+7a3c – 3a2c + 9ac3 — 662 (Art. 28). Now 4a3b - 2ab is the same thing as 2ab, and 7a'c - 3a2c is the same thing as 4a3c. Thus the expression may be put in the simpler form 2ab+ 4a3c + 9ac2 – 6b3.

ADDITION.

31. The addition of algebraical expressions is performed by writing the terms in succession each preceded by its proper sign.

this is the

Now if we

Hence

For suppose we have to add c-d+e to a-b; same thing as adding c+e-d to a-b (Art. 28). add ce to a-b we obtain a-b+c+e; we have however thus added d too much, and must consequently subtract d. we obtain a−b+c+e−d, which is the same as a−b+c−d+e; thus the result agrees with the rule above given. The result is called the sum.

We may write our result thus:

a − b + (c−d + e) = a − b + c − d+e.

32. When the terms of the expressions which are to be added are all unlike, the sum obtained by the rule does not admit of simplification. But when like terms occur in the expressions, we may simplify as in Art. 30. Hence we have the following rules:

When like terms have the same sign their sum is found by taking the sum of the coefficients with that sign and annexing the common letters.

Example; add 5a-36 and 4a -7b; the sum is 9a-106. For the 5a and the 4a together make 9a, and the 36 and 76 together make 106.

Again; add 4a2c - 10bde, 6a c-9bde and 11a c-3bde. The sum is 21a c-22bde.

When like terms occur with different signs their sum is found by taking the difference of the sum of the positive and the sum of

the negative coefficients with the sign of the greater sum and annexing the common letters as before.

and 5b-4a. The sum is 3a-4b.

Example; add 7a - 96

33. Suppose we have to take b+c from a.

Then as each of

the numbers b and c is to be taken from a the result is denoted by a-b-c. That is

We enclose the term b+c in brackets, because both the numbers b and c are to be taken from a.

Similarly a + d − (b + c + e) = a + d − b − c − e.

--

Next suppose we have to take b-c from a. If we take b from a we obtain a-b; but we have thus taken too much from a, for we are required to take, not b but, b diminished by c. Hence we must increase the result by c; thus

a − (b −c) = a − b + c.

Similarly, suppose we have to take b-c-d+e from a. This is the same thing as taking b+ e C d from a. Take away b+ e from a and the result is a-b-e; then add c+d, because we were to take away, not b+e but, b+e diminished by c+d; thus

a − (b − c − d + e) = a − b −e+c+d

= a−b+c+d-e.

34. From considering these cases we arrive at the following rule for subtraction. Change the sign of every term in the expression to be subtracted, and then add it to the other expression. Here as before, we suppose for shortness, that where there is no sign before a term, + is to be understood.

For example; take a-b from 3a+b.

3a + b − (a - b) = 3a+b-a+b= 2a + 2b.

Again; take 5a2+4ab - 6xy from 11a3+3ab - 4xy. 11a2 + 3ab - 4xy - (5a2 + 4ab - 6xy)

= 11a2 + 3ab – 4xy – 5a2 – 4ab + 6xy = 6a2 — ab + 2xy.

=

BRACKETS.

35. On account of the frequent occurrence of brackets in algebraical investigations, it is advisable to call the attention of the student explicitly to the laws respecting their use. These laws have already been established, and we have only to give them a verbal enunciation.

When an expression within brackets is preceded by the sign + the brackets may be removed.

Thus a-b+(c-d+e) = a − b + c − d+e, (Art. 31).

And consequently any number of terms in an expression may be enclosed by brackets, and the sign + placed before the whole.

Thus a-b+c-d+e may be written in the following ways, a-b+c+(-d+e), a-d+(c+e-b), a +(-d+c+ e − b), and so on.

When an expression within brackets is preceded by the signthe brackets may be removed if the sign of every term within the brackets be changed, namely + to· and to +.

Thus a-(b-c d+e) = a−b+c+d−e, (Art. 34).

And consequently any number of terms in an expression may be enclosed by brackets and the sign - placed before the whole, provided the sign of every term within the brackets be changed.

Thus a-b+c+d-e may be written in the following ways, a-b+c-(-d+e), a-(b-c-d+e), a+c-(b−d+e),

and so on.