16. Literal numbers, as has been stated, are used to represent numbers which may have any values whatever, or numbers whose values are, as yet, unknown. But it is frequently necessary to assign particular values to such numbers.

Substitution is the process of replacing a literal number in an algebraic expression by a particular value. See axiom (iii.). Ex. 1. If in a + b, a = 3 (read a has the value 3) and b

then

a+b=3+5=8, or a+b= 8.

=

5,

Notice that the last step involved an application of axiom (iv.). For we have a+b=3+5, and 3+ 5 = 8; therefore, by +58; axiom (iv.), a + b = 8.

Ex. 2. If in a (b+c), a = 11, b = 2, and c = 3, we have

=

Ex. 3. If, in a+b-2a+3b-c, we let a 6, b=11}, ğ, we have

C=

a+b-2a+3b-c=6+111 – 2 × 6 + 3 × 111 — §
=6+23-12 + 69 — § = 381.

Observe that in the work of the last example, the expression a+b2a+3b c is to be understood on the left of the symbol,, in the second line.

Ex. 4. If, in the last example, a = 3, b= 1, and c=1, we have a+b-2a+3b-c=3+1-6+3−1=4-6+3−1. We cannot further reduce 4-6+3-1, since we are unable, as yet, to subtract 6 from 4.

EXERCISES III.

What are the values of the following expressions when a = 6, b = 4, c = 2:

17. Some of the advantages of using literal numbers are shown by the following example:

are particular examples of the following arithmetical principle: The sum of two fractions which have a common denominator is a fraction whose denominator is that common denominator, and whose numerator is the sum of the two given numerators; or, 1st num. 2d num. 1st num.+ 2d num.

+

com. den. com. den.

=

com. den.

This principle can be stated still more concisely if the terms. of the fractions, which may be any numbers whatever, are represented by three symbols, say a, b, c. We then have

This equation states by means of signs and symbols all that is contained in the verbal statement of the principle. It is thus a symbolic statement of a general principle, and includes all particular cases that result from assigning particular values to a, b, c.

18. Notice the following advantages thus secured by introducing general numbers:

(i.) General laws and relations can be expressed with great brevity, and yet include all that the most general verbal statements can express.

(ii.) Such symbolic statements mass under the eye the various operations involved, and thus enable the eye to assist the understanding and memory.

EXERCISES IV.

Express in algebraic language (i.e., by means of the signs and symbols of Algebra) the following principles of Arithmetic :

1. If a, b, and c are any three numbers, their sum diminished by any one of them is equal to the sum of the other two.

If z is the result of subtracting y from x, express in algebraic language the following principles of subtraction :

2. The minuend is equal to the subtrahend plus the remainder.

3. The subtrahend is equal to the minuend diminished by the remainder. If z is the result of multiplying x by y, express in algebraic language the following principles of multiplication:

4. The multiplicand is equal to the product divided by the multiplier. 5. The multiplier is equal to the product divided by the multiplicand. If a is exactly divisible by b, and q is the quotient, express in algebraic language the following principles of division:

6. The dividend is equal to the divisor multiplied by the quotient.

7. The divisor is equal to the dividend divided by the quotient.

α

If is any fraction, and m is any integer, express in algebraic language b

the following principles of fractions :

8. If the numerator of a fraction is multiplied by any integer, the value of the fraction is multiplied by that integer.

9. If the denominator of a fraction is multiplied by any integer, the value of the fraction is divided by that integer.

Problems solved by Equations.

19. Another advantage of using literal numbers is shown by the following problem:

Pr. The older of two brothers has twice as many marbles as the younger, and together they have 33 marbles. How many has the younger?

The number of marbles the younger brother has is, as yet, an unknown number. Let us represent this unknown number by some letter, say x. Then, since the older brother has twice as many, he has x × 2, or 2x, marbles. The problem states, in verbal language: the number of marbles the younger has plus the number the older has is equal to 33;

in algebraic language: x+2x=33, or 3 x 33.

Dividing by 3 [Art. 15 (iv.)], x = 11, the number of marbles the younger has. The older has 2x, = 22.

EXERCISES V.

1. What number added to three times itself gives 28 ?

2. Divide 69 into two parts so that the greater shall be twice the less. 3. If twice a number be added to three times the number, the sum will be 63. What is the number?

4. If three times a number be subtracted from five times the number, the remainder will be 6. What is the number?

5. Divide 150 into two parts so that the less is one-fifth of the greater. 6. A and B together have $180, and A has five times as much as B. How many dollars has each ?

7. In a school are 120 pupils; in the second grade are twice as many as in the first, and in the third three times as many as in the first. many pupils are in each grade?

How

8. A, B, and C together invest $8000; A invests twice as much as B, and B five times as much as C. How many dollars does each invest?

9. Divide 30 into three parts, so that the second shall be one-half of the first, and the third one-third of the second.

10. Divide 52 into three parts, so that the second shall be one-half of the first, and the third one-fourth of the second.

11. Divide 85 into three parts, so that the first shall be four times the second, and one-third of the third.

12. If one-fourth of a number be subtracted from one-third of the number, the remainder will be 6. What is the number?

13. Three boys, A, B, and C, have together 27 pencils. B has twice as many as A, and C twice as many as A and B together. How many pencils has each ?

14. Three boys, A, B, and C, have together 32 pens; B has one-third as many as A, and C three times as many as A and B together. How many has each ?

15. Three boys, A, B, and C, have together 16 note books; A has five times as many as B, and the number that A has more than B is twice the number that C has. How many note books has each ?

20. General numbers are most frequently represented by the italicized letters of the English alphabet. But letters of other alphabets are sometimes employed, and there is often an advantage in using the same letter with some distinguishing marks to represent different numbers in the same discussion.

We add a list of the more common symbols.

Greek letters: a, ß, y, d, etc., read alpha, beta, gamma, delta, etc.;

with prime marks: a', a", a", a(n), read a prime, a two prime, a three prime, a n prime;

with subscripts: a1, a2, α, etc., read a sub-one, a sub-two, a subthree, etc., or simply a one, a two, a three, etc.

§ 2. POSITIVE AND NEGATIVE NUMBERS, OR ALGEBRAIC NUMBERS.

1. A still greater extension of the idea of number in passing from Arithmetic to Algebra is arrived at by the following considerations:

In ordinary Arithmetic we subtract a number from a greater or an equal number. We are familiar with such operations as 7-5=2, 6-5=1, 5-5 = 0.

But such operations as

4-5, 3-5, etc.,

(i.)

(ii.)

have not occurred in ordinary Arithmetic and cannot be carried out in terms of arithmetical numbers. For, from an arithmetical point of view, we cannot subtract from a number more units than are contained in that number. In general, the indicated operation ab can, as yet, be performed only when a is greater than b. But if a and b are to have any values whatever, the case in which a is less than b, that is, in which the minuend is less than the subtrahend, must be included in the operation of subtraction.

2. Now observe that, as the minuend in equations (i.) decreases by 1, 2, or more units (the subtrahend remaining the same) the remainder decreases by an equal number of units. When the minuend is equal to the subtrahend, the remainder is 0. If then, as in the indicated operations (ii.), the minuend becomes less than the subtrahend by 1, 2, or more units, the remainder must decrease by an equal number of units, and therefore become less than 0 by 1, 2, or more units.