The operation of subtracting a greater number from a less is therefore possible only when numbers less than zero are introduced. We then have from (i.) and (ii.):

Min. Subt. = Rem.

7-5=2

6-5=1

5-5=0

4 – 5 – a number one unit less than 0

=

3-5 a number two units less than 0

(iii.)

3. Numbers less than zero are called Negative Numbers. Numbers greater than zero are, for the sake of distinction, called Positive Numbers.

Positive and negative numbers are called Algebraic or Relative Numbers.

4. The Absolute Value of a number is the number of units contained in 'it without regard to their quality (i.e. whether positive or negative).

A positive number may be indicated by placing a small sign, +, to the left and a little above its absolute value; as, +5, +10, +16; read positive 5, positive 10, positive 16.

A negative number may be indicated by placing a small sign, -, to the left and a little above its absolute value; as, -5, -10, -16; read negative 5, negative 10, negative 16.

We must, as yet, carefully distinguish these symbols of quality, and, from the (larger) symbols of operation,+ and -.

+

5. Equations (iii.) can now be written as follows:

A negative remainder does not mean that more units have been taken from the minuend than were contained in it; such

a remainder indicates that the subtrahend is greater than the minuend by as many units as are contained in the remainder.

Thus, in +10 +155 and +87 — +92 =-5, the remainder, -5, indicates that the subtrahend is, in each case, 5 units greater than the minuend.

6. The results of the preceding articles, restated briefly, are : A positive number is a number greater than zero, by as many units as are contained in its absolute value.

E.g., +2 is two units greater than 0.

A negative number is a number less than zero by as many units as are contained in its absolute value.

E.g., -3 is three units less than 0.

Zero is the result of subtracting a number from an equal number.

E.g., 0 = +7 +7=-5--5+n+n= ̄n- ̄n, wherein n denotes any absolute number.

Since zero can be neither greater nor less than itself, it is neither a positive nor a negative number. It stands by itself, as the number from which positive and negative numbers are counted.

7. The Sign of Continuation, ..., read and so on, or and so on to, is used to indicate that a succession of numbers continues without end, as 1, 2, 3, ..., read, one, two, three, and so on; or that the succession continues as far as a certain number which is written after the sign..., as 1, 2, 3, ..., 10, read one, two, three, as far as, or to, 10.

We may now write the series of algebraic numbers:

... -4, -3, -2, -1, 0, +1, +2, +3, +4, •

...

In this series the numbers increase from left to right, and decrease from right to left; or a number is greater than any number on its left and less than any number on its right.

The numbers of ordinary Arithmetic are the absolute values of the positive and negative numbers of Algebra.

Relations between Positive and Negative Numbers and Zero. 8. From the results of the preceding article, we obtain the following general relations:

(i.) Of two positive numbers, that number is the greater which has the greater absolute value; and that number is the less which has the less absolute value.

(ii.) Of two negative numbers, that number is the greater which has the less absolute value; and that number is the less which has the greater absolute value.

For example, 3 >-5, or −5 <-3, since -5 is five units less than 0, and 3 is only three units less than 0.

9. Although negative numbers arise through the extension of the operation of subtraction, it is necessary to treat them as numbers apart from this particular operation.

As in Arithmetic, so in Algebra, any integer is an aggregate of like units.

SO

SO

+4 =+1++1++1++1, and −4= ̄1+ ̄1+‍1+−1.

+(})=+(})++(}), and −(})= ̄(})+ ̄(}).

Since letters are to represent numbers which may have any values whatever, they can represent either positive or negative numbers. Thus, in one case a may have the value +2, in another case the value -7; in the first case the absolute value of a is 2, in the second case the absolute value of a is 7.

EXERCISES VI.

1. What is the absolute value of +8? Of -11?_Of +(2 + y) ? For what values of x do the following expressions reduce to 0: 2. x+3?

3. x-18 ?

4. x- · +a?

5. x -(a + 6)? What values of a make the first members of the following equations identical with the second members:

What are the results of the following indicated operations:

6. a +7 +2 ?

How many units is each of the following numbers greater or less than 0:

Positive and Negative Numbers are Opposite Numbers.

10. The student is familiar with the principle of subtraction in Arithmetic that the remainder added to the subtrahend is equal to the minuend. This principle, like all principles of arithmetical operations, is retained in Algebra. Consequently, continuing equations (iv.), Art. 5, we have:

11. The last of equations (vi.), +5+5=0, furnishes an important relation between positive and negative numbers:

The sum of a positive and a negative number having the same absolute value is equal to zero; i.e., two such numbers cancel each other when united by addition.

E.g.,

+1+1=0, +3+3=0, -17++171 = 0.

For this reason, positive and negative numbers in their relation to each other are called opposite numbers. When their absolute values are equal, they are called equal and opposite

numbers.

12. Any quantities which in their relation to each other are opposite, may be represented in Algebra by positive and negative numbers; as credits and debits, gain and loss.

Ex. 1. 100 dollars credit and 100 dollars debit cancel each other. That is, 100 dollars credit united with 100 dollars debit is equal to neither credit nor debit; or,

100 dollars credit + 100 dollars debit = neither credit nor debit.

If credits be taken positively and debits negatively, then 100 dollars credit may be represented by +100, and 100 dollars debit by 100. Their united effect, as stated above, may then be represented algebraically thus:

+100+-100 = 0.

The result, 0, means neither credit nor debit.

Similarly for opposite temperatures.

Ex. 2. If a body is first heated 10° and then cooled down 8°, its final temperature is 2° above its original temperature; or, stated algebraically,

+10 +8=+2.

The result, +2, means a rise of 2° in temperature.

Similar reasoning applies to opposite directions.

Ex. 3. If a man walks 10 miles due north, and turning, walks 14 miles due south, he is then 4 miles south of his starting point; or,

+10+14= ̄4.

The result, 4, means that he is now 4 miles south of his starting point.

13. It is evidently immaterial which of two opposite quantities is taken positively and which negatively in any particular problem. Thus, we might call distances south positive and distances north negative. We have only to interpret results differently.

EXERCISES VII.

State algebraically in two ways each of the following relations (by Art. 13):

1. 100 dollars gain and 20 dollars loss is equivalent to 80 dollars net gain.

2. 250 dollars gain and 250 dollars loss is equivalent to neither gain nor loss.

3. A rise of 15° in temperature followed by a fall of 22° is equivalent to a fall of 7°.