CHAPTER II

POSITIVE AND NEGATIVE NUMBERS

ADDITION AND SUBTRACTION

20. In the conducting of business, capital is increased by gains and decreased by losses. Suppose that for a week's business the account stands:

The total losses of $250.50 neutralize $250.50 of the gains or credits, leaving a net gain of $71.00. Suppose that for another week's business the above losses and gains were reversed:

Arith

The credits, $250.50, are less than the losses, $321.50. metically the net loss is found by subtracting the credits, $250.50, from the total loss, $321 50, which gives $71.00.

This result can be calculated algebraically as follows: Designate

the credits plus (+) or positive dollars and the losses minus ( − ) or negative dollars, then the account can be represented:

where the relation between + dollars and - dollars is that, in balancing accounts, a given number of dollars cancels or neutralizes the same number of dollars; i. e., a loss of $75.00 neutralizes or cancels gains of $25.00 and $50.00. This result is expressed algebraically:

The gains and losses are said to balance, or there is a balance of zero dollars, $0.00.

Or if there should be a gain of $75.00 and two losses, one of $25.00 and one of $50.00, the result of the transactions would be expressed algebraically:

+$75.00 - $25.00 - $50.00

=

+$75.00 – $75.00 = $0.00.

Zero is defined as the difference between two equal numbers. Suppose a loss of $321.50. It may be separated into two losses, one of $250.50 and the other of $71.00, and the operation of balancing the dollars and dollars in the problem above would + be indicated algebraically as follows:

+$250.50

$321.50 = + $250.50 – $250.50 – $71.00

= $0.00 $71.00: = $71.00.

That is, there remains a debt of $71.00, or the balance is a negative number.

21. The Series of Natural Numbers. If from a fixed point 0, in a line A B, units of length are laid off to the right, the successive points so found can be designated by the series of natural numbers, 1, 2, 3, 4, 5, etc.

A

1 2 3 4 5 6

7 8 9 10 11 12

0

B

If the measurement proceeds from 0 toward B it is positive measurement; if the direction is reversed, say at the 11th division, and proceeds from 11 toward 0, this is another kind of measurement, namely negative.

If the measurement starts from the 5th and ends at the 9th division, 4 units are added to the original 5. The final distance from 0 is

+5+4+9.

This illustrates positive measurement.

If the measurement is from the 11th division to the 8th, the act of moving from the 8th to the 11th division is neutralized. This operation which neutralizes the operation of addition, is indicated by the sign, read minus. In moving from 0 to the right to the 11th division, then reversing the direction and moving to the left to the 8th division, the total result or distance from 0 is represented algebraically:

+11−3=+8+3−3+8+0+8 (see 21, definition of zero).

This may briefly be translated thus; move to the right 11 units from 0 and then move 3 units to the left from 11, stopping at a distance of 8 units from 0 to the right.

If the motion takes place from 0 to the right to the 6th division, and is then reversed over 6 divisions to the left, the final position is O and the distance from 0 is zero units. The result of the motion is indicated algebraically:

+66=0.

The operation of moving from zero to the right or from any point of division to the right can be expressed by addition of units, and of moving from the right to the left by the subtraction of units.

22. Positive Numbers. Suppose that it is desired to subtract 8 from 5. Move from the 5th division to the left over 5 unit spaces. The resulting position is 0, after but 5 of the 8 units have been subtracted. The act of moving over 8 unit spaces from the right to the left can be separated into one motion of 5 and another of 3 units. Therefore the operation of moving 8 units to the left from the 5th point of division may be indicated algebraically:

03 is represented by the simpler symbol 3, called a negative number. This result and the corresponding result -$71.00, derived

when the losses in a business transaction were $71.00 greater than the gains (20), necessitate the introduction of a system of nega tive numbers into Algebra.

23. Negative Numbers.-Begin at zero and lay off unit lengths to the left; by the repetition of the unit, a series of negative numbers is formed. These two series of numbers, the series of positive or natural and the series of negative numbers, are called series of algebraic numbers, and are represented on the line as follows:

-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 −2 −1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10+11+12+13 ml k jih g fe d с b a A B C D E F G H I J K L M

The subtraction of 8 from 5 will be expressed by a motion from E to the left over 8 unit divisions to c, and the result is —3; i. e., the final position is a place 3 units to the left of zero.

The result obtained by subtracting a greater number from a less when both are positive, is always a negative number.

In general, in case a and b are any two positive integers, the expression a b is a positive integer when a >b, is zero when a=b, and is a negative integer when a <b.

In a series of algebraic numbers, in counting from left to right, numbers are said to increase, in counting from right to left they are said to decrease in magnitude. Thus -4, -2, -1, 0, +2, +4 are arranged in ascending order of magnitude.

24. The Absolute Value of a Number.-The absolute value of a number is its value without its sign. Thus the absolute values of -4, -2, -1, +3, +5, are respectively 4, 2, 1, 3, 5.

-

25. Every algebraic number +5 or 5 consists of a sign + or and the absolute value of the number. The sign shows whether the number belongs to the positive series or the negative series of numbers; the absolute value of the number shows the place which the number has in the positive or negative series.

When no sign is written before a number, the sign + is understood. The sign

is always written.

26. Unlike Signs.--Two algebraic numbers which have respectively the signs and are said to have unlike signs. Thus +7

ADDITION OF ALGEBRAIC NUMBERS

27. Since algebraic numbers may be positive or negative, four different problems arise in the addition of them:

I. Addition of two positive numbers, as .

In particular +3+(+4)

II. Addition of a positive and a negative number, as +3+ (-4) III. Addition of a negative and a positive number, as 3+ (+4) IV. Addition of two negative numbers, as

- 3+ (-4).

-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10+11+12+13

A B C D E F G H I J K L M

ml k ji h g fed c b a I. The sum of +3 and +4 is found by counting from C, whose distance from 0 is +3, 4 units to the right, or in the positive direction, and is therefore +7, the number of units from 0 to G.

II. The sum of +3 and 4 is found by counting 4 units to the left, or in the negative direction, from C (or +3), and is therefore 1, the distance of a from 0.

III. The sum of 3 and

4 is found by counting from c (or 3) 4 units to the right, or in the positive direction, and is therefore +1, the distance of A from 0.

IV. The sum of 3 and 4 is found by counting from c (or 3) 4 units to the left, or in the negative direction, and is therefore - 7, the distance of 9 from 0.

28. If a and b represent any two integers, the results in 27 are therefore expressed as follows:

These four cases give rise to the following rules.

−a+b

a

- b.

29. RULES FOR THE ADDITION OF ALGEBRAIC NUMBERS

I. If two numbers have like signs, find the sum of their absolute values, and prefix the sign common to both numbers to the result.

II. If two numbers have unlike signs, take the difference between their absolute values, and prefix the sign of the number with the greater absolute value to the result.