Ex. 4. From +5 subtract -9.

Observe that, since the subtraction of a negative amounts to the addition of a positive, we may from the indicated subtraction +5 − −9, obtain

+595 + +9 = +(5 + 9) = +14.

Ex. 5. From -12 subtract -3.

Writing first as an indicated subtraction, and then transforming into an equivalent addition, we have -12 − −3 = −12 + +3.

Since in combination with the 12 negative units the 3 positive units diminish the number to 9 negative units, we may write

From another point of view, 12 negative units may be obtained by combining 9 negative units with 3 negative units.

Hence -12 -3 -9 +-3--3; and since successively adding and subtracting 3 negative units produce no final change in the value of the original 9 negative units, the second member of the identity reduces to -9. Ex. 6. From -4 subtract -11.

Replacing the indicated subtraction by an equivalent addition, we have −4 − −11 = −4 + +11.

In combination with the 11 positive units the 4 negative units produce a decrease in number to 7 positive units. Hence, -4++11 may be written as +(11 - 4) = +7.

From another point of view, the subtraction of 11 negative units in one operation amounts to the two separate operations of subtracting 4 and 7 negative units successively. Hence, we may write -4 --11 = −4 — −4 ——−7.

Since the subtraction of 4 negative units from 4 negative units produces zero, we have as a remainder the expression --7, which may be transformed into the equivalent expression + +7.

Hence 4-11 =+ +7, as above.

Ex. 7. From -1 subtract +14.

The subtraction of 14 positive units may be looked upon as amounting to the addition of 14 negative units.

Hence 1+14= −1 + −14.

In combination by addition, 14 negative units and 1 negative unit amount to 15 negative units.

Hence −1 +14 may be expressed as −(1 + 14) = −15.

Ex. 8. From -16 subtract +10.

Observe that the subtraction of 10 positive units is equivalent to the addition of 10 negative units. We may write 16+10-16 + −10. We have an expression for 16 negative units increased by 10 negative

units, resulting in 26 negative units. Hence, 16 +-10 may be written as −(16 + 10) = −26.

37. Inequality. To agree with the ordinary arithmetic notions of inequality, mathematicians have agreed to call one algebraic number, a, greater or less than another, b, according as the reduced value of ab is positive or negative.

From the definition it appears that any positive number (represented by *a) must be considered greater than any negative number (represented by b) since

+ab+a++b= (a + b)

which is a positive number.

From the same point of view 0 is to be regarded as being greater than any negative number, b, since

0 − b = 0 + +b = +b

which is a positive number.

Hence, corresponding to the expression "greater than 0,” there follows directly, by application of the Principle of No Exception, also the idea "less than 0."

38. Again, from our previous definition, one negative number, c, is to be regarded as being greater than another negative number, -d, according as the reduced value of

The reduced value of ̄c + d will be negative if c be numerically greater than d, and positive if d be numerically greater than c.

Ex. 9. Compare −3 and −4.

−3 − −4 = −3 + +4 = +1, a positive number.

Hence 3 is greater than -4.
Ex. 10. Compare −5 and −1.
From the definition we have

−5 − −1 = −5 ÷ +1 = −4, a negative number.

Therefore -5 is less than -1.

39. Using the symbol, "infinity," to represent any number which is numerically greater than any assignable number, it follows

from the reasoning above that the series of extended number may be regarded as being arranged in order of increasing magnitude, from left to right, as follows:

Observe that O may be regarded as belonging to both the positive and negative parts of the series; it is all that they have in common. (See Chapter II. § 23.)

EXERCISE IV. 3

Perform the following indicated subtractions :

1. From +6 subtract +4. 2. From +8 subtract +1.

Find the values of the following expressions when the given values are substituted for the letters appearing in them.

If a 1, b = 5, c=3, d = +4.

=

+1,

CHAPTER V

MULTIPLICATION AND DIVISION OF ALGEBRAIC NUMBERS

I. MULTIPLICATION

1. THE multiplication of abstract numbers is first defined in arithmetic as the taking of one number as many times as there are units in another.

E. g. To multiply 4 by 3, we take as many 4's as there are units in 3. We have 4 x 3 = 4+4 + 4 = 12.

2. In algebra, as in arithmetic, we call the number multiplied the multiplicand, the number which multiplies it the multiplier, and the result of the operation the product.

Whenever the first of two numbers such as 2 × 3 is regarded as the multiplier, it is customary to read the product as "2 times 3," while if 3 is regarded as the multiplier and 2 as the multiplicand, we may say "2 multiplied by 3."

3. Our first idea of multiplication is that it is an abbreviated addition.

From this point of view the multiplier must, in the original sense of the word, be the result of counting; that is, it must be a positive whole number.

The multiplicand may be any number previously defined, that is, it may be abstract or concrete, positive or negative, or even zero, but the multiplier must be an abstract number.

In a product consisting of two abstract numbers, the one at the right is usually regarded as the multiplier. However, since we speak commonly of 2 books, 3 apples, etc., mentioning the multiplier first, mathematicians find it convenient to arrange a given product containing numbers and letters so that the numerical parts shall occur in the first place.