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5a2-2a and 6x are integral expressions.

Expressions like x3 + 4x2 + x + 1 are sometimes regarded as integral, since the literal numbers are not in fractional form.

43. An expression that can be written without using a root sign is called a Rational Expression.

1, 2, 3, ..., a + {},

x2

x- y

1

and (a - b)2 are rational expressions.

√25 is rational, since it can be written 5 without a root sign.

44. An expression that cannot be written without using a root sign is called an Irrational Expression.

a + √ò, a + 2 √ā +1, and VI are irrational expressions.

Va2 is not irrational, however, since it may be written a.

POSITIVE AND NEGATIVE NUMBERS

45. For convenience, arithmetical numbers may be arranged in an ascending scale:

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The operations of addition and subtraction are thus reduced to counting along a scale of numbers. 2 is added to 3 by beginning at 3 in the scale and counting 2 units in the ascending, or additive direction; and consequently, 2 is subtracted from 3 by beginning at 3 and counting 2 units in the descending, or subtractive direction. In the same way 3 is subtracted from 3. But if we attempt to subtract 4 from 3, we discover that the operation of subtraction is restricted in arithmetic, inasmuch as a greater number cannot be subtracted from a less. If this restriction held in algebra, it would be impossible to subtract one literal number from another without taking into account their arithmetical values. Therefore, this restriction must be removed in order to proceed with the discussion of numbers.

To subtract 4 from 3 we begin at 3 and count 4 units in the descending direction, arriving at 1 on the opposite, or subtractive side of 0. It now becomes necessary to extend the scale 1 unit in the subtractive direction from 0.

To subtract 5 from 3 we begin at 3 and count 5 units in the descending direction, arriving at 2 on the opposite, or subtractive side of 0. The scale is again extended, and may be extended indefinitely in the subtractive direction in a similar way.

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For convenience, numbers on opposite sides of 0 are distinguished by means of the small signs and −, called signs of quality, or direction signs, being prefixed to those which stand in the additive direction from 0 and to those which stand in the subtractive direction from 0.

The former are called Positive Numbers, the latter Negative Numbers.

Hence, the scale of algebraic numbers may be written:

-5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5,

...

46. By repeating +1 as a unit any positive number may be obtained, and by repeating 1 as a unit any negative number may be obtained. Hence, positive numbers are measured by the positive unit, +1, and negative numbers by the negative unit, −1, or by parts of these units.

47. If +1 and −1, or +2 and −2, or any two numbers numerically equal but opposite in quality are taken together, they cancel each other. For counting any number of units from 0 in either direction and then counting an equal number of units from the result in the opposite direction, we arrive at 0. Hence,

If a positive and a negative number are united into one number, any number of units or parts of units of one cancels an equal number of units or parts of units of the other.

48. Two concrete quantities of the same kind are sometimes opposed to each other in some sense so that, if united, any number of units of one cancels an equal number of units of the other. For convenience, such quantities are often distinguished as positive and negative.

If money gained is positive, money lost is negative, for any sum gained is canceled by an equal sum lost. If a rise in temperature is positive, a fall in temperature is negative. If distances north or west or upstream are positive, distances south or east or downstream are negative.

ADDITION

49. 1. If a man has 10 dollars in one pocket and 15 dollars in another, how much money has he?

2. If in algebra money in hand is considered a positive quantity, indicate his financial condition algebraically. What is the sum of 10 positive units and 15 positive units, that is, of +10 and +15? of +4 and +8? of +a and +b?

3. If a person owes one man 10 dollars and another 15 dollars, how much does he owe both? Indicate his financial condition algebraically, regarding a debt as a negative quantity.

4. What is the sum of 10 negative units and 15 negative units, that is, of 10 and -15? of 6 and -14? of a and b?

5. What sign has the sum of two algebraic numbers that have like signs?

6. If a man has 25 dollars and owes 15 dollars, how much of his money will be required to cancel the debt? How many

dollars will he have after settlement?

7. What is the result when 15 is united with +25, that is, what is the algebraic sum of 15 and +25? of -20 and +10? of +8 and 3? of +6 and −10?

50. The aggregate value of two or more algebraic numbers is called their Algebraic Sum.

The process of finding the simplest expression for the algebraic sum of two or more numbers is called Addition.

51. PRINCIPLES. -1. The algebraic sum of two numbers with like signs is equal to the sum of their absolute values with the common sign prefixed.

2. The algebraic sum of two numbers with unlike signs is equal to the difference between their absolute values with the sign of the numerically greater prefixed.

By successive applications of the above principles any number of numbers may be added.

Only similar terms can be united into a single term.

Principle 1 may be established as follows:

The sum of 5 positive units and 3 positive units is evidently (5+ 3) positive units, or 8 positive units; that is,

+5+ +3= +(5+3)= +8.

Similarly, whatever absolute values a and b represent, since a times the unit +1 plus b times the unit +1 is equal to (a + b) times the unit +1,

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Again, the sum of 5 negative units and 3 negative units is (5 + 3) negative units, or 8 negative units; that is,

−5 + −3 = −(5 + 3) = −8.

Similarly, whatever absolute values a and b represent, since a times the unit 1 plus b times the unit -1 is equal to (a + b) times the unit -1, ̄a + b = −(a + b).

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Principle 2 may be established as follows:

The sum of 5 positive units and 3 negative units is 2 positive units, since, § 47, the 3 negative units cancel 3 of the positive units and leave 2 positive units; that is,

+5 + −3 = +(5 − 3) = +2.

The sum of 5 positive units and 7 negative units is 2 negative units, since, § 47, the 5 positive units cancel 5 of the negative units and leave 2 negative units; that is,

+5 + −7 = −(7 − 5) = −2.

Similarly, whatever absolute values a and b represent,

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for, § 47, the ₺ negative units will cancel b of the a positive units and leave (a - b) positive units;

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for, § 47, the a positive units will cancel a of the b negative units and leave (ba) negative units.

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52. To conform with the ideas already presented, the terms 'greater' and 'less' must be interpreted as follows:

An algebraic number is increased, or made greater, when a positive number is added to it, and decreased, or made less, when a negative number is added to it.

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Since, by 51, 3+1=-2, −2+1=1, −1 + 1 = 0, +1 + +1 = +2, etc., in the scale of algebraic numbers

..., -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, ...,

each number is greater than the number on its left and less than the number on its right; that is,

..., 3-2, 2<-1, -1 <0, 0 < +1, +1 < +2, +2 < +3, .... NOTE. 32 may be read -3 is less than -2' or '-2 is greater than -3.' Hence, it follows that:

1. Any positive number is greater than zero and any negative number is less than zero.

2. Of two positive numbers that which has the greater absolute value is the greater, and of two negative numbers that which has the less absolute value is the greater.

53. Abbreviated notation for addition.

Referring to the scale of algebraic numbers, it is evident that adding positive units to any number is equivalent to counting them in the positive direction from that number, and adding negative units to any number is equivalent to counting them in the negative direction from that number. Hence, in addition, the signs and denoting quality have primarily the same meanings as the signs and denoting arithmetical addition. + and subtraction. For example, by the definition of positive and negative numbers,

also

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+1 means 0+1 and 1 means 0-1;

5; etc.

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-

+5 means 0 +5 and 5 means 0 Hence, in addition, but one set of signs, and is necessary, and in finding the sum of any given numbers, the signs + and may be regarded either as signs of quality or as signs of operation, though it is commonly preferable to regard them as signs of operation.

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