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+49. 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: +6 60. 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. 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: Suppose that it is desired to subtract 8 0-3 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 negative 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 mik jih g fe dc ba 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 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 ab is a positive integer when a>b, is zero when ab, 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 and the absolute value of the number. -5 consists of a sign + or 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 and - 9 have unlike signs. ADDITION OF ALGEBRAIC NUMBERS 27. Since algebraic numbers may be positive or negative, four different problems arise in the addition of them: II. Addition of a positive and a negative number, 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 ml k ji h g fed c ba ABCDEFGHIJKLM 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 g 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. 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. The results in the several cases, I, II, III, IV, are called the algebraic sums in distinction from the arithmetical sum, which is simply the sum of the absolute values of the numbers. III. If there are more than two numbers to be added, add two of the numbers, then this sum to the third, and so on; when the numbers to be added are positive and negative, take the difference between the absolute values of the sum of the positive numbers and the sum of the negative numbers and prefix the sign of the greater sum to the result, which will be the algebraic sum of the numbers. 30. The Addition of Similar Monomials.-1. Find the sum of 3a+a+4a+7a=(3+1+4+7)a [Law V, 7] = 15 a. Hence, the sum of the monomials is 15 a. 2. Find the sum of 3b, 5b, -- By I, 129, find the sum of the coefficients, which is -3-5-7-11-26 Hence, the sum of the monomials is 26b. The same result would be obtained by assuming Law V, 7 to hold for negative numbers, thus: -3b-56-76-116(-3-5-7-11)b =26b, by I, 29. 3. Find the sum of 6 ax, 5 ax2, -2ax, +13 ax, 19 ax2, +ax2. By II and III, 29, the sum of the coefficients of the positive terms is and the sum of the coefficients of the negative terms is 5-2 -19= - 26. The difference between 26 and 20 is 6, and the sign of the greater Hence, the sum is - 6 ax2. is REMARK.-The divisions AB, BC, etc., in the figure in 27 would, in case of example 1, be a; in case of example 2, be b; and in example 3, be ax2. Here, as in example 2, the same result would be obtained by assuming Law V, 7 to hold for positive and negative numbers, thus: 6 ax2 — 5 ax2 — 2 ax2 + 13 ax2 — 19 ax2 + ax2 = Therefore, (6-5-213-19+1)ax = 6 ax2, by II, 29. To find the sum of similar monomials, find the algebraic sum of the coefficients and prefix this sum to the letters common to the several terms. |
