ADDITION AND SUBTRACTION 36. A third way of finding the perimeter. We have seen that the perimeter may be found in two ways. 1. Arithmetically. Each side is measured and the measures are added. The resulting arithmetical sum is the perimeter. 2. Algebraically. This means that the sides are added without finding the length of each. Thus, p=a+b+c. The algebraic sum a+b+c denotes the perimeter. The equation p=a+b+cis a formula for finding the value of the perimeter when the values of a, b, and c are given. The following is a third way of finding the perimeter. a A Draw a line of indefinite length, as OX (Fig. 63). Open the compass a distance a (Fig. 62) b and from O lay off on OX the segment OD Fig. 62 equal to a. Similarly, from D lay off a distance DE equal tob in the direction of X, and from E lay -a+b+c 4 off EF equal to c. X OF is the perF-----a + 中 -- imeter of triangle FIG. 63 ABC. The line segment OF may now be measured to determine the arithmetical value of a+b+c. E F с 2. Find in three ways the perimeter of the quadrilateral ABCD (Fig. 65). 3. Draw two unequal segments a and b. Construct the sums a+b and bta. Meas- A ure each sum and show that a+b=b+a. B FIG. 65 37. The law of order in addition. Exercise 3 illustrates the principle that the value of a sum remains unchanged when the order of the addends is changed. This is called the law of order in addition. This law helps us to simplify the adding of numbers by arranging them first in an advantageous order. For example, the terms in the sum 475+210+25 may be taken in the order 475+25+210= 500+210=710 To indicate that we wish to add first 475 and 25 and then 210, the last statement 'may be written in the following forms: (475+25)+210=500+210=710, or [475+25]+210=500+210 = 710, or {475+25}+210=500+210 = 710. 38. Parentheses. The symbols (),[], { } have been used to show the order in which numbers are to be added, subtracted, or multiplied. Thus, 4X(2X3) means that first 2 is to be multiplied by 3 and the product is then multiplied by 4. EXERCISES each of the following exercises state the meaning of the symbols and then add, subtract, and multiply in the order indicated. 1. (324+15)+75. 4. 5+{7–3}. 2. (24+8) - 22. 5. 16–17+3]. 3. (15-2) +4. 6. 18—[10–6+2]. 7. Rearrange in the most advantageous way and then add: 240+325+60; 736+298 +64; 350+287+50. 8. State the meaning of each of the following expressions: (6+7)+3; 26+(65+125); (a+b)+(c+d); (10+15)+50+13. The symbols ( ), [ ], and { } are called parentheses, brackets, and braces respectively. b a a-b B A a 39. Finding the difference between two segments. Draw two segments aandb (Fig. 66), making a greater than b. X Draw an indefinite line, OX. On OX lay off OA=a. From A lay off, in the direction A0, the segment AB=b. The segment OB is the difference between a and b. In the form of an equation this may be written OB=a-6 EXERCISE с Draw three segments a, b, c (Fig. 67) making a>b>c. Then construct the following sums and differences, marking all segments as shown in Figs. 63 and 66: a+b-c; a+c+b; atc-b; b-cta; FIG. 67 a-c+b. b a 40. Monomials. Terms. Numbers like 30 Xn, 20 Xt, rXt, are monomials, or terms. The numbers 30 and n are factors of 30 Xn, 20 and t are factors of 20 Xt, r and t are factors of rXi. 41. Similar terms. Monomials having a common (the same) literal factor, as 4Xa and 8Xa, are similar terms. 42. Coefficient. In the monomials 2xn, 30 XX, 20 Xt, the numerical factors 2, 30, 20, åre called coefficients of the literal factors n, x, and t respectively. It is customary to write products like 2xn, 30 Xw, 20 Xt, briefly 2n, 30w, 20t, omitting the multiplication sign. It must be remembered that such numbers have two meanings. For example, 3m means m+m+m, and 3 multiplied by m. State two meanings of each of the products above. When no coefficient is stated, as in the numbers a, x, y, the coefficient is understood to be 1. Thus a means la, x means 1x. EXERCISES 1. State the following products in a brief form: 6Xs, 5Xm, 80 Xd, pXr. 2. Give several examples of monomials; of similar monomials; of dissimilar monomials. 3. State two meanings for each of the following: 5y, 4x, 7a. What is the meaning of ab, xy, rt? 4. Find the values of 15t when t=1, 2, 6, 7.5, 10.75, 123. 43. Polynomials. Numbers like a+b+cand x+y+z, which consist of two or more monomials (terms), are called polynomials (having many terms). A polynomial with only two terms, as a+b, is a binomial. A polynomial containing three terms, as a+b+c, is a trinomial. 44. Adding and subtracting similar terms. We have seen that 2a means a ta, and that 3a means atata, Hence, 2a+3a means (ata)+(atata), or atatatata, or 5a. It follows that 2a+3a=5a. This shows that the similar terms 2a and 3a may be combined, or collected, by adding the coefficients 2 and 3, and then multiplying the sum by the common factor a. Numbers not having a common factor cannot be combined. Thus in the sum of a and b the addition must remain indicated, as a+b. EXERCISES 1. By combining similar terms, reduce each of the following polynomials to a simpler form. Arrange the work as follows: 4m- jm=(4-})m=3 m. a b 2. The lengths of the sides of the recb tangle (Fig. 68) are respectively a, b, a, and b. Write the perimeter in the simplest form. a 3. Find the perimeter of each of the figures FIG. 68 below (Fig. 69) and write the results in the simplest form. Find the value of each of the resulting polynomials for a=6, b=3, c=2, d=4, f=1. 13 a Fig. 69 |