24. A geometrical line has length, but neither breadth nor thickness. NOTE. — Lines drawn upon paper or upon the blackboard are not geometrical lines, since they have breadth and thickness. They represent geometrical lines. 25. A straight line is the shortest distance from one point to another point. 26. A curved line changes its direction at every point. 27. A broken line is not straight, but is made up of straight lines. 1. The line AB is a 5. The line EK is a line made up of equal lines. 7. The perimeter of a regular pentagont is a -- line made up of equal lines. 8. The circumference of a circle is a line, every point in which is equally distant from a point called the center of the circle. 9. The diameter of a circle is a line. 10. Imagine a straight line drawn upon the surface of a stovepipe. Can you draw a straight line upon the surface of a sphere? * See Book I., p. 53. + See Book I., p. 63. See Book II., p. 256. 28. MISCELLANEOUS REVIEW. 1. If a equals one side* of a regular pentagon, the perimeter of the pentagon is — 2. If b equals the perimeter of a square, the side of the square equals b: 3. If a equals a straight line connecting two points and b equals a curved line connecting the same points, then a is † than b. 4. Find the difference between two hundred seven thousandths, and two hundred and seven thousandths. I 5. How many zeros in 1 million expressed by figures ? 1 billion ? 1 trillion ? 6. How many decimal places in any number of millionths ? billionths? trillionths? 7. How many decimal places in 5 thousandths ? in 25 thousandths ? in 275 thousandths? in 4346 thousandths ? 8. A figure in the second integral place represents units how many times as great as those represented by a figure in the second decimal place? 9. If a= 6,b=2, and d=8, what is the numerical value of the following ? 12 a +36-5 d. 10. John had a certain amount of money and James had 5 čirnes as much; together they had 354 dollars. How many dollars had each? Let x= the number of dollars John had. 6 x= 354 dollars. 5x= * The expression “a equals one side" means that a equals the number of units in one side. Remember that in this kind of notation the letters employed stand for numbers. # Longer or shorter ? See p. 15, Exercise 14. ADDITION. 29. Addition (in arithmetic) is the process of combining two or more numbers into one number. NOTE 1.-— The word number, as here used, stands for measured magnitude, or number of things. NOTE 2.- Addition (in general) is the process of finding the sum of two or more magnitudes. 30. The sum is the number obtained by adding. 31. The addends are the numbers to be added. 32. The sign, +, which is read plus, indicates that the numbers between which it is placed are to be added; thus, 6+4, means, that 4 is to be added to 6. 33. The sign, =, which is read equal or equals, indicates that that which is on the left of the sign equals that which is on the right of the sign; thus, 3+4= 7. 5++2= 6 +5. 34. PRINCIPLES. 1. Only like numbers can be added. 2. The denomination of the sum is the same as that of the addends. 35. PRIMARY FACTS OF ADDITION. There are forty-five primary facts of addition. They are given in the Werner Arithmetic, Book II., p. 273. The nine primary facts which many pupils fail to memorize perfectly are given below. 7 8 9 8 9 8 9 9 9 6 5 7 6 7 8 17 د حراج 14 16 7. In each of the above examples, tell which are addends and which is the sum. 37. Observe that in written problems in addition the figures that stand for units of the same order are usually written in the same column. 1. In example 2, what figures represent units of the second decimal order? Of the third decimal order? 2. In example 5, what figures represent units of the first integral order? 38. Observe that in written problems in addition of denominate numbers, the figures that stand for units of the same denomination and order are usually written in the same column. 1. In example 5, what figures represent units of gallons ? of quarts ? 2. In example 5, what figures represent tens of gallons ? 3. In example 4, what figures represent hundreds of acres ? Addition - Simple Numbers. 39. Find the sum of 275, 436, and 821. Operation. Explanation. 275 The sum of the units of the first order is 12; this is 436 equal to one unit of the second order and 2 units of the 821 first order. Write the 2 units of the first order, and add the 1 unit of the second order to the other units of the 1532 second order. The sum of the units of the second order is 13; this is equal to 1 unit of the third order and 3 units of the second order. Write the 3 units of the second order, and add the 1 urit of the third order to the other units of the third order. The sum of the units of the third order is 15; this is equal to 1 unit of the fourth order and 5 units of the third order, each of which written in its place. The sum of 275, 436, and 821 is 1532. 40. PROBLEMS. 8. 564 693 684 502 264 726 877 112 (a) Find the sum of the eight sums. TO THE TEACHER. — Impress upon the pupil the fact that in arithmetic nothing short of accuracy is commendable. One figure wrong in one problem in ten is failure. The young man or the young woman who cannot solve ten problems like those on this page, without an error, is worthless as an accountant. |