SCHOOL ARITHMETICS BOOK TWO CHAPTER I I. FUNDAMENTAL OPERATIONS What you have Learned of Arithmetic. You have already learned much about arithmetic. You have learned how to count and to write numbers, at least to a million; how to add, subtract, multiply, and divide whole numbers or integers; how to solve many problems; how to keep simple accounts; how to do some easy work with fractions; how to write United States money; how to measure lines and rectangles; and how to use the ordinary tables needed in making these measurements. What there is still to Learn. There is, however, much more to learn about the uses of arithmetic and more to learn about how to work with numbers. Every one of us is interested in what the world is doing and is interested in our country and its products. We shall therefore review some of the most important things about arithmetic and shall then study some new problems relating to business and to our everyday life. Notes and explanations in this type are intended primarily for the teacher, to be used as needed. Numerals. Symbols used to represent numbers are called numerals. The common numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are also called figures, Arabic numerals, or Hindu-Arabic numerals. The symbol 0 is called zero, naught, cipher, or o. Little attention is now given to the distinction between notation (the writing of numbers) and numeration (the reading of numbers) or to the memorizing of such definitions as those given on this page. Schools should recognize the modern reading of numbers in telephoning, as 'three-nine-six-o" for 3960, and in entries in bookkeeping, "nineteen seventy-five point fifty" for $1975.50. The nine figures, 1, 2, 3, 4, 5, 6, 7, 8, and 9, are called digits. Period. For ease in reading we separate the figures of a large number into groups of three, beginning at the right. Each of these groups is called a period. We indicate the periods by commas or, in print, by spaces, thus: 14,375,286 or $756 283.75. In an automobile number a hyphen is sometimes used, as in 45-784. In separating the figures of a number into periods, the pupil should begin at the right (or at the decimal point if there is one) and count toward the left. The left-hand period may have one, two, or three figures. Periods are not commonly indicated in numbers of four figures, and in this book we shall use commas to indicate periods only in numbers of five figures or more. Place Value. The value of a period or figure depends upon the place it occupies. Therefore each period has a place value. In the number 21,235,406,146 we have four periods as here shown. In each period there are three orders: units, tens, and hundreds. Billions Millions Thousands Units 21, 235, 406, 146 The number is read "twenty-one billion, two hundred thirtyfive million, four hundred six thousand, one hundred forty-six." A thousand million is called a billion. READING AND WRITING NUMBERS Numbers 1 to 16, oral 1. Looking at your fingers, do you see why people came to count by tens? Why was it? Give the proper name to each of the following: 2. 3 tens. 3. 10 tens. 4. 1000 thousand. 5. 1000 million. Read the following numbers: 6. $26.06. 8. 427,000. 10. 5,000,000. 12. 3,125,000,000. 7. $37.50. 9. 400,027. 11. 7,275,060. 13. 4,000,125,000. 14. What advantage has our method of writing nine (9) over the method the Romans used, as shown on a clock face (IX)? 15. In writing 738 which place has the 8? the 3? the 7? 16. In the number 34,628,059,483 what name has the period 483? the period 059? the period 628 ? the period 34? Write the following in figures: 23. Seven million, twenty-five thousand, fifty-nine. 26. One billion, two million, one hundred sixty. Exercises are for written work unless the contrary is stated. The motive for this work having been well established, the exercises in the present review are largely abstract. Roman Numerals. Many years ago the Romans had a system of numerals which we still see used on clocks and for numbering the chapters of books. Sometimes we see them used in dates, as in the number of the year. Everyone knows the first twelve numerals as they appear on a clock face. They are as follows: At present in writing Roman numerals we make use of seven letters, usually capitals, as follows: These letters are combined according to the following rules: 1. If a letter of less value is written after a letter of greater value, the sum of the values is to be taken. Thus, V = 5, I = 1, and so VI = 5 + 1 = 6. 2. If a letter of less value is written before a letter of greater value, the difference of the values is to be taken. Thus, X = 10, I= 1, and so IX = 10 −1 = 9. 3. Repeating a letter repeats its value. Thus, C = 100, and so CC = 100 + 100 = 200. Usually letters are taken not more than three times, but, as with IIII, this is not always the case. In ancient times CCCC was commonly used instead of CD for 400. Sometimes a dash is placed over a letter to increase its value a thousandfold, as in V for 5000, but this is too rarely used to be learned. |