THE NEW YORK STATE SYLLABUS The following synopsis shows how the work is arranged to meet the requirements of the Syllabus. Schools not feeling the necessity for reviewing the work in the fundamental operations with integers should begin with page 57. NEW YORK STATE ARITHMETIC YEARS FIVE AND SIX CHAPTER I WRITING AND READING NUMBERS 1. Units. The standards by which we count or measure are called units. Thus in speaking of $4 the unit is $1; in speaking of 5 eggs the unit is 1 egg, but in speaking of 10 dozen eggs the unit is 1 dozen eggs; in speaking of 40 thousand bricks the unit is a thousand bricks; in counting 1, 2, 3, 4, and so on, the unit is 1; in measuring long distances the unit may be 1 mile. 2. Numbers. That which shows how many times a unit is taken, or what part of a unit is taken, is called a number. Thus 3 indicates that 1 has been taken 3 times; $6 indicates that $1 has been taken 6 times; indicates that four fifths of 1 is taken; hence 3, $6, and are all numbers. 3. Abstract and Concrete Numbers. A number used without reference to any particular unit is called an abstract number; if the unit is named, the whole expression is called a concrete number. For example, 5 is an abstract number, but $5 and 5 bushels are concrete numbers. It must be understood, however, that the true number part is always abstract. That is, in $5 the true number part is 5. It is convenient, however, in explaining the solution of problems to make the distinction between abstract numbers and concrete numbers. 4. Integers. Numbers applied to whole units are called whole numbers, integral numbers, or integers. Fractional numbers, already familiar to the class from work in the earlier grades, and mentioned in § 2, are treated later. 5. Notation and Numeration. The writing of numbers is called notation; the reading of numbers is called numeration. 6. Kinds of Notation. The class is already familiar with the common or Arabic notation (so called because it came to Europe through the Arabs), and the Roman notation. The Arabic characters, sometimes called figures or ciphers, originated in India. They were employed without the zero more than 2000 years ago, and the zero or naught has been in use over 1200 years. These figures have been known in Europe more than 900 years, but did not become generally used until about 400 years ago, when they finally displaced the clumsy notation of the ancient Romans. 7. Numerals. The characters used in any system of notation are called numerals. Thus we speak of Arabic numerals and of Roman numerals. The Arabic numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The Roman numerals are I, V, X, L, C, D, M. The Arabic characters, excluding 0, are sometimes called digits. 8. Place Value. The value of a figure in a number written in Arabic numerals depends upon its position in the number and is called the place value of the figure. Thus, in the number 53, the 3 occupies units' place and means 3 units; the 5 occupies tens' place and means 5 tens. In the Roman system VIII, although made up of V (five) and III (three), like 53, means only eight, because the numerals have no such place value. 9. Orders of Units. Each successive place in a number that may be occupied by a figure is called an order of units. These orders increase from right to left in a tenfold ratio. Hence the Arabic system is called a decimal system (from the Latin decem, ten). The ones (or units) are called units of the first order; the tens, units of the second order; and so on. х 10. Names of the Orders. As already learned in school, the number names of the first few orders are as follows: 11. Periods. When the figures of a number are five or more, we separate them into groups of three figures each by commas or by spaces, beginning at the right. These groups are called periods. Thus we write thirty-two thousand, seven hundred fifty-six, 32,756 ; it is often printed with a space instead of the comma, thus, 32 756. 12. Separatrix. The comma separating two periods of a number is called the separatrix. 13. Names of the Periods. Beginning to name the periods at the right, the first period is called the units' period; the second, the thousands' period; the third, the millions' period; and the fourth, the billions' period. We rarely use number names above millions. The names of the periods above billions are trillions, quadrillions, quintillions, sextillions, septillions, and so on; but these need not be learned. 14. Reading Numbers. In reading numbers we read each period by itself and add the name of the period. Avoid all useless words. Thus we read 2,341,406 "two million, three hundred forty-one thousand, four hundred six." It is customary to omit the name of the units' period. Read aloud the following sentences : 31. The area of Alabama is 52,250 square miles. 41. The area of New York State is 49,170 square miles. 42. The area of Texas is 265,780 square miles. 43. The area of the United States is 3,616,484 square miles. |