COMMON SCHOOL ARITHMETIC. SECTION I. PRELIMINARY DEFINITIONS. RTICLE 1.-A Unit is one, or a single thing of any kind. 2. A Number is a unit, or a collection of units. Thus, one, two, three, four, five, are numbers. 3. The Unit of a number is one of the collection forming that number. Thus, one is the unit of six, one book is the unit of six books. 4. An Integer is a number formed wholly of entire units. Thus, three, five, six, nine, are integers. Integers are also called integral or whole numbers. 5. Similar Numbers are those which have the same unit. Thus, three yards and five yards are similar numbers. 6. Dissimilar Numbers are those which do not have the same unit. Thus, three yards and three books are dissimilar numbers. 7. A Concrete Number is one that names the kind of unit numbered. Thus, five bushels, in which the kind of unit is named, is a concrete number. 7 8. An Abstract Number is one that does not name the kind of unit numbered. Thus, five, in which the kind of unit is not named, is an abstract number. 9. Arithmetic is the science of numbers and the art of computing by them. 10. A Solution in Arithmetic is the process of answering a question which requires computation. 11. A Proof of a solution is the process of testing its cor rectness. 12. A Problem is a question for solution. 13. A Principle is a general truth. 14. A Rule is a concise statement of the method of solving a problem. 15. An Example is a problem which is used to illustrate a principle or rule. 16. An Exercise is a problem which is intended to render knowledge familiar by drill or practice. EXERCISES. 1. How many units in one? In one dollar? Three is a collection of how many units? 2. What is the unit of two books? Of four? Of five pounds? Of seven houses? 3. Are two cents and five cents similar or dissimilar numbers? Why are three men and five books dissimilar numbers? 4. Is four yards a concrete or an abstract number? Three? Two boys? 5. Why is one mile the unit of four miles? Why is one the unit of six? 6. Why is two houses a concrete number? Why is four an abstract number? 7. Of the two numbers, two miles and ten miles, what is the unit? 8. Of the two numbers, seven dollars and nine dollars, what is the unit? SECTION II. NUMERATION AND NOTATION. 17. The Naming of numbers requires only a small number of different words. A single thing is named one; one and one is named two; one and one and one is named three; and so we have the separate names, One, two, three, four, five, six, seven, eight, nine, ten. 18. Ten, by being regarded as forming a set or collection of units, may be treated as a single thing, or as a unit equal to ten ones. One and ten, two and ten, three and ten, four and ten, etc., by change of form, give the familiar names, Eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. Two tens, three tens, four tens, etc., by change of form, give the names, Twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. Twenty and one, twenty and two, etc.; thirty and one, thirty and two, etc., to ninety and nine, by change of form, give the names, Twenty-one, twenty-two, etc.; thirty-one, thirty-two, etc., to ninety-nine. 19. One Hundred is the name given to a collection of ten tens. One hundred and one hundred, two hundred and one hundred, etc., form Two hundred, three hundred, etc., to nine hundred. FIGURES. 20. Figures are the characters commonly used to represent numbers. They are as follows— PRINTED, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. WRITTEN, 0 1 2 3 4 5 6 7 8 9 Zero, one, two, three, four, five, six, seven, eight, nine. The figure 0 is sometimes called a Cipher, or Naught, because when written alone it expresses no value, or the absence of number; and the figures 1, 2, 3, 4, 5, 6, 7, 8, 9 are called Numerals, or Significant Figures, because each expresses as many ones as are denoted by its name. Numbers greater than nine are expressed by repeating or combining two or more of the ten figures. 21. Exact Tens are written with the figure expressing the number of tens at the left of 0, which marks the absence of ones; and tens and ones are written with the figure expressing the tens at the left of the figure expressing the ones. Thus, is written 10, Ten, or 1 ten, Twelve, or 1 ten and 2 ones, 12, and so on to ninety-nine, or 9 tens and 9 ones. 22. Exact Hundreds are written with the figure expressing the hundreds at the left of two zeros. Hundreds, tens and ones are expressed together in a number by writing the figure expressing the tens at the left of the figure expressing the ones, and the figure expressing the hundreds at the left of that expressing the tens. Thus, One hundred, or 1 hundred 0 tens 0 ones, is written 100. Two hundred, or 2 hundreds 0 tens 0 ones, is written 200. Four hundred ten, or 4 hundreds 1 ten 0 ones, is written 410. Five hundred six, or 5 hundreds 0 tens 6 ones, is written 506. Nine hundred seventy-eight, or 9 hundreds 7 tens 8 ones, is written 978. 23. Numeration is the method of naming numbers, and of reading numbers expressed by figures. 24. Notation is the method of writing numbers, or of expressing numbers by figures. Write in figures— WRITTEN EXERCISES. 1. One hundred thirty-six; two hundred thirteen. 9. Five hundred twenty-two; nine hundred ninety-nine. ORDERS AND PERIODS OF UNITS. 25. Orders of Units are denoted by the successive figures used in expressing a number. Thus, in 365, the 5, which expresses 5 ones, denotes units of the First Order; the 6, which expresses 6 tens, denotes units of the Second Order; and the 3, which expresses 3 hundreds, denotes units of the Third Order. 26. In naming numbers, the first three orders of units are regarded as forming a group, called the Class, or Period, of Units, having ones, tens and hundreds. Thus, 425 forms a period composed of 425 units. 27. Ten hundreds form One Thousand; ten thousands form One Ten-Thousand; and ten ten-thousands form One HundredThousand. These three orders of units form a group, called the Period of Thousands, having ones, tens and hundreds. Thus, 363425 is composed of 363 thousands 425 units, or of two periods, and is read three hundred sixty-three thousands four hundred twentyfive. 406007 is composed of 406 thousands 007 units, or of two periods, and is read four hundred six thousands seven. In like manner are formed and read other periods. |