4. Is three yards a concrete, or an abstract number? Why? Three? Five hours? Two? Seven gallons? Ninety? 5. Are two dozen and four dozen similar, or dissimilar numbers? Why? Fourteen and sixty? Five minutes and seven bushels? Six tens and eight ones? Eight tens and eighty? 6. Name three units. Three numbers. Three concrete numbers. Three abstract numbers. Three similar numbers. Three dissimilar numbers. Three numbers that are concrete and similar. Three abstract and similar. Three concrete and dissimilar. Section II. NOTATION AND NUMERATION. 1. How many single things are one and one? Two and one? Three and one? Six and one? Eight and one? 2. How many single things, or units, are in a group of ten? Of ten and one? Ten and two? Ten and three? five? Ten and seven? Ten and nine? Ten and 3. How many single things, or units, are in two groups of ten single things each? In three groups of ten each? In four groups? In six? Eight? Nine? 4. How many single things, or units, are in ten groups of ten single things each? In ten groups and one group of ten each? In ten groups and five groups? 15. In naming numbers, objects are considered as being collected into groups of ten, the groups increasing in size or value tenfold, and receiving different names. 16. In naming numbers, a single thing is called one. A collection of ten ones considered as a single group is called one ten. A collection of ten tens considered as a single group is called one hundred. Ten hundreds taken together are called one thousand; ten thousands are called one ten-thousand; ten tenthousands are called one hundred-thousand; etc. 17. A group of a thousand ones is called a thousand; a group of a thousand thousands is called a million; a group of a thousand millions is called a billion; etc. 18. Orders of Units are kinds or classes of units formed by grouping together ten lower or smaller units. Thus, ones are units of the first order; tens, of the second order; hundreds, of the third order; thousands, of the fourth order; etc. 19. A Period is a group of three orders of units, beginning with ones. 20. The First, or Ones' Period, is formed of the first, the second, and the third orders of units, and comprises ones, tens, and hundreds. 21. The Second, or Thousands' Period, is formed of the fourth, the fifth, and the sixth orders, and comprises thousands, ten-thousands, and hundred-thousands. 22. The Third, or Millions' Period, is formed of the seventh, the eighth, and the ninth orders, and comprises millions, ten-millions, and hundred-millions. 23. In writing numbers, or expressing numbers by written characters, three methods are used: 1. By words; as one, five, ten, etc. 2. By letters; as I, V, X, etc.; called the Roman Method. 3. By figures; called the Arabic Method. FIGURES.-0, 12, 3, 4, 5, 6, 7, 8, 9. NAMES.-naught, one, two, three, four, five, six, seven, eight, nine. 24. The name of a figure denotes the number of units which it represents. Thus, O, named naught, or cipher, denotes nothing, or the absence of number; 2 is called two, and denotes two units; 3 is called three, and denotes three; etc. 25. Units of the first order, or ones, are expressed by writing the figures having the same name as the number itself. Thus, one is expressed by 1; two, by 2; six, by 6; etc. 26. A unit of the second order, or one ten, is expressed by writing the figure 1 with a cipher after it; thus, 10. In the same manner are expressed 27. The numbers between ten and twenty are expressed by writing 1 with the number of ones after it; thus, 11. Fourteen, 14. Seventeen, Eleven, 17. The numbers between twenty and thirty are expressed by writing 2 with the number of ones after it. Thus, twenty-one, 21; twenty-five, 25; twenty-nine, 29; etc. 28. The numbers between ten and one hundred are expressed by writing the figure denoting the number of tens in the left-hand or second place, and the figure denoting the number of ones in the right-hand or first place. 29. When a number is expressed by two figures written side by side, the right-hand figure denotes ones, and the lefthand figure denotes tens. 30. A unit of the third order, or one hundred, is expressed by writing the figure 1 with two ciphers after it; thus, 100. 31. The numbers between one hundred and two hundred are expressed by writing 1 with the number of tens and ones 32. The numbers between one hundred and one thousand are expressed by writing the figure denoting the hundreds in the third place, the tens in the second place, and the ones in the first place. 33. When a number is expressed by three figures written side by side, the right-hand figure denotes ones, the middle figure denotes tens, and the left-hand figure hundreds. 34. A unit of the fourth order, or one thousand, is expressed by writing 1 with three ciphers after it; thus, 1000. In the same manner are expressed 35. The numbers between one thousand and ten thousand are expressed by writing the figure denoting the number of thousands in the fourth place, the hundreds in the third place, the tens in the second, and the ones in the first; thus, One thousand two hundred and five is written 1205. Five thousand four hundred and sixty 36. A unit of the fifth order, or one ten-thousand, is expressed 10000; one hundred-thousand, 100000; one million, 1000000; etc. 37. The orders of units are denoted by the places in which the figures used to express a number stand. 38. A figure in the first place denotes ones, or units of the first order; in the second place tens, or units of the second order; in the third place hundreds, or units of the third order; etc. Thus, in the number 987, 7 denotes 7 ones, or units of the first order; 8 denotes 8 tens, or units of the second order; and 9 denotes 9 hundreds, or units of the third order. The number is read nine hundred and eighty-seven. Definitions. 39. Notation is the method of expressing numbers by figures. Thus, the expressing of the number three hundred and fifty-seven by the figures 357 is the notation of three hundred and fifty-seven. 40. Numeration is the method of reading numbers that are expressed by figures. Thus, the naming or reading of 468 as four hundred and sixty-eight is the numeration of 468. 41. Significant Figures are the figures which signify or denote a number of units. The significant figures are 1, 2, 3, 4, 5, 6, 7, 8, 9. 42. The Value of a figure is the number of units which it represents. Thus, the value of 3 is three ones; of 9 is nine ones; etc. 43. Significant figures have two values: a simple value and a local value. 44. The Simple Value of a figure is the value which it expresses when taken alone. 2 |