SERIES SECOND, composed of tens. § 4.-1. One ten equals ten units. 2. The figures that express units are made to represent tens by placing a cipher on the right hand of each. § 5.-1. Each number in the above table expresses tens; that is, each of them represents ten times as many single objects as when written without a cipher, as in the table of units. 2. It must also be remembered that the cipher has given each figure this increased value. Hence, the cipher, although it has no value when standing alone, increases the value of the other figures in a tenfold ratio,—that is, makes them ten times as large, when placed on their right. 3. Hence, also, any figure, having a cipher on its right, is said to occupy the place of tens, instead of the place of units, as when standing alone. 4. This increased value is called the local value of a figure. § 6. How many single objects or units, and how many tens are represented by 10? Ans. Ten objects, but only one ten. How many objects and how many tens are repre 4. What does 1 ten equal? What figures represent tens? How made to represent tens? One ten, how written? Two tens, how written? Three tens, how written? Four tens, how written? Five tens, how written? Six tens? Seven tens? Eight tens? Nine tens? 5. What does each number express? How many times more objects than in table of units? What has increased the value? In what ratio does the cipher increase the value of figures? What is meant by tenfold ratio? What place is a figure said to occupy when one cipher stands on its right? What is the increased value a figure receives, as it changes its place, called? By sented by 20? Ans. Twenty objects and two tens. 30? by 40? by 50? by 60? by 70? by 80? by 90? Write on the slate 1 ten, (10,) 2 tens, 7 tens, 9 tens, 2 tens, 3 tens, 5 tens, 4 tens, 6 tens, 8 tens. § 7.-1. In the table of units we learn to express units only, and in the table of tens to express tens only. 2. We shall next investigate the manner of expressing numbers between ten and twenty, twenty and thirty, thirty and forty, forty and fifty, &c. 3. Figures when standing alone, that is, when there is no figure on their right, express units only, (§ 2;) when, however, they are made to occupy a second place towards the left, by placing ciphers on their right, they invariably represent tens. 4. Hence the first, or right-hand place, is called units' place, and the second place, or place next to the left, is called tens' place. 5. Obviously, therefore, to express any number composed of both units and tens, we unite in one expression the given units and tens, placing the units always on the right of the tens, that is, in the place occupied by the cipher in the table of tens. What number is produced by APPLICATION OF § 7. uniting 1 ten and 1 unit? Ans. 11. 1 ten and 2 units? Ans. 12. 1 ten and 3 units, &c. What number is produced by uniting 2 tens and 5 units? Ans. 25. 2 tens and 7 units? 2 tens and 8 units? 2 tens and 9 units? 3 tens and 3 units? 3 tens and 4 units? 3 tens and 7 units? 3 tens and 9 units? 4 tens and 7 units? 4 tens and 8 units? 4 tens and 1 unit? 4 tens and 5 units? 4 tens and 3 units? 4 tens and 9 units? 5 tens and 7 units? 5 tens and 6 units? 5 tens and 9 units? 5 tens and 7 units? 6 tens and 8 units? 6 tens and 5 units? 6 tens and 3 units? 7 tens and 4 units? 7 tens and 7 units? 7 tens 7. What do we learn to express in table of units? What, in table of tens? Figures standing alone express what? What when a cipher stands on their right? What is the first place on the right called? What is the second place on the left called? How do we express a number composed of both units and tens? and 5 units? 8 tens and 6 units? 8 tens and 9 units? 8 tens and 3 units? 9 tens and 9 units? 9 tens and 7 units? 9 tens and 3 units? 9 tens and 4 units? 9 tens and 1 unit? 9 tens and 0 units? Combine in one expression (use the slate) 3 tens, 7 units; 4 tens, 2 units; 1 ten, 9 units; 5 tens, 7 units; 6 units, 3 tens; 8 units, 7 tens; 3 units, 9 tens; 1 unit, 8 tens; 7 units, 3 tens; 1 unit, 6 tens. SERIES THIRD, Composed of hundreds. § 8. One hundred equals ten tens, or a hundred units. Two hundred equals twenty tens, or two hundred units. Three hundred equals thirty tens, or three hundred units, &c. § 9.-1. Each number in the above table represents ten times as many objects as the corresponding numbers in the table of tens, and a hundred times as many as in the table of units. This increase in value is produced by annexing another cipher. 2. The figures 1, 2, 3, 4, 5, &c., standing alone, express units or single objects; with one cipher on the right, they express tens; and with two ciphers, they express hundreds. 3. Hence, any figure, when it has two ciphers on its right, is said to occupy the place of hundreds. 8. One hundred equals how many tens?-how many units? Two hundred, how many? Three hundred, how many? 9. What does each number of series 3d represent? How is this increase produced? What do the figures 1, 2, 3, 4, &c., standing alone, express? What, with one cipher on the right? With two ciphers? What place does a figure, having two ciphers on the right, occupy? § 10. How many hundreds, how many tens, and how many units are represented by 100, (that is, by 1 and 00?) Ans. 1 hundred, 10 tens, or 100 units. How many of each by 200? How many of each by 300? by 400? by 500? by 600? &c. § 11.-1. Our next object will be to combine units, tens, and hundreds in one expression. By this combination we shall be able to express any number from one to nine hundred and ninety-nine, inclusive. 2. Each figure standing alone expresses units, (§ 2,) and with a cipher on its right, it expresses tens. 3. When, also, one significant figure is placed on the right of another, they together express a combination of units and tens. (§ 7.) 4. Each figure also expresses hundreds, when two ciphers are placed on its right. Hence, the first, or righthand place, is the place of units; then, advancing to the left, the second is the place of tens, and the third the place of hundreds. 5. To combine units, tens, and hundreds in one expression, we have then only to give each figure its appropriate place; viz. to place the ten's figure on the left of the unit figure, and the hundred's figure on the left of the tens. § 12. What number is produced by combining in one expression 1 hundred, 7 tens, and 6 units? 2 hundreds, 8 tens, and 5 units? 4 hundreds, 9 tens, and 7 units? 7 hundreds, 4 tens, and 9 units? 3 hundreds, 5 tens, and 4 units? 2 hundreds, 6 tens, and 4 units? 5 hundreds, 7 tens, and 9 units? 6 hundreds, 4 tens, and 8 units? 1 hundred, 9 tens, and 6 units? 4 units, 3 tens, and 7 hundreds? 3 tens, 7 hundreds, and 5 units? 6 units, 2 hundreds, and 5 tens? 4 tens, 9 hundreds, and 7 units? 4 tens, 11. What numbers can be expressed by a combination of units, tens, and hundreds? When does a figure express units? When tens? When do figures express units and tens combined? When does each figure express hundreds? What is the right-hand place? the second place? the third? How do we combine units, tens, and hundreds in one expression? 8 units, and 9 hundred ? 8 tens, 9 units, and 5 hundred ? 6 units, 5 hundred, and 7 tens? SERIES FOURTH, composed of thousands. § 13. One thousand equals ten hundred, a hundred tens, or a thousand units. Two thousand equals twenty hundred, two hundred tens, or two thousand units. Three thousand equals thirty hundred, three hundred tens, or three thousand units, &c. § 14.-1. The numbers presented in the former tables are here again increased tenfold, and this increase is effected in the same manner as before; viz., by placing a cipher on the right of each of those numbers. Each cipher placed on the right of any number removes each figure of that number one place further to the left, and by so removing it, increases its value in a tenfold ratio,—that is, makes its value ten times as great as before. 2. We hence perceive, that each and every figure has a local value, that is, a value depending upon the place it occupies, besides its own specific value, as shown in Table of Units, (2) also that by employing only ten characters, viz., 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, any number, however large, can be expressed. 15. What number is produced by combining in one expression 1 thousand, 8 hundred, 9 tens, and 3 units? Three 13. One thousand equals how many hundred? tens? units? Two thousand equals how many hundred? tens? units? thousand equals how many hundred? tens? units? how written? two, how written? three, how written? &c. One thousand 14. How are previous numbers here increased? how is this increase effected? What is the effect of each cipher? how is the value affected? What value has each figure depending on what? what other value? How many characters required to express all num bers? |