How many single books make the single bundle? How many units of the first order in one unit of the second order? If we make ten bundles with ten books in each bundle and then place the ten bundles together, making one large bundle, the large bundle will also be one thing-a unit. How many small bundles in the large bundle? This last mentioned unit is a unit of the third order. How many units of the second order make one unit of the third order? Write the figure one, and at the left of it write another figure one. The first one is a unit of the first order. The second one is a unit of the second order. How many second order? units of the first order make one unit of the Write another figure one at the left of the second. This last one expresses a unit of the third order. How many units of the second order make one unit of the third order? Write another figure one at the left of the last. This is a unit of the fourth order. How many units of the third order make one unit of the fourth order? Now write, without assistance, a unit of the fifth order and a unit of the sixth order. Write with one figure two units of the first order; two of the second; two of the third; two of the fourth; two of the fifth; and two of the sixth. NOTE.-Always remember that one order of figures occupies but one place, and the largest number of any one order is nine. Write in figures, as one number, three units of the first order, four units of the second order, two units of the third order, five units of the fourth order, one unit of the fifth order, and nine units of the sixth order. These numbers are integers. 12. An Integer is a whole number. 13. Units of the First Order either stand alone, or occupy the right-hand place. 14. Units of the Second Order occupy the second place from the right. 15. Units of the Third Order occupy the third place. You may now tell what place units of the fourth order occupy; units of the fifth order, sixth order, seventh order, eighth order, ninth order, tenth order. Units of the second order may be expressed without a unit of the first order, by putting a cipher in the place of the unit of the first order. Thus, 10. The orders of units are indicated by the relative positions of the figures. Units of any order may be written without expressing the units of other orders by putting ciphers in the place of the other units. Thus, two units of the third order are written 200; two units of the first order and four units of the fifth order are written thus, 40002, ciphers taking the place of the absent units. Write two tens. What number have you written? Write three tens, and at the right of it two units. What number have you now? Write three units of the fourth order, and in the same line one unit of the third order, five units of the second order, and nine units of the first. What number do they express? Units of the First Order express single things, and are called simply units. Units of the Second Order express collections of ten single things, and are therefore called tens. Units of the Third Order express collections of ten tens, or one hundred, and are therefore called hundreds. Units of the Fourth Order express collections of ten hundreds or one thousand, and are therefore called thousands, as shown in the following NOTE TO TEACHERS.-Minds are not all formed in the same mould. The teacher will find an unlimited variety, and must be ready to vary his methods of teaching accordingly. SIMPLE AND LOCAL VALUES. Another method of presenting this subject is the following. It is inserted to meet the wants of those who fail to comprehend the first method. 16. Figures have two values, called a Simple Value and a Local Value. The Simple Value of a Figure is its value when standing alone, or when used as the right hand figure of a number. The Local Value of a Figure is its value arising from the place in which it stands. When 2 stands alone or at the right hand, it denotes 2 units; when it stands in the second place from the right, it denotes 2 tens, as in 24; when it stands in the third place from the right, as in 234, it denotes 2 hundreds. The local values of figures increase from the right to the left by a scale of tens. TEST QUESTIONS, What do units of the first order express? Units of the second order? Third? Fourth? How many values have figures? What is the simple value of a figure? What is the local value? Write 2 so as to show its simple value. Write 2 so as to denote 2 tens ; to denote 2 hundreds. How do the local values increase? Places of figures and orders of units are counted from right to left, but numbers are read from left to right. EXAMPLES IN NOTATION AND NUMERATION. 17. 1. Numbers from one to nine inclusive are collections of simple units, and are expressed by a single figure. 2. Numbers from ten to ninety-nine inclusive are composed of tens and units. Thus, twenty-seven is composed of 2 tens and 7 units; forty-eight is composed of 4 tens and 8 units. Write the following numbers by means of figures: 3. Numbers from one hundred to nine hundred and ninety-nine inclusive are composed of hundreds, tens, and units. Thus, the number four hundred and sixty-five is composed of 4 hundreds, 6 tens, and 5 units; two hundred and three is composed of 2 hundreds, 0 tens, and 3 units. EXAMPLES. Write the following numbers by means of figures: 1. Two hundred and sixty-five. 2. Three hundred and ninety. 8. One hundred and ninety-nine. 9. Five hundred and seventy-three. 10. Six hundred and sixty-six. 4. To read a number of three figures, we name the number of hundreds and then read the tens and units, as though they were by themselves. Thus, 512 is read, five hundred and twelve; 874 is read, eight hundred and seventy-four; 209 is read, two hundred and nine. Read the following numbers: |