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NUMERATION AND NOTATION.
One, Two, Three, Four,
§ 1 A single thing is called
One and one more are called
&c. Each word, one, two, three, four, five,
six, &c., points out how many things are spoken of. These words are called NUMBERS. Hence, NUMBERS are the expressions for several things of the same kind.
Questions. What is a single thing called? One and one? Two and one? Three and one ? Four and one ? Five and one ? Six and one? Seven and one ? &c. What are Numbers ?
$ 2. The unit of a number is one of the equal things which the number expresses.
Thus, if the number be six apples, one apple is the unit; if it be five pounds of tea, one pound of tea is the unit; if it be ten feet of length, one foot is the unit; if it be four hours of time, one hour is the unit.
Q. What is the unit of a number? What is the unit of the number six apples? Of the number five pounds of tea ? Of the number ten feet? Of the number four hours ?
§ 3. ARITHMETIC treats of numbers. Numbers are expressed by certain characters, called figures. There are ten of these characters. They are
O which is called a cipher, or Naught,
Nine. Q. Of what does arithmetic treat ? How are numbers expressed ? How many figures are there ? Name them.
§ 4. The character 0 is used to denote the absence of a thing. As, if we wish to express by figures that there are no apples in a basket, we write, the number of apples in the basket is 0. The nine other figures are called. significant figures, or Digits. 1 expresses a single thing, or a unit of a number. 2
two things of the same kind, or two units. 3 three things
or three units. 4 four things
or four units. 5
or five units. 6 six things
or six units. 7 seven things
or seven units. 8 eight things
or eight units. 9 nine things
or nine units. Q. What does 0 express? What are the nine other figures called ? How many things does 1 express? How many things does 2 express? How many units in 3? In 4 ? In 5 ? In 6 In 7? In 8? 'In 9 ?
§ 5. If we wish to express the number ten, we have no separate character for it. We must combine the characters already known. This we do by writing 0 on the right hand of the 1; thus, 10, which is read ten.
This 10 is equal to ten of the units expressed by 1. It is, however, but a single ten, and in this sense may be regarded as a unit, the value of which is ten times greater
than the unit expressed by 1. It is called a unit of the second order.
Q. Have we a separate character for ten? How do we express ten? To how many units 1 is it equal? May we consider it a single unit? Of what order ?
§ 6. When two figures are written by the side of each other, the one on the right is called the place of units, the other, the place of tens, or units of the second order; and each unit of the second order is equal to ten units of the first order.
When units simply are named, units of the first order are always meant.
Two tens, or twenty, are written 20
60 Seven tens, or seventy,
70 Eight tens, or eighty,
80 Nine tens, or ninety,
90 The intermediate numbers between 10 and 20, between 20 and 30, &c., may be readily expressed by considering the tens and units of which they are composed. For example, the number twelve is made up of one ten and two units. It must therefore be written by setting 1 in the place of tens, and 2 in the place of units; thus, - 12 Eighteen has 1 ten and 8 units, and is written
18 Twenty-five has 2 tens and 5 units, and is written
25 Thirty-seven has 3 tens and 7 units, and is written
37 Fifty-four has 5 tens and 4 units, and is written
54 Eighty-nine has 8 tens and 9 units, and is written
89 Ninety-nine has 9 tens and 9 units, and is written
99 Hence, any number greater than nine and less than one hundred, may be expressed by two figures.
Q. When two figures are written by the side of each other, what is the place on the right called? The place on the left? When units simply are named, what units are meant? How many units of the second order in 20? In 30 ? In 40 ? In 50 ? In 60 In 70? In 80 ? In 90 ? Of what is the number 12 made up ? Also, 18, 25, 37,