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NUMERATION AND NOTATION.
$1 A single thing is 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 five things
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 ?
§ 3. 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 limes 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 unils of the second order; and each unil
. 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
40 Five tens, or fifty,
50 Six tens, or sixty,
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
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 ihe right called? The place on the leti? When units simply are named. what units are meant ? How many units of the fecond 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, We are,
54, 89, 99? What numbers may be expressed by a single figure ? What numbers may be expressed by two figures ?
§ 7. In order to express one hundred, or ten units of the second order, we have to form a new combination. It is done thus,
100 by writing two ciphers on the right of 1. This number is read, one hundred. Now this one hundred expresses 10 units of the second order, or one hundred units of the first order. But the one hundred is but an individual hundred, and in this light may be regarded as a unit of the third order.
We can now express any number less than one thousand.
For example, in the number three hundred and seventyfive, there are 3 hundreds, 7 tens, and 5 units therefore, to express 3 units of the 3d order, 7 units of the second order, and 5 of the 1st.
Hence, we write and we read from the right, units, tens, hundreds.
In the number eight hundred and ninety-nine, there are 8 units of the 3d order, and 9 of the 2d, and 9 of the 1st.
It is written and read, units, tens, hundreds.
In the number four hundred and six, there are 4 units of the 3d order, 0 of the 2d, and 6 of the Ist. It is written
406 and in a similar manner we may express, by three figures, any number greater than ninety-nine and less than one thousand.
Q. How do you express one hundred? To how many units of the 2d order is it equal? To how many of the 1st order? May it be considered a single unit? Of what order is it? How many units of the 3d order in 200? In 300 ? In 400? In 500? In 600 ? In 700 ? In 800? In 900 ? Of what is the number 375 composed ? The number 406? What numbers may be expressed by three figures ?
§ 8. To express ten units of the 3d order, or one thousand, we form a new combination by writing three ciphers on the right of 1; thus,
Now, although this thousand expresses one thousand units of the 1st order, it is, nevertheless, but one single thousand, and may be regarded as a unit of the 4th order.
Proceeding in this way, we may place as many figures in a row as we please. When so placed, we conclude:
1st. That the same figure has different values according to the place which it occupies.
2d. That counting from the right hand towards the left, the first is the place of units; the second, the place of tens; the third, the place of hundreds; the fourth, the place of thousands ; &c.
3d. That ten unils of the first place are equal to one unit of the second place; that ten unils of the second place are
; equal to one unit of the third place; that ten units of the third place are equal to one unit of the fourth place; and so on, for places farther to the left.
Q. To what are ten units of the 3d order equal? How do you express them? May this be considered a single unit ? Of what order ? May any number of figures be written in a row? When so placed has the same figures different values ? On what does the value of the same figure depend? What is the first place on the right called? What is the second called? What is the third called ? What is the fourth called? What are ten units of the first place equal to? What are ten units of the second place equal to? To what are ten units of the third place equal.
§ 9. Expressing or writing numbers by figures, is called NOTATION. Reading the order of their places, correctly, when written, is called NUMERATION.
Q. What is Notation? What is Numeration? Which way do you numerate ? 1. Write three tens.
Ans. 30. 2. Write one hundred and fifty.
Ans. 3. Write twelve tens.
Ans. 120. 4. Write 4 units of the first order, 5 of the 2d, 6 of the 3d, and 8 of the 4th.
Ans. 5. Write 9 units of the 5th order, none of the 4th, 8 of the 3d, 7 of the 2d, and 6 of the 1st.
Ans. 90876. 6. Write 1 unit of the 6th order, 5 of the 5th, 4 of the 4th, 9 of the 3d, 7 of the 2d, and none of the 1st.