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THE WERNER ARITHMETIC.
BOOK THREE-PART ONE.
NOTATION. 1. The expression of numbers by symbols is called notation.
2. In mathematics two sets of symbols are employed to represent numbers; namely, ten characters—1, 2, 3, 4, 5, 6, 7, 8, 9, 0-called figures, and the letters, a, b, c, d, x, y, 2.
NOTE.— The figures from 1 to 9 are called digits. The term significant figures is sometimes applied to the digits. The tenth character (0) is called a cipher, zero, or naught.
The Arabic Notation. 3. The method of representing numbers by figures and places is called the Arabic Notation. It is the principle of position in writing numbers that gives to the system its great value.
4. A figure standing alone or in the first place represents primary units, or units of the first order; a figure standing in the second place represents units of the second order; a figure standing in the third place represents units of the third order; a figure standing in the first decimal place represents units of the first decimal order, etc.
5. The following are the names of the units of eight orders: Fourth deciinal order
ten-thousandths. Third decimal order
thousandths. Second decimal order
hundredths. First decimal order.
tens. Third order
hundreds. Fourth order .
6. What are the names of the units represented by each figure in the following ? 3265.8419.
7. In a row of figures representing a number, the figure on the right represents the lowest order given; the figure on the left, the highest order given. In general, any figure represents an order of units higher than the figure on its right (if there be one), and lower than the figure on its left (if there be one).
8. Ten units of any order equal one unit of the next higher order ; thus, ten hundredths equal one tenth; ten tenths equal one primary unit, etc.
9. The naught, or zero, is used to mark vacant places; thus, the figures 205 represent 2 hundred, no tens, and 5 primary units.
Note 1.-Observe that a figure always stands for units. If it occupies the first place, it stands for primary units; if it occupies the second place, it stands for tens (that is, units of tens); the third place, for hundreds; the first decimal place, for tenths; the second decimal place, for hundredths, etc. Thus, a figure 5 always stands for fivefive primary units, five thousand, five hundredths, five tenths, according to the place it occupies.
Note 2.-In reading integral numbers the primary unit should be, and usually is, most prominent in consciousness. Thus, the number 275 is made up of 2 hundreds, 7 tens, and 5 primary units; but 2 hundreds equal two hundred (200) primary units, and seven tens equal seventy (70) primary units; these (200 + 70 + 5) we almost unconsciously combine in our thought, and that which is ordinarily present in consciousness is 275 primary units. So in the number 125, 246, there are units of six orders, which we reduce in thought to primary units and say, one hundred twenty-five thousand two hundred forty-six primary units.
NOTE 3.—In reading decimals, too, the primary unit should be prominent in consciousness. Thus, .256 is made up of 2 tenths, 5 hundredths, and 6 thousandths; but 2 tenths equal 200 thousandths, and 5 hundredths equal 50 thousandths; these (200 + 50 + 6) we combine in our thought, and that which should be present in consciousness is 256 thousandths of a primary unit.
10. EXERCISE. Write in figures :
1. Two hundred fifty-four thousand one hundred sixty.
2. One hundred seventy-five and two hundred six thousandths.
3. Eighty-four and three hundred twenty-five thousandths.
4. One hundred ninety-seven and twenty-seven hundredths.
5. Seven thousand four hundred twenty-four and six tenths.
6. Twenty-four thousand six hundred fifty-one. 7. One hundred thirty-five thousand two hundred fifty. (a) Find the sum of the seven numbers.
11. EXERCISE. Read in two ways as suggested in the following:
324.61, (1) 3 hundreds, 2 tens, 4 primary units, 6 tenths, 1 hundredth. (2) Three hundred twenty-four and sixty-one hundredths. 1. 2746.2
6. 2651.4 2. 546.85
7. 80.062 3. 24.006
8. 2085.7 4. 1.6285
9. 120.08 5. 286.35
10. 32.405 Use the word and in place of the decimal point only.
12. EXERCISE. Observe that any number may be read by giving the name of the units denoted by the right-hand figure, to the entire number : thus, 146 is 146 primary units; 21.8 is 218 tenths; 3.25 is 325 hundredths.
1. 27 = 2 tens + 7 primary units = 27 primary units. 2. 2.7 = 2 primary units + 7 tenths
tenths. 3. .27 = 2 tenths + 7 hundredths
hundredths. 4. .027 = 2 hundredths + 7 thousandths
thousandths. 5. .436 4 tenths + 3 hundredths + 6 thousandths
thousandths. 6. 5.247 5 primary units + 2 tenths + 4 hundredths + 7 thousandths = 5247 ths. 7. 3.24
hundredths. 8. 5.206
thousandths. 9. 25.13
hundredths. 10. 14.157
thousandths. 11. 275.4
tenths. 12. 275.12
NOTE.-Exercise 12 and Exercise 13 are important as a preparation for the clear understanding of division of decimals. Do not omit them nor permit the work to be done carelessly.
13. EXERCISE. Observe that any part of a number may be read by giving the name of the units denoted by the last figure of the part, to the entire part; thas, 24.65 is 246 tenths and 5 hundredths; 14.275 is 1427 hundredths and 5 thousandths. In a similar manner read each of the following: 1. 2.75
tenths and hundredths. 2. 32.46
tenths and hundredths. 3, 1.425
hundredths and thousandths. 4. 24.596
tenths and thousandths. 5. 321.45
tenths and hundredths. 6. 14.627
hundredths and thousandths. 7. 2.6548
hundredths and ten-thousandths. 8. 32.479
tenths and thousandths. 9. 21467
hundreds and primary units. 10. 16853
tens and primary units.
14. EXERCISE. Observe that in reading a mixed decimal in the usual way, we divide it into two parts and give the name of the units denoted by the last figure of each part to each part; thus, 2346.158 is read 2346 (primary units) and 158 thousandths.
Read the following in the usual manner. Do no not use the word and in reading the numbers in the second column: 1. 200.006 .206
6. 800 and 24. 824. 2. 400.0005 .0405 7. 9000 and 6. 9006. 3. 500.025 .525
8. 2400 and 8. 2408. 4. 200 and 40. 240. 9. 17000 and 4. 17004. 5. 700 and 35. 735, 10. 46500 and 40. 46540.
15. EXERCISE. Write in figures:
1. Two hundred and eight thousandths.