« ΠροηγούμενηΣυνέχεια »
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.
ORDER OF PROCEDURE IN TRAINING FOR ARITH:
STEP I. Since arithmetic is concerned mainly with the comparison of measured magnitudes and their numerical expression, it is of prime importance that the work should begin in the actual comparison and measurement of sentient objects.
STEP II. Since by far the greater part of the actual work in arithmetic must be done without the presence of the sense magnitudes compared, it is equally important that the pupil should early learn to image measured magnitude and to compare the images of measured magnitudes.
NOTE:--The imaging of magnitude, as the words are here used, means the 'mental reproduction of that which has been in the mind during an act of sense perception with special attention to quantity.
STEP III. Since most of the magnitudes compared by the mathematician never have been to him objects of sense perception, and some of them never can be, it is no less important that the pupil should be constantly trained in the creation of imaginary measured magnitudes.
Note 1.-A most serious defect in ordinary arithmetical training is, that pupils image figures instead of magnitudes - that they are expected to see relation when the magnitudes compared are not present in consciousness.
NOTE 2.—The multiplication of sense-experiences in the comparison of magnitudes will not alone give the desired power. To learn to work with ideal magnitudes, one must work with ideal magnitudes. 'Object teaching” must be vigorously begun and as vigorously laid aside.
NOTE 3.- In the lower grades, great emphasis must be put upon the actual measurement of sense-magnitudes. In the higher grades, the chief part of the work will be in the comparison of imaginary magnitudes. But there is no grade from which Step I. can be entirely omitted, and no grade (in which formal number work should be taught) where Step I. should occupy as much time as Step II., or where Step II. should occupy as much time as Step III.
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 decimal 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 decima! 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 thou sandths.
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.