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I. Given, two measured magnitudes, to find their relation.
II. Given, one measured magnitude and its relation to another magnitude, to find the measurement of the second magnitude.
Relation may be (1) by difference or (2) by quotient.
4. Every branch of mathematics is concerned with number and its expression; but it is the special province of arithmetic (1) to present the common method of numerical notation, (2) to investigate the properties and laws of number, and (3) to teach the art of computation in the simple processes of reduction, addition, subtraction, multiplication, and division.
NOTE.—For the double purpose of giving practice in simple com. putations and of preparing pupils for the ordinary number work of life, applications of these processes to certain classes of business transactions are presented in all arithmetical text-books.
5. It is the province of Algebra, by means of literal and figure notation combined, (1) to abridge and (2) to generalize the various processes of computation.
6. A unit is one.
7. A unit of measurement is a standard of measurement; as, a foot, a quart, an hour, a dollar, etc.
NOTE.—Counting is measuring, though sometimes inexact, owing to lack of uniformity in the magnitudes counted.
8. A unit of number is one of number.
Note 1.-A unit of number is, in a sense, a unit of measurement. It is a standard for measuring number.
NOTE 2.—The unit of number may be simply, one or one ten, or one hundred, or one tenth, or one thousandth; thus, we may have units of the first order, units of the second order, units of the first decimal order, etc. We may have, too, fractional units other than decimal; as, one fourth, one fifth, one twenty-first, etc. A unit is a one, or any
group regarded for the time as one. Any unit, however great, may become a part of a major unit, and any unit, however small, may be regarded as made up of minor units.
NOTE 3.-It is sometimes said that a unit is a standard by which we couat of measure, and that a number is a unit or a collection of units. If this be true, sir boys, six marbles, six books, etc., are numbers !
9. A number that is joined to the name of a standard of measurement is said to be a concrete number.
NOTE.— Neither four feet, six hours, por eight apples is a concrete number; the four, six, and eight of these expressions are concrete numbers. If eight apples is a concrete number, then a red apple is a concrete color!
10. A number that is not joined to the name of a standard of measurement, as four, five, eight, ten, is said to be an abstract number.
Note 1.-An abstract number may suggest simply a ratio, or it may suggest both a ratio and a measured magnitude. In the expression, 8 feet 2 feet = 4, the four suggests pure number, i.e., simply a ratio -the ratio of the magnitude 8 feet to the magnitude 2 feet. The expression 6+5=11, may suggest that 6 inches increased by inches is equal to 11 inches. One may also think the 6 or the 5 or the 11 as the ratio of a certain magnitude to the standard of measurement.
NOTE 2.-Strictly speaking, number is ratio and, from its very nature, is always abstract. But the expressions, “concrete number," “ denominate number," etc., have been accorded a place in the language. They are convenient terms to express certain uses of abstract number and hence are freely employed on the pages of this book.
ORDER OF PROCEDURE IN TRAINING FOR ARITH
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.
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.
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.
THE WERNER ARITHMETIC.
BOOK THREE-PART ONE.
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, z.
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:
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.