PRACTICAL ARITHMETIC. NOTATION AND NUMERATION. 1. PRELIMINARY DEFINITIONS. A number is one or a collection of ones of the same kind. 2. The unit of any given number is one of the collection which constitutes the number. The unit of 16 oranges is 1 orange; of 16 dozen oranges is 1 dozen oranges; of 16 centuries is 1 century; of 16 thousand bricks is 1 thousand bricks; of 16 tens is 1 ten; of 16 thirds is 1 third. The unit of 16 is 1. 3. "How many?" "What?"- A number answers the question, "How many?" This is sometimes given. as the definition of number. But our conception of a number is incomplete until another question is answered, namely, "What?" or "What kind?" "Five" has only half its meaning till we know of what kind the units are boys, dozens, hundreds, tenths, or what not. Five sevenths is not a number that can be used in solving a problem till we know what it is five sevenths of"John's money," the sheep in the pasture,” etc. - 66 So it is of the greatest importance that we get into the habit of answering the question, "What?" or "Of what kind?" What is seventeen? It may be seventeen men, tenths, hundreds; or it may be one ten and seven ones of any one of these kinds, if we choose so to regard it. One hundred twenty-three may be thought of as so many ones, or as twelve tens and three ones; that is, it may be thought of as one number or two numbers. What is four and two sixths? This can not be thought of as one number; it is four units of one kind joined with two units of another kind. 4. Primary and Relative Units. Every integral number, if expressed by more than one figure, and every fractional number of any kind, will include the thought of more than one kind of unit. In every such case the units are of two classes. For convenience we call them primary and relative units. The primary unit always has two distinguishing marks: first, when the number is applied to objects is made concrete the primary unit takes the name; second, when the number is expressed by figures the primary unit is always expressed in the first place at the left of the decimal point. Any relative unit is some exact part of its primary unit, or it is some number of times that primary unit. ILLUSTRATIONS: In the number one hundred twenty-three, there are three primary units, two relative units of one kind, and one of another kind. If the number were applied (made concrete), the name apple, for instance - would be applied to the primary units only. In a fractional number the primary unit may or may not be written. In such examples as four and two sixths, or five and three tenths, the primary unit is written if the expression be made in figures. But there would be no figures to express primary units if two sixths or three tenths were written alone in figures. 5. Concrete and Abstract Numbers.- When the objects to which numbers belong are named, the number is said to be concrete; as 2 apples, 27 cents, 368 miles. When no objects are named, the number is abstract; as 2, 27, 368. 6. Simple and Compound Numbers.- When the relation between the primary and the relative units of a number is invariably expressed by ten, or a power of ten, the number is Simple; if any other number expresses that relation the given number is Compound. Hence, all the numbers in Federal Money, and in the Metric system, are simple numbers. Hence, also, a socalled mixed number, like four and one seventh, is just as truly a compound number as numbers expressing shillings and pence, or bushels and pecks. 7. What is Arithmetic? Arithmetic teaches the theory of numbers and the art of computing by them. 8. Notation is the writing of numbers by means of characters. There are two systems of notation in general use, the Roman and the Arabic. 9. Numeration is the reading of numbers expressed by characters. ROMAN NOTATION. 10. The Romans employed seven letters to express numbers. They are: 11. Expressing Other Numbers.- Other numbers are expressed by putting two or more of these letters together, according to the following principles: I. When a letter is written after another letter of the same or greater value, they express together the sum of their values. Thus, II=two; XV-fifteen; XXX=thirty; CX one hundred ten. = II. When a letter is written before another letter of greater value, the two together express the difference of their values. Thus, IV four; IX=nine; XL=forty; XC=ninety. III. A bar placed over a letter multiplies its value by one thousand. Thus, V-five thousand; L-fifty thousand; M=one million.⚫ EXERCISES. 12. Read the following expressions: III, VII, XIII, XIV, XVI, XIX, XX, XXV, XXVIII, XXXII, XXXIX, XL, XLIV, XLVI, L, LV, LXIII, LXVII, LXXI, LXXVIII, LXXX, LXXXIV, LXXXIX, XC, XCV, XCVII, C, CCC, D, DC, DCCCC, M, MM, MX, MC, MD, X, C, MDCCCXCVI. 13. Express the following numbers by the Roman notation: Twelve, fifteen, twenty-four, thirty-seven, forty-five, fifty-nine, sixty-three, seventy-six, eighty-eight, ninety-one, one hundred, one hundred nine, one hundred twenty-five, one hundred forty, two hundred seventyfour, four hundred fifty, five hundred ninety-nine, one thousand one hundred ten. Roman numerals are generally used to number lessons, chapters, and volumes. They are used on the dials of time-pieces and in the titles of kings. |