6. Abstract Numbers. Numbers standing alone, as 4, 7, 13, which mean 4 units, 7 units, 13 units, but do not specify the kind of objects counted or the kind of units of measure taken, are called abstract numbers. They signify simply the number of repetitions of some unit. 7. Concrete Numbers. Expressions that give the name of the objects counted or of the unit of measure employed, and the number of such objects, or of such units of measure, are called concrete numbers. Thus, 5 horses, 7 feet, 6 pounds, 5 days, are called concrete numbers. Such expressions consist of two parts, the number proper, and the kind of units taken, and should, strictly speaking, be called quantities. 8. Arithmetic treats of the simple properties of numbers, and the art of computing by numbers. NOTATION AND NUMERATION. 9. The first numbers have special names, as follows: one, two, three, four, five, six, seven, eight, nine, ten. 10. The first nine of these numbers are called Simple Units, or units of the first order. 11. The group of ten units has received the name of a Ten, or a unit of the second order; and we count by tens as by units; thus: one ten, two tens, three tens nine tens, ten tens. ... 12. The group of ten tens has received the name of a Hundred, or a unit of the third order; and we count by hundreds, as by tens and units; thus: one hundred, two hundreds ... ten hundreds, 13. A group of ten hundreds is called a Thousand, or a unit of the fourth order. 14. From ten units of the fourth order is formed a ten thousand, or a unit of the fifth order; and from ten units of the fifth order is formed a hundred thousand, or a unit of the sixth order. 15. Units of the seventh order are called Millions; of the eighth order, ten millions; of the ninth order, hundred millions. Finally, units of the tenth order are called Billions; units of the thirteenth order, Trillions; and so on. 16. The table of units of different orders is as follows: 17. The group of the first three orders is called the first class of units, and the group of the three following orders, the second class, and so on. 18. The unit of the second class is equal to a thousand units of the first class, and a unit of the third class is equal to a thousand units of the second class, and so on. 19. To read a number we decompose it into units of the different orders, and state how many groups there are of each kind, commencing with the highest order. Thus, for example, two millions, three thousands, five hundreds, seven tens, and four units. 20. It is clear that the names of all numbers up to a billion are formed by combining the names of the first nine numbers with the words ten, hundred, thousand, million. 21. Usage sanctions the following irregularities: I. Instead of saying two tens, three tens, four tens, five tens, six tens, seven tens, eight tens, nine tens, we say twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. II. The names of the numbers between ten and twenty are eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. 22. The names of the numbers between twenty and a hundred are: twenty-one, twenty-two, twenty-three... twenty-nine, thirty-one, thirty-two, thirty-three... thirty-nine, ... ... ... ninety-one, ninety-two, ninety-three ... ninety-nine. 23. The names of the numbers between a hundred and a thousand are: hundred one, hundred two ... hundred ninety-nine, nine hundred one ... nine hundred ninety-nine. 24. The common system of notation employs ten figures or digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine of these figures represent the first nine numbers; the last, which is called Zero, Naught, or Cipher, is used to denote the absence of units of the order in which it stands. It is possible to express all numbers by these ten digits by making the value of each figure increase tenfold for every place that it is moved to the left. 25. If we have given a number written in figures, the position of each figure counting from the right indicates the order of units that the figure represents. If we divide the number into periods of three figures each, the first period on the right will be the period of simple units, the second period will be the period of thousands, the third will be the period of millions, and so on. In each period the first figure on the right expresses the units of that class, the second figure the tens, and the third the hundreds. Thus: Thus, the number 21,334,334 means and is read 21 millions, 334 thousands, 334 units. If the number is applied to dollars, it means and is read 21 million, 334 thousand, 334 dollars. The next period is the billions' period. NOTE. The fundamental principle of forming and expressing numbers should be illustrated by making little bundles of wooden toothpicks, ten in each bundle, and then making bundles of hundreds by taking for each hundred ten bundles of ten each. When the pupil has become familiar with forming and expressing numbers consisting of hundreds, tens, and units, he should be shown that the method of forming and expressing numbers of hundreds, tens, and units of thousands is precisely the same, the only difference being that the unit of this period is not a single toothpick, but a pile of ten bundles of a hundred each, which is a thousand. 26. To write a number in figures we write successively the number of units of each order from left to right, beginning at the highest order and taking care to supply by zeros orders of units that may be lacking. 27. To read a number written in figures we divide the number into periods of three figures each from right to left: this done, we begin to read at the left-hand period and read as if the figures of that period stood alone, adding the name of the period; then the next period to the right is read with the name of that period, and so on. 28. The number 1256 may be read one thousand two hundred fifty-six, or it may be read twelve hundred fifty-six. The number 5004 may be read five thousand four, or it may be read fifty hundred four. The shortest method is the best method of reading any number. Twelve hundred fiftysix is shorter than one thousand two hundred fifty-six; five thousand four is shorter than fifty hundred four. 29. It will be seen that the value of each figure, in any number expressed in figures, depends on two things: First, the value attached to the figure without regard to its position. And, secondly, the value it acquires from the place it holds in the number. The value of a figure, without regard to its position, is called its absolute value; and the value it acquires by its position is called its local value. 30. The art of expressing numbers by means of figures is called Notation, and the art of expressing in words a number written in figures is called Numeration. 31. The unit of money is the dollar. Instead of writing 'the word dollars, this mark $ is used, which is called the |