ARITHMETIC. (PART 1.) DEFINITIONS. 1. Arithmetic is the science of numbers and the art of computation. 2. A unit is one, or a single thing, as one boy, one horse, one, one dozen. 3. A number is a unit or a collection of units, as three apples, five boys, seven. 4. The unit of a number is one of the units included in the collection of units forming the number. Thus, the unit of twelve is one, of twenty dollars is one dollar. 5. A concrete number is a number applied to some particular kind of object or quantity, as three horses, five dollars, ten pounds. 6. An abstract number is a number not applied to any object or quantity, as three, five, ten. 7. Like numbers are numbers that express units of the same kind, as 6 days and 10 days, 2 feet and 5 feet. 8. Unlike numbers are numbers that express units of different kinds, as ten months and eight miles, seven dollars and five feet. For notice of copyright, see page immediately following the title page. 21 NOTATION AND NUMERATION. 9. Numbers are expressed in three ways: (1) by words; (2) by figures; (3) by letters. 10. Notation is the art of expressing numbers by figures or letters. 11. Numeration is the art of reading numbers expressed by figures or letters. ARABIC NOTATION. 12. The Arabic notation is the method of expressing numbers by figures. This method employs ten characters, called figures, to represent numbers, viz.: Figures 0 1 2 3 4 5 6 7 8 9 Names naught, one two three four five six seven eight nine cipher, or zero The first figure (0) is called naught, cipher, or zero, and, when standing alone, has no value. The other nine figures are called digits, and each one has a value of its own. An integer is any whole number. 13. Since there are only ten figures used in expressing numbers, each figure must have different values, determined by the way in which it is used. 14. The value of a figure depends upon its position in relation to others. 15. Figures have simple values, and local, or place, values. 16. The simple value of a figure is the value it expresses when standing alone. 17. The local, or place, value of a figure is its value as determined by its position in a number. Thus, 6 standing alone means six ones In the sixth place it denotes six hundred-thousands 6 60 600 6,000 60,000 600,000 6,000,000 18. The value of a figure increases tenfold with each remove to the left. 19. The cipher has no value in itself, but it is useful in fixing the place of other figures. To represent the number four hundred five, only two significant figures are necessary, one to denote four hundred, and the other to denote five; but if these two figures are placed together, as 45, the 4, being in the second place, will mean 4 tens. To denote 4 hundreds it should be in the third place. A cipher, therefore, must be inserted in the tens place to show that the number is composed of hundreds and units only, and that there are no tens. Four hundred five is therefore written 405. If the number were four thousand five, two ciphers would be inserted; thus, 4,005. If it were four hundred fifty, the cipher would be in the units place to show that there are no units, but only hundreds and tens; thus, 450. Four thousand fifty is written 4,050, the ciphers indicating that there are no hundreds and no units. 20. In reading numbers, it is usual to divide them by commas into groups of three figures each, called periods, beginning at the right. The first figure is said to belong to the first order, the second to the second order, etc. Each period contains three orders, named as shown in the following table: |