STANDARD ARITHMETIC. NOTATION AND NUMERATION. 1. A single thing is called a Unit. 2. A unit or a collection of units is a Number. A number answers the question "how many?" A number may be expressed by words or other characters, viz. figures and letters. 3. The method of expressing numbers by figures or letters is called Notation. The method of expressing numbers by figures is called the Arabic Notation, from the Arabs who first introduced it into Europe. The method of expressing numbers by letters is called the Roman Notation, because it was used by the ancient Romans. 4. The method of reading numbers expressed by figures or letters is called Numeration. THE ARABIC SYSTEM. 5. In counting a large number of objects, it is natural to arrange them in equal groups. When the number of the first groups becomes large they may be gathered into larger groups, and these again into larger groups, and so on. By general agreement the system of grouping by tens, called the decimal system, has been adopted. 6. The Arabic system of notation, which is a decimal system, employs ten figures to express numbers, viz.: 0 1 2 3 4 5 6 7 8 9 Naught One Two Three Four Five Six Seven Eight Nine Naught is also called zero and cipher. By combining these figures in accordance with certain principles, any number can be expressed. 7. PRINCIPLE. When figures are written side by side, the one at the right expresses units, the next tens, and the next hundreds. EXERCISES. 8. Tell what each figure in the following expresses: 9. Figures in units' place express units of the first order; those in tens' place, units of the second order; those in hundreds' place, units of the third order; etc. 10. The units of the second order, or tens, are named ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety. The suffix ty means ten. Thus forty means four tens. 11. The numbers between 1 ten and 2 tens are named eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. Thirteen means three and ten; fourteen, four and ten, etc. 12. The other numbers between 20 and 100 are read without the word and between the tens and the units. Thus, 35 is read thirty-five, not thirty and five. Three units of the second order, five of the first order. Five units of the second order, seven of the first order. Seven units of the first order, nine of the second order. Write all numbers below twenty. Write all numbers between twenty and forty. Write all numbers between fifty and seventy. 14. In reading numbers expressed by three figures, the tens are read after the hundreds, and the units after the tens, without the word and. Thus, 346 is read three hundred forty-six. Five units of the third order, two of the second, three of the first. Two units of the third order, two of the second, two of the first. Four units of the third order, four of the second, four of the first. 16. From the previous examples the following general principle is deduced: PRINCIPLE. The representative value of a figure is increased ten-fold by each removal one place to the left, and decreased ten-fold by each removal one place to the right. 17. In writing and reading numbers, the figures are separated into groups of three figures each, called periods. These periods contain the hundreds, tens, and units of each denomination. 18. The following table illustrates the system of notation: HUNDREDS N TENS 01 UNITS ✪ HUNDREDS TENS UNITS TENS O UNITS W HUNDREDS 0 HUNDREDS O) TENS UNITS QUADRILLIONS. TRILLIONS. BILLIONS. MILLIONS. THOUSANDS. UNITS. 25, 673, 210, 040, 385, 861 The number is read twenty-five quadrillion, six hundred seventy-three trillion, two hundred ten billion, forty million, three hundred eighty-five thousand, eight hundred sixty-one. 1. Each period, except the one at the left, must contain three figures. 2. The periods are separated from each other by commas. 3. In reading numbers the name of units' period is omitted. 4. The periods above quadrillions in their order are quintillions, sextillions, septillions, octillions, nonillions, decillions, etc. RULE FOR NUMERATION. ·Beginning at the right, sepa rate the numbers into periods of three figures each. Beginning at the left, read each period as if it stood alone, adding its name. EXERCISES. 19. Copy, point off into periods, and read: |