four hundreds, and that of the 6 is six tens, or sixty. The 5 has its simple value, five. ART. 20. When the nine digits occupy the second, or ten's place, each will then express the same number of tens that it did units, when occupying the place of units. When they occupy the third, or hundreds' place, each will then express as many hundreds as it did units when in the place of units. NOTE. This will be rendered plain by inspecting the following NOTE. The terms thirteen, fourteen, fifteen, sixteen, &c., are obviously contractions of three and ten, four and ten, five and ten, six and ten, &c. In a similar way, by contracting the expressions two tens, three tens, four tens, &c., the expressions twenty, thirty, forty, &c., are derived. Twenty-one, Twenty-two, Twenty-three, Twenty-four, Twenty-five, Twenty-six, Twenty-seven, Twenty-eight, and Twenty-nine, are respectively expressed by .placing in regular order the digits in the place of the cipher in the number 20. In a similar manner, numbers from Thirty to Forty, from Forty to Fifty, from Fifty to Sixty, &c, are expressed by placing the digits in the place of the cipher in the numbers Thirty, Forty, Fifty, Sixty, &c. The terms twenty-one, twenty-two, &c., are compounded of twenty and one, twenty and two, &c. Other numbers expressed by two figures are similarly formed. ART. 21. By inspecting the above table, it will be observed, that a figure standing in the second place, or place of tens, is ten times as great as though it were in the first or units' place. A figure that stands in the third place, or place of hundreds, is ten times as great as though it were in tens' place, and one hundred times as great as though it were in the place of units. Hence, ten units make one ten, and ten tens make one hundred. We therefore infer universally, that ART. 22. Figures increase in value from right to left in a ten-fold ratio; that is, each removal of a figure one place towards the left increases its value ten times. ART. 23. As figures in the Arabic system of notation increase in ten-fold ratio from right to left, and decrease in the same ratio in an opposite direction, it is called the Decimal system of notation. The word decimal is derived from the Latin, lecem, ten. ART. 24. NUMERATION is the art of reading numbers, expressed by figures. NOTE.-By carefully studying the following Table, the pupil will soon be able to read any number which requires not more than nine figures to ex press it. TABLE. Hundreds of Millions, or units of the 9in order. Hundreds of Thousands, or units of the 6th order. Tens, or units of the 2d order. Units, or units of the 1st order. 4 4 5 4 5 3 4 5 3 2 4 5 8 2 6 4 5 8 2.67 4 5 3 2 678 Figures occupying the place of units, are sometimes called units of the first order, -those occupying the place of tens, units of the second order,-those accupying the place of hundreds, units of the third order, &c., as shown by the Table. To read these numbers, first denominate each figure by the names units, tens, &c., as shown by the table, and then read from left to right as follows: Four. : Four hundred and fifty-three. Four thousand five hundred and thirty-two Forty-five thousand three hundred and twenty-six. Four hundred fifty-three thousand two hundred and sixty-seven. Four millions, five hundred thirty-two thousand, six hundred and seventy-eight. 4 5 3 2 6 7 8 1 Forty-five millions, three hundred twentysix thousand, seven hundred and eighty-one Four hundred fifty-three millions, two hun4 5 3 2 6 7 8 1 9 dred sixty-seven thousand, eight hundred and nineteen. REMARK. It would be well to write the figures of this Table on the black board, and have the pupils read them individually as well as collectively. This Table shows plainly the simple and local values of figures. Each figure, except those in the place of units, has a local value, which may be named by the pupil as the teacher points to them separately. ART. 25. In the United States and continental Europe the French method of numeration is in general use. In this method of numeration a different name is given to every three figuers, counting from the right. The first period contains units, tens of units, hundreds of units, and is therefore called the period of Units. For a similar reason the next left hand period is called the eriod of Thousands, &c. ART. 26. In the following Table the words above the row of figures express the particular denomination of the figures over which they are placed. To read a number expressed by figures :— Denominate each figure from right to left, remembering the name of each period, then read the figures of each period, beginning at the left hand, in the same manner as those of the period of units are read, and at the end of each period give its name. FRENCH METHOD OF NUMERATION. "Hunds. of Decillions. -Tens of Decillions. Hunds. of Nonillions. Hunds. of Octillions. Tens of Octillions. . Hunds. of Septillions Hunds. of Quintillions. Hunds. of Sextillions. Hunds. of Quadrillons. Hunds. of Trillions. Hunds. of Billions. Hunds. of Millions. Hunds. of Thousands. Tens of Thousands. Hundreds. Tens Period Period Period Period Period Period Period Period Period Period Period Perica of De- of No- of Oc of Sep- of Sex- of Quin- of Quad- of Tril eillions. nillions. tillions. tillions. tillions. tillions. rillions. lions. of Bil- of Mil- of Thou- of Units, EXERCISES IN NUMERATION. Read the following numbers : EXERCISES IN NOTATION. ART. 27. To express numbers by figures. Begin at the left, and write the figures of the highest order mentioned, observing to place in each order, the figures belonging to it, and when no digit is mentioned, to fill the place with a cipher. Express the following numbers by figures :1. Forty-three. 2. Eighty-nine. 3. Three hundred and eight. 4. Four thousand, one hundred and four. 5. Seventy-five thousand and seventy-five. 6. Six hundred and five thousand, one hundred and twenty-three. 7. Eight hundred and seventy-two thousand, five hundred and twelve. 8. Nine millions, seven hundred and sixty-five thousand, four hundred and thirty-two. 9. Three hundred and forty millions, forty-three thou sand, five hundred and sixty-seven. 10. Three hundred and seventy-four billions, four hundred and thirty-eight millions, eight hundred and sixty-two thousand, eight hundred and forty-seven. FUNDAMENTAL RULES OF ARITHMETIC. ART. 28. Notation and Numeration are the Primary principles of the four Fundamental Rules of arithmetic; namely, ADDITION, SUBTRACTION, MULTIPLICATION, and DIVISION. These are called Fundamental Rules, because all other arithmetical operations are dependent on them. A Rule, in Arithmetic, is a prescribed method of performing an Arithmetical operation. |