ARITHMETIC. ARITHMETIC is the science of numbers, and the art of computation. Arithmetic is a science, so far as it explains and illustrates the principles and properties of numbers; it is an art, so far as it furnishes rules for the solution of numerical problems, or for calculation. Art. 1. Number is either a unit, or a collection of units. Any single whole thing is called a unit, or one. One and one are called two. Two and one are called three. Three and one are called four. Four and one are called five. Five and one are called six. Six and one are called seven. Seven and one are called eight. Eight and one are called nine. Nine and one are called ten. Thus, each successive number may be formed by the continual addition of a unit. Art. 2. Numbers are the expressions and measures of quantity. Quantity signifies as much of anything as there is, as a quantity of cloth; or as many things as there are, as a quantity of pens. In measuring quantities of magnitude, some known quantity, as a yard, is taken for the unit of measure. In measuring quantities of multitude, one of the things which compose the quantity is its unit of measure; and the measure of any quantity is the number of times it contains its unit of measure. Art. 3. When the unit indicates the name of anything, as 1 dollar, it is called a denominate unit; and numbers composed of denominate units are called denominate numbers; as 5 dollars, 10 pounds. When the unit expresses no idea except number, as 1, it is called an abstract unit; and numbers composed of abstract units are called abstract numbers; as 5, 10. NOTATION AND NUMERATION. Art. 4. NOTATION is the writing of numbers. Ten characters, or figures, are used in writing numbers. These figures are 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, O zero, or cipher. Each of these figures, except the cipher, always expresses the same simple number; but in writing numbers greater than nine, the law or scale of notation assigns also a local number to each figure, which depends upon its place. In the number 111, the 1 in the first place on the right expresses only its simple number one, and it is called a unit of the first order; - the 1 in the second place expresses ten times one, or ten, and it is called a unit of the second order; and the 1 in the third place expresses ten times ten, or one hundred, and it is called a unit of the third order. If we use the 0, which does not express a number of itself, we can express each of the above numbers singly; thus, 100, 10, 1; all of which are equal to 111. The established law or scale of notation is, that each successive removal of the same figure one place to the left increases or repeats its number ten times. Art. 5. The units of the second order are written thus: 10, 20, 30, 40, 50, 60, 70, 80, 90. The units of the third order are written thus: 100, 200, 300, 400, 500, 600, 700, 800, 900. The numbers between 10 and 20 are written thus: 11, 12, 13, 14, 15, 16, 17, 18, 19. The numbers between 20 and 30 are written thus: 21, 22, 23, 24, 25, 26, 27, 28, 29. The numbers between all other tens are written in a similar manner. The units of the fourth order are written thus: 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000. The numbers between the thousands are expressed by their appropriate figures. The higher orders of units may be written in a similar manner, to an unlimited extent. Art. 6. NUMERATION is the reading of numbers. The following table exhibits twenty-four places, or orders, in the scale of numbers, divided into eight periods of three figures each, by points. It also exhibits the name of each order and period. NOTATION AND NUMERATION TABLE. Billions. 8th period. 7th period. 6th period. 5th period. 4th period. 3d period. 2d period. 1st period. Sexullions. Quintillions. Quadrillions. Trillions. Millions. Thousands. Units. 5 5.5 5 5. 5 5 5.5 5 5.5 5 5.5 5 5.5 5 5.5 5 5.5 24th order, Hunds. of Sextillions. 23d order, Tens of Sextillions. 22d order, Units of Sextillions. 21st order, Hunds. of Quintillions. 20th order, Tens of Quintillions. 17th order, Tens of Quadrillions. 15th order, Hunds. of Trillions. 14th order, Tens of Trillions. 12th order, Hunds. of Billions. 11th order, Tens of Billions. 9th order, Hunds. of Millions. 6th order, Hunds. of Thousands. 2d order, Tens. 3d order, Hundreds. 4th order, Units of Thousands. 1st order, Units. The division of numbers into periods of three figures each, enables us to read large numbers as easily as we can read a number containing only three figures. The figures of each period are to be read in the same manner as we read the figures of the period of units. Beginning at the left, the several periods in the table are read, in succession, as follows: 555 sextillion, 555 quintillion, 555 quadrillion, 555 trillion, 555 billion, 555 million, 555 thousand, 555. Art. 7. From the arrangement and illustration of the preceding table, we have the following rule for reading numbers: RULE. Divide the given numbers into periods of three figures each, by points, counting from the right. Then begin at the left, and read the number of hundreds, tens, and units of each period, in succession, and conclude each period, except the period of units, with its name. Learners should be required to read the following numbers : 234178 87654321 987654321 1234567891 9080706050 8707605400 4050634031 Art. 8. In teaching young children to write numbers, dictate a small number, and require them to name the number of figures necessary to write the number; also, to name each figure, in succession, from left to right. Continue this exercise, until they can readily name the required number of figures, also each figure in succession. In teaching children to write large numbers on the slate or blackboard, first require them to name the number of figures necessary to write the proposed number, then the number of periods, and, lastly, the figures of each period, in succession, requiring them to write each figure at the time of naming it. ILLUSTRATION. Let it be required to write eight hundred and seventy-five trillion, fifty billion, forty-five million, five hundred and forty. Hundreds of trillions is the highest order named in this number, therefore fifteen figures, or five periods of figures, will be necessary to write it. Beginning at the left, with the highest order and period, we write 8 in the place of hundreds, 7 in the place of tens, and 5 in the place of units. The orders of hundreds and units of billions being omitted, we write a 0 in the place of hundreds, 5 in the place of tens, and a 0 in the place of units. The order of hundreds of millions being omitted, we write a 0 in the place of hundreds, 4 in the place of tens, and 5 in the place of units. The three orders of thousands being omitted, we write a 0 in the place of each of the orders, thus, 000. In the period of units, we write 5 in the place of hundreds, 4 in the place of tens, and a 0 in the place of units. Arranging all the periods and figures in their proper order, the number is written thus: 875.050.045.000.540. From the preceding remarks and illustration, we derive the following rule for writing numbers: RULE. First, ascertain the number of figures necessary to write the proposed number; also, the number of periods. Then begin at the left, and write a figure expressing the number of hundreds, tens, and units, of each period, in succession, observing to write a cipher in the place of each omitted order. Art. 9. All questions throughout this work which are arranged in double columns, are to be performed both mentally and orally, unless otherwise directed. 1. How many and what figures 2. How many and what figures are necessary to write seventy- are necessary to write ninetyfive? nine? 3. One hundred and twentyone? 5. Three hundred and fiftyone? 7. Five hundred and thirtytwo? 9. Seven hundred and seventythree? 4. Two hundred and eighty one? 11. Nine hundred and ninetynine? 13. Three thousand four hundred and ten? 15. Five thousand and seventyfive? 17. Seven thousand and seventy-five? 19. Nine thousand and ninetyseven? 21. Fifteen thousand and twenty-one? 23. Twenty-five thousand and thirty-five? 20. Ten thousand and nine hundred? 22. Twenty thousand and one hundred? 24. Thirty thousand and fortyseven? 25. Forty-five thousand, seven hundred and fifty-nine. 26. Seventy-nine thousand, nine hundred and ninety-nine. 27. Six hundred and ten thousand, five hundred and forty seven. 28. Seven hundred and six thousand, four hundred and thirty-two. 29. Nine hundred and ninety-nine thousand, nine hundred and ninety-nine. 30. Two million, one hundred and fifty thousand, three hundred and twenty-one. 31. Thirty-five million, twenty-four thousand, one hundred and fifty-nine. 32. Seventy-six million, five hundred and forty-three thousand, three hundred and twenty-one. 33. Ninety-nine million, eight hundred and seven thousand, six hundred and fifty-four. 34. Three hundred million, two hundred and ten thousand, one hundred and ten. 35. Five hundred and forty million, three hundred and twenty thousand, two hundred and one. 36. Nine hundred and eighty-seven million, six hundred and fifty-four thousand, three hundred and twenty-one. 37. Ševenty-five billion, sixty-four million, three hundred thousand, two hundred and seventeen. 38. Five hundred and forty trillion, four hundred and thirty billion, three hundred and twenty million, two hundred and ten thousand, one hundred and one. 39. Eight hundred and nineteen sextillion, seven hundred and sixteen quintillion, six hundred and fifteen quadrillion, five hundred and fourteen trillion, four hundred and thirteen billion, three hundred and twelve million, two hundred and eleven thousand, one hundred and ten. |