116 ARITHMETIC.. Part 2. Ith or hdrs. of mills'. place. 2d or tens' place. si .1 or units' place. Seven units or seven 7 Three tens, or thirty 3 Four tens and six units, or forty-six 4 Eight hundreds 8010 Seven hundreds and three units, or seven hundred and three 703 Five hundreds and four tens, or five hundred and forty 5 40 Six hundreds, five tens, and eight units, or six I undred and fifty-eight 658 Six thousands 600 Six thousands and five units 6005 Six thousands and four tens, or six thousand and forty 610140 Six thousands and four tens and five units, or six thousand and forty-five 6045 Six thousands and seven hundreds 67100 Six-thousand, seven hundred, and five 617 5 Six thousand, seven hundred, and forty Six thousand, seven hundred, and forty-five 67145 l'our tens of thousands, or forty thousand 01010 Forty thousand and three 41010103 F'erty thousand, five hundred and three 503 Forty-seven thousand, five hundred, and eighty three 14 7583 Four hundred and twenty-six thousand, eight hundred and fifty-three 261853 Three hundred and twenty-eight millions, four hundred and thirty-five thousand, six hundred and eighty-seven 321814 3 516 817 Three hundred millions 3 010 0100101010 Twenty millions 210 00101010 Eight millions 0101010 Four hundred thousand 0101010 Thirty thousand 30101010 Five thousand 5 Six hundred 610 Eighty 8 Seven In looking over the above examples it will be observed, that the three first places on the right have distinct names, viz. units, tens, hundreds; and that the three next places are all called thousands, the first being called simply thousands; the second, tens of thousands; the third, hundreds of thousands. In the same manner there are three places appropriated to millions, and distinguished in the same way, viz. millions, tens of millions, hundreds of millions. The same is true of all the other names, three places being appropriated to each name. From this circumstance it is usual to divide the figures into periods of three figures each. This division very much facilitates the reading and writing of large numbers. Indeed it enables us to read a number consisting of any number of figures, as easily as we can read three figures. This is illustrated in the following example. cu Ilundreds Hundreds Hundreds Units Units 3 8 5, 6 7 9,2 5 8,6 7 3,4 6 2,9 2 7,6 4 8 We have only to make ourselves familiar with reading and writing the figures of one period, and we shall then be able to read or write as many periods as we please, if we know the names of the periods. It is to be observed that the unit of the first period is simply one ; the unit of the second period is a collection of a thousand simple units; the unit of the third period is a collection of a thousand units of the second period, or a mil lion of simple units; and so on as we proceed towards the left, each period contains a thousand units of the period next preceding it. The figures of each period are to be read in precisely the same manner as the figures of the right hand period. At the end of each period, except the right hand period, the name of the period is to be pronounced. The right hand period is always understood to be units without mention being made of the name. In the above example, the right hand period is read, six hundred and forty-eight (units being understood.) The second period is read in the same manner, nine hundred and twentyseven,--but here we must mention the name of the period at the end ; we say, therefore, nine hundred and twenty-seven thousand. If we would put the two periods together, we begin on the left and say, nine hundred and twenty-seven thousand, six hundred and forty-eight. The third period is read four hundred and sixty-two,-adding the name of the period, it becomes four hundred and sixty-two millions : and the three periods are read together, four hundred and sixty-two millions, nine hundred and twenty-seven thousand, six hundred and forty-eight. Beginning at the left hand of the above example, the several periods are read separately as follows-three hundred and eignty-five ; six hundred and seventy-nine ; two hundred and fifty-eight ; six hundred and seventy-three; four hundred and sixty-two; nine hundred and twenty-seven ; six hundred and forty-eight. Giving each period its name and putting all together as one number, it becomes three hundred and eighty-five quintillions ; six hundred and seventy-nine quadrillions ; iwo hundred and fifty-eight trillions ; six hundred and seventy-three billions ; four hundred and sixty-two millions ; nine hundred and twenty-seven thousand; six hundred and forty-eight. The names of the periods are derived from the Latin numerals, by giving them the termination illion and making some other alterations, so as to render the pronunciation easy. After quintillions come sertillions, septillions, octillions, nonillions, decillions, undecillions, duodecillions, f'c. A number dictated or enunciated, is written by beginning at the left hand, and proceeding towards the right, care being taken to give each figure its proper place. If any place is omitted in the enunciation, the place must be supplied with a zero. If, for example, the number were three hundred and twenty-seven thousand, and fifty-three; we observe that the highest period mentioned is thousands, which is the second period, and that there are hundreds mentioned in this period, (that is, hundreds of thousands,) this period is therefore filled, and the number will consist of six places. We first write 3 for the three hundred thousand, then 2 im mediately after it for the twenty thousand, then 7 for the seven thousand ; there were no hundreds mentioned in the enunciation, we must put a zero in the hundreds' place, then 5 for the tens, and 3 for the units, and the number will stand thus, 327,053. Let the number be fifty-three millions, forty thousand, six hundred and eight. Millions is the third period, and tens of millions is the highest place mentioned, hence there will be but two places occupied in the period of millions, and the whole number will consist of eight places. We first write 53 for the millions. In the period of thousands there is only one place mentioned, that is, tens of thousands, we must put a zero in the hundreds of thousands' place, then 4 for the forty thousand, then a zero again in the thousands' place; in the next period we write 6 for the six hundred, there being no tens in the example we put a zero in the tens' place, and then 8 for the eight units, and the whole number will stand thus, 53,040,608. Whole periods may sometimes be left out in the enunciation. When this is the case, the places must be supplied by Great care must be taken in writing numbers, to use precisely the right number of places, for if a mistake of a single place be made, all the figures at the left of the mistake, will be increased or diminished tenfold.* zeros. ADDITION. II. We have seen how numbers are formed by the successive addition of units. It often happens that we wish to put together two or more numbers, and ascertain what number they will form. A person bought an orange for 5 cents, and a pear for 3 cents ; how many cents did he pay for both ? * The custom of using nine characters, and consequently the tenfold ratio of the places, is entirely arbitrary; any other number of figures might be used by giving the places a ratio corresponding to the number of figures, if we had only the seven first figures for example, the ratio of the places would be eight fold, and we should write numbers, in every other respect, as we do now. It would be necessary to rejert the names eight and nine, and use the name of ten for eight. Twenty would correspond to the present sixteen: and one hundred, to the present sixty-four, &c. The following is an example of the eight fold ratio, with the nuinbers of the ten fold ratio corresponding to shem. To answer this question it is necessary to put together the numbers 5 and 3. It is evident that the first time a child undertakes to do this, he must take one of the numbers, as 5, and join the other to it a single unit at a time, thus 5 and 1 are 6, 6 and 1 are 7, 7 and 1 are 8; 8 is the sum of 5 and 3. A child is obliged to go through the process of adding by units every time he hạs occasion to put two numbers to gether, until he can remember the results. This however he soon learns to do if he has frequent occasion to put numbers together. Then he will say directly that 5 and 3 are 8, 7 and I are 11, &c. Before much progress can be made in arithmetic, it is necessary to remember the sums of all the numbers from one to ten, taken two by two in every possible manner. These are all that are absolutely necessary to be remembered. For when the numbers exceed ten, they are divided into two or more parts and expressed by two or more figures, neither of which can exceed nine. This will be illustrated by the examples which follow. A man bought a coat for twenty-four dollars, and a hat for eight dollars. How much did they both come to ? Operation. Coat 24 dolls. In this example we have 8 dolls. to Hat 8 dolls. add to 24 dolls. Here are twenty dolls. and four dolls. and eight dolls. Eight Both 32 dolls. and four are twelve, which are to be joinEight fold Ten fold Eight fold Ten fold One 1 corresp. to 1 Fifteen 15 13 'Two 2 2 Sixteen 16 14 Three 3 3 Seventeen 17 15 Four 4 Twenty 16 Fivo 5 5 Thirty 30 24 Six 6 Forty 32 Seven 7 7 Fifty 40 Ten 10 8 Sixty 48 Eleven 11 9 Seventy 56 Twelve 12 10 One hundred 100, &c. Thirteen 13 11 One thousand 1000 512 Fourteen 14, 12 In the same manner if we had twelve figures, the places wond have heen in a thirteen fold ratio. The ten fold ratio was probably suggested by counting the fingers. This is the most convenient ratio. If the ratio were less, it would require a larger number of places to express large numbers. If the ratio were larger, it would not require so many places indeed, but it would not be so easy to perform the operations as at present on account of the numbers in each place being so large. corresp. to 4 4 6 |