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septillions, octillions, nonillions, decillions, undecillions, duodecillions, &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, ihe 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 immediately 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 tkousands' 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 enun
ciation. When this is the case, the places must be sup- plied by zeros. 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 fig
ures at the left of the mistake, will be increased or diminished tenfold. *
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 ten. fold 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 bad 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 reject 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 numbers of the ten fold ratio corresponding to them. Eight fold Ten fold Eight fold
Ten fold One 1 corresponds to Fifteen 15 corresp. to 13 Two
14 Three 3 Seventeen 17
16 Four 4
24 Six 6
40 Ten 10 8. Sixty 60
48 Eleven 11
56 Twelve 12
10 One hundred 100, &c. . 64 Thirteen 13 11 One thousand 1000
512 Fourteen 14
12 In the same manner we had twelve figures, the places would have been 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.
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 upit at a time, thus, 5 and I 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 has occasion to put two numbers together, 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 4 are 11. &c.
Before much progress can be niade 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 pine. 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. Hat 8 dolls. to add to 24 dolls. Here are twen
ty dolls. and four dolls., and eight Both 32 dolls. dolls. Eight and four are twelve, which are to be joined to twenty. But iwelve is the same as ten and two, therefore we may say twenty and ten are thirty and two are thirty two.
A man bought a cow for 27 dolls. and a horse for 68 dolls. How much did he give for both ?
Operation. Cow 27 dolls. In this example it is proposed Horse 68 dolls. to add together 27 and 68. Now
27 is 2 tens and 7 units; and 68 Both 95 dolls. is 6 tens and 6 units. 6 tens and 2 tens are 8 tens; and 8 units and 7 units are 15,
which is 1 ten and 5 units; this joined to 3 tens makes 9 tens and 5 units, or 95.
A man bought ten barrels of cider for 35 dolls., and 7 barrels of flour for 42 dolls., a hogshead of. molasses for 36 dolls., a chest of tea for 87 dolls., and 3 hundred weight of sugar for 24 dolls. What did the whole amount to?
Operation. Cider 35 dolls. In this example there are five Flour 42 dolls. numbers to be added together. Molasses 36 dolls. We observe that each of these Tea 87 dolls. numbers consists of two figures. Sugar 24 dolls. It will be most convenient to add
together either all the units, or Amount 224 dolls. all the tens first, and then the other. Let us begin with the tens. 3 tens and 4 tens are 7 tens, and 3 are 10 tens, and 8 are 18 tens, and 2 are 20 tens, or 200. Then adding the units, 5 and 2 are 7, and 6 are 13, and 7 are 20, and 4 are 24, that is, 2 tens and 4 units; this joined to 200 makes 224.
It would be still more convenient to begin with the units, in the following manner : 5 and 2 are 7, and 6 are 13, and 7 are 20, and 4 are 24, that is, 2 tens and 4 units; we may now set down the 4 units, and reserving the 2 tens add them with the other tens, thus : 2 tens (which we reserved) and 3 tens are 5 tens, and 4 are 9 tens, and 3 are 12 tens, and 8 are 20 tens, and 2 are 22 tens, which written with the 4 units make 224 as before.
A general has three regiments under his command ; in the first there are 478 men; in the second 564 ; and in the third 593. How muny men are there in the whole?
In this example, each of 564 men
the numbers is divided nto 593 men
three parts, hundreds, tens
and units. To add these In all 1,635 men together it is most conven
First reg. Second reg. Third reg.
ient to begin with the units as follows : 8 and 4 are 12, and 3 are 15, that is, 1 ten and 5 units. We write down the 5 units, and reserving the 1 ten, add it with the tens. 1 ten (which we reserved) and 7 tens are 8 tens, and 6 are 14 tens, and 9 are 23 tens, that is, 2 hundreds and 3 tens. We write down the 3 tens, and reserving the 2 hundreds add them with the hundreds. 2 hundreds (which we reserved) and 4 bundreds are 6 hundreds, and 5 are 11 hundreds, and 5 are 16 hundreds, that is, 1 thousand and 6 hundreds. We write down the 6 hundreds in the hundreds' place, and the 1 thousand in the thousands' place.
The reserving of the tens, hundreds, &c. and adding them with the other tens, hundreds, &c. is called carrying: The principle of carrying is more fully illustrated in the following example.
A merchant had all his money in bills of the following description. one-dollar bills, ten-dollar bills, hundred-dollar bills, thousand-dollar bills, &c. each kind he kept in a separate box. Another merchant presented three notes for payment, one 2,673 dollars, another 849 dollars, and another 756 dollars. How much was the amount of all the notes ; and how many bills of each sort did he pay, supposing he paid it with the least possib'e number of bills.
The first note would re6 7 3
quire 2 of the thousand-dol8 4 9
lar bills; 6 of the hundred7 5 6
dollar bills; 7 ten-dollar 4 2 7 8 bills; and 3 one-dollar bills. The second note would require 8 of the hundred-dollar bills, 4 ten-dollar bills; and 9 one-dollar bills. The third note would require 7 of the hundred-dollar bills ; 5 ten-dollar-bills; and 6 one-dollar bills. Counting the one-dollar bills, we find 18 of them. This may