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66 This is the fundamental law of our written numeration, that a removal of one place towards the left increases the value of the figure ten times."
3. When we wish to write any number of tens, there are none of the above characters with which we can do it; if, for instance, we wish to write ten; or twenty ; we cannot, by any of the above characters, do it. write 1, with any of the characters at its right, they will make more than ten; and the same of the other figures. To remedy this we have a character which signifies nothing, and is called a cipher, or nought; this is the character, .
If we wish to write ten, we first write the figure 1, and at the right of it, write the cipher; thus, 10 expresses ten, the 1 being removed to the left, by this character which signifies nothing. If it be required to write one hundred, we must write 1, with two ciphers at its right, thus, 100 ; to write one thousand, three ciphers must be written, 1000 ; to write twenty, 2 with one cipher at its right, and so on with all the figures ; any number of tens, hundreds, thousands, &c. being expressed, by first writ. ing the significant figure, and then at its right place the proper number of ciphers.
Pup. I do not understand you when you say significant figure. I should like to have you explain it.
Tut. By significant figures are meant the figures which express some number of themselves, or all the figures except the cipher.
When we wish to write any number between one hundred and one hundred and ten, care must be taken that we place the figures according to their values. If it be required to write one hundred and one, we write it thus, 101 ; to write one hundred and two, thus, 102; and so on, the cipher in the middle carrying the significant figure at the left into the place of hundreds. Ciphers are frequently placed in the middle, and all other places except at the left of numbers ; thus in writing any number in figures we should take care that we do not give a wrong expression by omitting the ciphers.
In general' when a number is dictated, it is written by beginning at the left, and placing, one after another, the figures according to their values.
When a number is written in figures, in expressing it in language, it is necessary to substitute for each figure the words which it represents, and then mention the value of its place according to the following table.
Tens of trillions .
Tens of millions,
Tens of thousands,
The figures here you see are divided by commas, into periods of three figures each, which enables us more readily to enumerate them. This division is performed by beginning at the right and placing a comma at the right of every fourth figure ; sometimes the left hand figures do not amount to a full period. In the above table the left hand period consists of only two figures, and the whole table is read as follows ;-thirty-four trillions, six hundred and eighty-five billions, six hundred and seventyfour millions, three hundred and twenty one thousands, and nineteen. You may express the following numbers in figures. Twelve, Twenty,
20 One hundred,
100 Five hundred and sixty-four,
564 Six hund. & fourteen thọus. five hundred & one, 614,501 Nineteen millions and one,
Express the following numbers in words. 112, 684, 5001401, 8647236, 10109824, 1800014601.
Pup. I think that I now understand Numeration, and shall be able to attend to Addition with success.
Tut. You must not be too positive in thinking that you onderstand it, for it is ten to one if you do not meet with
numbers which you cannot enumerate. If ever you are at a loss to know how to enumerate any numbers, you must review Numeration and make every thing perfectly familiar. Before you proceed further you must examine the following characters, and make them familiar to your. self as fast as you come to the rules to which they are applied.
The sign of equality ; as 4 and 4 added together are
+ $ The sign of addition ; as, 2 + 2 = 4; meaning that
2 added to 2 is equal to 4. S The sign of subtraction; as, 8 — 4= 4; meaning,
that 4 taken from 8 leaves 4. S The sign of multiplication ; as, 3 X 3 = 9; meaning, that 3 multiplied by three is equal to 9. The sign of division ; as, 4 = 2 = 2; or, 2) 4( 2 ;
meaning, that 4 divided by 2, gives 2 for a quotient; ) (or that 2 is contained in 4 twice.
The sign of proportion ; as, 2 4::8 : 16 ;
These signs are all that will be necessary in your pres. ent course of instruction.
You are now prepared to be instructed in Simple Addition, and what do you think Simple Addition is ?
Pup. Simple Addition is a rule which teaches how to · put together simple numbers, so as to express them in one whole number,
Tut. You have formed a very correct idea of it. It is a rule by which two or more numbers of the same denomination are put together so as to be expressed in one sum.
4. In the addition of numbers the most natural way is to begin at unity, and count till you arrive at the full expression of one of those numbers, and then proceed to add a unit at a time, until you have added
as many units B 2
as the other number contains. If you were to add 8 and 4, you would begin and count, one, two, three, four, five, six, seven, eight, and ascend 4 places above 8, which would be twelve, the amount of 8 and 4. In this manner all small numbers are added together, and the amount of each addition committed to pemory.
But in the addition of large numbers this process is too tedious, and the endless number of sums which may be produced by addition, render it impossible to retain them in the memory. To avoid this difficulty we have recourse to the combination of units into tens, hundreds, thousands, &c. The following is the rule for
Place the numbers under each other so that units may stand under units, tens under tens, hundreds under hundreds, &c, and draw a line underneath.
Begin at the right hand figure of the lower number, and find the amount of all the figures of that column, and if their sum* does not exceed 9, set it down ; but if it exceeds 9, set down what the sum exceeds ten or any even number of tens ; and if it be ten or an even number of tens, write a cipher instead of the excess, and carry as many to the next figure towards the left as the sum contained tens. Proceed in this manner with every column through the whole, and at the last or left hand column, set down the whole amount of that column.
You may add together the following numbers ; 268, 346, 792, and they must be placed as follows :
2 6 8 3 4 6 7 9 2
1 4 0 6
Pup. The work is all very plain, but I do not understand why I should carry one for every ten ; I should like to have you give the reasons for carrying in this way.
* The amount of all the numbers, expressed in one whole pummp, is called the sum of those numbers.
Tut. I shall now have an opportunity to see whether you understand Numeration ; for if you do not, you cannot understand the explanations which I shall give you.
According to Numeration, (3. Con. I.) you will recollect that ten units are equal to one in the place of tens ; and that ten in the place of tens are equal to one in the place of hundreds, and so on, increasing towards the left in a tenfold proportion. In the above example, you first add the right hand column, or the column of units, and find their sum is 16 ; this means 16 units, and as ten units are equal to 1 in the place of tens, by setting down what 16 exceeds 10, and carrying 1 to the next figure, you retain the true value. If you should begin with the second column, at 9, and add up that column, which is the col. umn of tens, you would get 19, which expresses 19 tens, or 190 units. Now if the second column contains 19 tens, and the column of units contains 16 units, and 10 units are equal to one in the place of tens, it is plain that by taking the ten which 16 contains, and adding it to the 19 tens, you express the true number of tens there are in the two right hand columns.
Pup. But how do we know how many tens are expressed, if the cipher will express any number of tens.
Tut. The cipher itself does not express any number, but the figure at the left of it. In the above example the cipher stands for the sum of that column, and the ten which was added from the column of units, which make 20 ; and 20 containing 10 twice, a cipher must be written, and 2 carried to the next figure ; then add the left hand column, and set down the whole amount, because there are no figures at the left to which you can carry.
Again, suppose you were requested to add together four dollars and six dimes,* and six dollars and seven dimes, you would place them thus :
Here you first add together the dimes at the right, and And they amount to 13 ; now the figures at the left are
* A dime is a ten cent piece.