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these nine characters all numbers whatever inay
To express ten, we make use of the first character 1. But to distinguish it from one unit, it is written in a new place, thus 10; the 0, which is called zero or a cipher, being placed on the right. The zero 0 has no value, it is used only to cccupy a place, when there is nothing else to be put in that place.
Numbers expressed with the Roman Letters.
XXVII Four *IIII Twenty-eight
XXVIII Five V Twenty-nine
*XXVIIII Six VI Thirty
XXXII,&c Nine *VIIII Forty
*XXXX Ten х
*LXXXX Fifteen XV One hundred
DCCCC Twenty-four *XXIIII Oue thousand
M One thousand, eight hundred, and twenty-six MDCCCXXVI
A man has a carriage worth seven hundred and sixty-eight dollars and two horses, one worth two hundred and seventy-three dollars, and the other worth two hundred and forty-seven dollars; how many dollars are the whole worth? These numbers may be written as follows:
CCLXXIII dolls. to see that it will be the most convenient to
commence on the right, and count the Is
first. We find eight of them, which we MCCLXXXVIH dolls. should write thus VIII, but observing that
" It is usual to write four IV, instead of III, and nine IX, instead of VIIII, and foriy XL, iustead of XXXX, and ninety XC, instead of LXXXX, &c. in which a sinall character before a large, takes out its value from the large. This is more convenient when no calculation is to be made. But when they are to be used in calculation, the method given in the text is best.
Eleven is written thus, 11, with two ls. The I on the left expresses one ten ; and the one on the right expresses one unit, or one added to ten. Twelve is written 12; the 1 on the left signifies one ten, and the 2 on the right signifies two units, and the whole is properly read ten and two. there are inore Vs we set down only III, reserving the V and counting it with the other Vs. Counting the Vs we find two, and the one which we reserved makes three. Three Vs are equivalent to one X and one V. We write the V and reserve the X. Counting the Xs, we find seven of them, and the one which was reserved makes eight. Eight Xs are equivalent to LXXX. We write the three Xs and reserve the L. Counting the Ls, we find two of them, and the one which was reserved makes three. Three Ls are equivalent to CL. We write the L and reserve the C. Counting the Cs, we find six of them, and the one which was reserved makes seven. Seven Cs are equivalent to DCC. We write the CC and reserve the D. Counting the Ds we find one, and the one which was reserved makes
Two Ds are equivalent to M. The whole sum therefore is MCCLXXXVIII dollars.
The general rule for addition, therefore is, to begin with the characters which express the lowest numbers and count all of each kind together without regard to their value, only observing that five Is make one V, and that two Vs make one X, and that five Xs make one L, &c., and setting them down accordingly.
A man having one hundred and seventy-eight dollars, paid away seventy-nine dollars for a horse ; how many had he left?'
Vs from the Vs, &c. But a difficulty immeLXXXXVIIII dolls. diately occurs, for we cannot take IIII from III ; it is necessary therefore to take the IIII from VIII, that is, from IIIIIIII, which leaves IIII; these we get down. Since we have used the V in the upper line, it will be necessary to take the V in the lower line from one of the Xs, that is from VÝ. V from VV, leaves V, which we set down. Having used one of the Xs, there is but one left. We cannot take XX from X, we must therefore use the L, which is equivalent to five Xs, which, added to the one X, make XXXXXX; from these we take XX and there remain XXXX, which we set down. Since the L in the upper line is already used, it is necessary to take the L in the lower line from the C which is equivalent to LL; one L taken from those, leaves L, which we set down. The whole remainder therefore is LXXXXVIIII dolls.
Hence the general rule for taking one number from another, ex pressed by the Roman characters, is, to begin with the characters expressing the lowest numbers, and take those of the same kind from each other, when practicable, but if any of the numbers to be subtracted exceed those from which they are to be taken, a character of the next highest order must be taken, and reduced to the order required, and joined with the others from which the subtraction is to be made
This process is called subtraction.
The following is the manner of writing the numbers from nine to ninety-nine, inclusive.
The first column contains the figures, the second shows the proper mode of expressing them in words and the way in which they are always to be understood, and the third contains the names which are commonly applied. The common names are expressive of their signification, but not so much so as those in the second column. Figures.
Proper mode of expressing Common Names.
them in words. 10.
One Ten or simply Ten. Ten. 11. Ten and one.
Eleven. 12. Ten and two.
Twelve. 13. Ten and three.
Thirteen. 14. Ten and four.
Fourteen. 15. Ten and five.
Fifteen. 16. Ten and six.
Sixteen. 17. Ten and seven.
Seventeen. 18. Ten and eight.
Eighteen. 19. Ten and nine.
Nineteen. 20. Two tens.
Twenty. 21. Two tens and one.
Twenty-one. Two tens and two. Twenty-two. 23.
Two tens and three. Twenty-three. 24.
Two tens and four. Twenty-four. 25. Two tens and five.
Twenty-five. 26. Two tens and six.
Two tens and seven. Twenty-seven. 29.
Two tens and eight. Twenty-eight. 29.
Two tens and nine. Twenty-nine. 30. Three tens.
Thirty. Three tens and one. Thirty-one. 32, &c. Three tens and two. Thirty-two. 40. Four tens.
Forty. 41, &c. Four tens and one. Forty-one. 50. Five tens.
Fifty. &c. Five tens and one.
Fifty-one. 60. Six tens.
Sixty. 61, &c. Six tens and one.
Sixty-one. 70. Seven tens.
Seventy. 71, &c.
Seven tens and one. Seventy-one. 20. Eight tens.
Eighty B1, &c.
Eight tens and one, Eighty-one.
Preper mode of expressing Common Names.
them in words. 90. Nine tens.
Ninety. 91, &c. Nine tens and one.
Nine tens and ninė. Ninety-nine. Nine tens and nine or ninety-nine is the largest number that can be expressed by two figures. If one be added to nine tens and nine, it makes ten tens, or one hundred. To express one hundred we use the first figure again ; but in order to show that it has a new value, it is put in another place, which is called the hundreds' place. The hundreds’ place is the third place counting from the right. One hundred is written, 100; two hundred is written, 200.; three hundred is written, 300. The zeros on the right have no value; their only purpose is to occupy the two first places, so that the figures 1, 2, 3, &c. may stand in the third place.
The figures in the second place, we observe, have the same value whether the first place be occupied by a zero or by a figure : for example, in 20 and in 23 the 2 has precisely the same value ; it is two tens or twenty in both. In the first there is nothing added to the twenty, and in the second three is added to it.
It is the same with figures in the third place. They have the same value, whether the two first places are occapied by zeros or figures. In 400, 403, 420, and 435, the 4 has the same value in each, that is four hundred. The value of every figure, therefore, depends upon its place as counted from the right towards the left. A figure standing in the first place signifies so many units; the same figure standing in the second place signifies so many tens; and the same figure standing in the third place signifies so many hundreds. For example, 333, the three on the right signifies three units, the three in the second place signifies three tens or thirty, and the 3 in the third place signi'es three hundreds. The number is read three hundreds, three tens, and three, or three hundred and thirty-three. We have seen that all the numbers from ten to twenty, from twenty to thirty, &c. are expressed by adding units to the tens; in the same manner all the numbers from one hundred to two hundred, from two hundred to three hundred, &c. are expressed by adding tens and units to the hundreds. For example, to express five hundred and eighty-two, we write five hundreds eight tens, and two units thus, 592.
The largest number that can be expressed by three figures is 999, nine hundreds, nine tens, and nine units, or nine hundred and ninety-nine. If to this we add one unit more, we have a collection of ten hundreds, which is called one thousand. To express this, the 1 is used again ; but to show that it expresses 1 thousand it is written one place farther to the left, that is, in the fourth place, thus 1000. Two thousand is written 2000, and so on, to nine thousand, which is written 9000. The intermediate numbers are expressed by adding hundreds, tens, and units to the thousands.
It is easy to see that this manner of expressing numbers may be continued to any extent. Every time a figure is removed one place to the left its value is increased ten-fold, and since nothing limits the number of places which we may use, there can be no number conceived, however large, which cannot be expressed with these nine characters.
We sometimes call the figures in the first place or right hand place, units of the first order ; those in the second place, or the collection of tens, units of the second order ; those in the third place, or the collection of hundreds, units of the third order, &c.
The following table exhibits the first nine places or orders, with their names, and contains a few examples to illustrate them.