(10) Nine hundred and nine millions, nine thousand and ninety-nine. As far as practical utility is concerned, we shall seldom or never have occasion to express by figures, numbers exceeding Hundreds of Millions; but the system of Notation admits of being extended so as to represent any nuinber whatever. Thus, instead of supposing that each division consists of three figures, if we include six figures as far as we can in each division, the first may be regarded as SO many hundreds of thousands of Units; the next as so many hundreds of thousands of Millions; the next as so many hundreds of thousands of what are called Billions, and the succeeding divisions, of so many hundreds of thousands of what are termed Trillions, Quadrillions, &c. Ex. To represent Ten thousand millions by figures; for the first division we have, according to this view, 000000, and for the second 10000, so that the representation required is 1 0.0 0 0,0 0 0.0 0 0. 15. It will readily be observed, from what has already been said, that each of the nine figures or digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, has an absolute value of itself, whereas the auxiliary digit O has no such value; and on this account the former are sometimes termed significant figures, in contradistinction to the last. It will moreover have occurred to the reader, that every one of these significant digits, in addition to its absolute value, which is fixed and certain, possesses also a local value dependent upon the situation in which it is placed; thus, in the expression of the number Four thousand, three hundred and twenty-one, which will be 4 3 2 1, the 1 in the first place on the right hand, retains its absolute value; the second figure 2, in its situation denotes two tens or twenty; the third is three hundred, and the fourth is four thousand; so that the local values of 2, 3, and 4, are respectively, ten times, a hundred times and a thousand times, as great as their absolute values: and it is the circumstance of assigning to each of the significant figures a local as well as an absolute value, which confers upon the system, the immense powers it possesses of being adequate to the representation of any number, however great, as already shewn. NUMERATION. 16. DEF. Numeration is the art of reading or estimating the value of any number, expressed by means of the numeral characters in whatever manner combined or repeated, and is therefore the reverse of Notation. 17. From the circumstance of every figure possessing a local as well as an absolute value, it follows that the value of each must be estimated by the place which it occupies: hence, therefore, a figure standing by itself expresses so many units; a figure in the second place from the right denotes so many tens; a figure in the third place, so many hundreds, and so on, according to articles (10) and (11): consequently, if we suppose any numerical expression to be divided into portions, each consisting of three figures as far as they go, the figures of the division on the right will be units, and tens and hundreds of units; those of the next division will be units, tens and hundreds of thousands; those of the third will be units, tens and hundreds of millions, and so on. Thus, 25 is Twenty-five. 304 is Three hundred and four. 5287 is Five thousand, two hundred and eighty-seven. 60539 is Sixty thousand, five hundred and thirty-nine. 207385 is Two hundred and seven thousand, three hundred and eighty-five. 1739204 is One million, seven hundred and thirtynine thousand, two hundred and four. 35024376 is Thirty-five millions, twenty-four thousand, three hundred and seventy six. 275008005 is Two hundred and seventy-five millions, eight thousand and five. In every one of these instances we conceive the expression to be separated into portions of three figures each as far as they go, beginning at the right hand: as in 275008005, we observe that 005 is the first portion, 008 the second, and the third portion is 275, each consisting of three figures: that is, 275 denotes two hundred and seventy-five millions, 008 eight thousand and 005 five units, and the expression will be read as above. 18. The substance of the last article will be rendered still more clear by means of the following scheme, which is called the Numeration Table: Tens of Millions. wherein the local value of every figure in each of the horizontal rows is pointed out by the name written upwards at the top of the whole: thus, in the third horizontal line from the bottom, the figures will be read Nine hundred and eighty-seven; and in the second line from the top, Ninety-eight millions, seven hundred and sixty-five thousand, four hundred and thirty-two. 19. For practice, the student is advised to write down in words at length, the following numerical expressions. 20. The principles of Notation, or the expressing of any number by means of the ten numeral characters, and those of Numeration, or the reading of numerical magnitudes so expressed, being once established and understood, we proceed to the consideration of the four funda 9 mental Arithmetical Operations that can be performed upon numbers, which are those of Addition, Subtraction, Multiplication and Division, each of which will be defined, explained and exemplified in order. I. ADDITION. 21. DEF. Addition is the first of the fundamental operations of Arithmetic, and consists in finding a number equal to the aggregate of two or more numbers taken together, and this number is called their Sum. Ex. 1. To find the sum of the simple numbers 2, 5 and 9; we see that two units and five units taken together make seven units, and this with nine units more, will manifestly amount to sixteen units, which is written 16. The operation may stand as follows: therefore 2 Ex. 2. Add together the numbers expressed by 254, 893 and 487. Here it would be absurd to collect immediately into one sum, numbers of different local values, as for instance, to say that three units and five tens amount together to either eight units or eight tens, and we therefore place the numbers to be added together in such a form that each of the figures of the same denomination may be in the same vertical line, as on the left of the page: Explanation of Operation. Common Form. 254 200 and 50 and 4 4 8 7 3 80 7 ... 220 1 4 10 1600 230 and then, as is seen in the operation on the right, we have first added the units together and thus have 14 units, or 1 ten and 4 units: we have next found the sum of the tens to be 22, which with the 1 ten before obtained amount to 23 tens, or 2 hundreds and 3 tens; and lastly, we have by the same kind of process ob tained 14 hundreds, which together with the 2 hundreds last found make 16 hundreds, or 1 thousand and 6 hundreds: whence the entire sum is 1 thousand, 6 hundreds, 3 tens and 4 units, or 1634. The reasoning here used is thus applied to the figures on the left of the page: the numbers of tens and hundreds found by adding the vertical columns of units and tens are annexed, or carried to the columns of tens and hundreds respectively, and they are here put down under them just above the horizontal line; but in practice they are generally omitted altogether by mentally adding them to the lowest figures of the next vertical rows, and then proceeding as before. 22. To effect the operation of Addition, as appears from the two instances just considered, it is therefore merely necessary to know from memory or by practice, the sums of every two numbers expressed by single figures, and the reasoning above employed leads to a general conclusion which is comprised in the following Rule. Rule for performing Addition. Place the numbers under one another in such a manner that units may stand under units, tens under tens, hundreds under hundreds, and so on, and draw a line below all the horizontal rows of figures: then add up the figures in the first vertical row on the right hand, find the numbers of tens and units in their sum, and put down the number of units, whether it be zero or any of the nine other digits: carry as many units as there are tens thus found to the next vertical row, and add them up as before, observing the numbers of tens and units contained in the sum: place the number of units under the row added, and carry the number of tens to the next; proceed in the same manner till the last row is added, when put down both the numbers of tens and units, as there are no more figures of higher denominations. 23. To ascertain whether the operation is correctly performed, various expedients might be resorted to; as for instance, that of adding the numbers downwards instead of upwards, which, because the same set of numbers cannot have two different sums, must give the same result as before: but the only one, with this exception, which does not involve principles hereafter to be explained, |