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PRINCIPLES OF ARITHMETIC.
Prop. 1.--To explain the common system of Notation.
The numbers from one to nine are denoted by the nine symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, and a tenth 0, called a cipher, is used to denote the absence of number, or naught. It remains to be shewn, how by means of these symbols all numbers may be denoted.
Suppose a number of counters in a bag, which we will call A, and suppose other bags B, C, D, &c. in a line with A, A being on the right hand, let the counters be counted out by tens, and for every ten counted let one be placed in the bag B. When all the tens have been counted, the counters in B will shew the number of tens in the whole, and those that remain in A will shew the number of single counters besides. Now let the counters in B be counted in the same way, and for every ten counted let one be placed in C. When all the tens have been counted, the counters in C will shew the number of tens of tens, or hundreds, in the whole, and those that remain in B will shew the number of single tens besides. Let the same operation be repeated with the counters in C, &c. till the number in every bag is less than ten. Let it be observed, that each counter in A represents one, each in B represents ten, each in C one hundred, &c. Remembering this, we can by looking at the counters in the several bags tell how many counters there are altogether; i.e. the number is denoted by the counters in the bags.
Now let each bag be labelled, and on the label let there be written the figure denoting the number of counters in the bag, and over the figure also the number of counters represented by each single one in the bag, viz. on A let there be written units, on B tens, &c. Knowing then, as we do, the number of counters in each bag from the figure on the label, we can form the same idea of the whole number by looking at the figures, as by looking at the counters themselves, and more easily, because we are saved the M
trouble of counting the number in each bag. Thus the number is denoted by the figures on the labels. Now if we look at these figures, and the writing above them, we perceive that, as they are arranged in a line, the number denoted by any figure increases ten-fold with every increase of its distance from the right. Thus a 2 on A signifies 2 units; on B 2 tens, or ten 2's; on C 2 hundreds or ten times ten 2's; and so on. If then we recollect this fact, we may dispense with the writing over the figures, and we have left only the figures themselves, arranged in a certain order, each figure also having a signification depending on its position in the rank. In this way any number whatever may be denoted by the ten symbols, or figures, by making the same agreement respecting the local value of the figures.
Thus we have arrived at the ordinary system of Notation, or the system by which any number may be denoted by the ten symbols, which of themselves cannot represent a number greater than nine. And the system is this:-The number is divided into collections of single units, tens, hundreds, &c. the number of collections of each kind being less than ten; and then the figures, denoting the number of each kind of collection, are written in order from right to left, beginning with the units.
Cor. Hence it appears that the addition of a number of units, tens, &c. less than 10 is performed by writing the figures, which represent their numbers one after the other, the highest denomination being on the left, the lowest on the right, the others being arranged in order according to their denomination, from the highest to the lowest, as we progress from left to right, a cipher being supplied, where any denomination is wanting.
Prop. 2.-To shew how to increase a number by an exact number of units, tens, &c.
Since the figures denoting a number shew the number of units, tens, &c. which the number contains, and since, if the number be increased by an exact number of units, tens, &c. it must contain so many more units, tens, &c. than it did before, therefore in general the figure in the units,' tens', &c. place must be increased by the given number of units, tens, &c. which are to be added.
If the increased figure be not greater than 9, the increased number will be denoted by putting this figure instead of the former. Thus if 563 be increased by 2 tens, the increased number will be denoted by 583. But,
If the increased figure be greater than 9, it will be denoted by two figures, which cannot occupy the place of the former one, without entirely changing the number denoted.
Thus if 563 be increased by 9 units, the increased number cannot be denoted by 5612, though 9 and 3 are 12; because 5612 denotes 5 thou
sands, 6 hundreds, 1 ten, and 2 units, whereas the increased number is only 5 hundreds, 6 tens, and 12 units.
To determine what is to be done in this case, we have only to consider, that a number of units, tens, &c. greater than 9 are equivalent to two parts, viz. to some number of tens, hundreds, &c. and a number of units, tens, &c. less than 9. Thus 12 units are equivalent to 1 ten and 2 units. Now it will not alter the number, if we take away from one part and add to another the same number. Therefore we may remove the tens, hundreds, &c. from the units, tens, &c. and add them to the other tens, hundreds, &c. in the number. Thus the number 563 increased by 9 units becomes 5 hundreds, 6 tens, 12 units; but 12 units are equivalent to 1 ten and 2 units; remove then the 1 ten from the 12 units, and add it to the 6 tens, thus making 5 hundreds, 7 tens, 2 units, or the number 572.
Hence we conclude that, to increase a number by an exact number of units, tens, &c., we must increase by this number the figure in the units', tens', &c. place; and if the figure so increased be not greater than 9, the increased number will be denoted by putting it in the place of the original figure; but if the figure so increased be greater than 9, we must change it into tens and units, and, writing the units in the place of the original figure, increase the figure of the next higher denomination by the number of tens.
Cor. Similarly it may be shewn that the method of increasing a compound quantity by a number of any one of its denomination is this. Add the given number to the number of the same denomination in the given quantity; if the sum be less than the number of units, which make up one of the next higher denomination, write it in the place of the original number; but if the sum be not less than this number, convert it into a number of units of the next higher denomination, add it to the number of this denomination in the given quantity, and write the remainder in the place of the number first increased.
Prop. 3.-To shew how to diminish a number by an exact number of units, tens, &c. less than ten.
Since the figures, denoting a number, shew the number of units, tens, &c. which the number contains, and since, if the number be diminished by an exact number of units, tens, &c. it will contain so many fewer units, tens, &c. than it did before, therefore in general the figure in the units', tens', &c. place must be diminished by the given number of units, tens, &c. to be taken away.
If the figure to be diminished be not less than that by which it is to be diminished, the diminished number will be denoted by putting the dif
ference instead of the original figure. Thus if 765 be diminished by 5 tens, the diminished number will be denoted by 715. But,
If the figure to be diminished be less than that by which it is to be diminished, since we cannot take a greater from a less number, we must consider in what way the result is to be found. Now it does not alter the number to take away from one part and add to another the same number. Therefore we may take away 1 from the number of the next higher denomination to that with which we are dealing, and add it to the number of this one, thus increasing it by 10. Now we are able to diminish it by the required number; and the result will be denoted by writing the difference in the place of the original figure, and diminishing the figure of the next higher denomination by 1. Thus let it be required to diminish the number 765 by 9 tens: we consider that the number 765 contains 7 hundreds, 6 tens, and 5 units; or six hundreds, 16 tens, 5 units; and therefore the diminished number will contain 6 hundreds, 7 tens, 5 units, or will be 675.
Hence we conclude that, in order to diminish a number by an exact number of units, tens, &c. wè must, if possible, diminish the figure in the units', tens', &c. place by the given number; but if not, we must increase it by 10, and then diminish it by the number, and diminish also the figure of the next higher denomination by 1.
Cor. Similarly it may be shewn that to diminish a compound quantity by a number of any denomination, we must, if possible, diminish the units of this denomination in the given quantity by the given number. But if this be not possible, we may diminish the units of the next higher denomination by 1, and having added its equivalent in units of the lower denomination to the number of lower denomination, we may then effect the subtraction. Thus to diminish £9: 10: 6 by 15s. we may diminish the £9 by 1, and adding the equivalent of £1, viz. 20s. to the 10s. may subtract 15s. from the sum; thus obtaining the remainder £8: 15: 6.
Prop. 4.-To prove and explain the Rule for Addition of Numbers.
Let the numbers be represented by heaps of counters, then their sum will be represented by all the counters together, and might be found by putting them together, and counting them. But provided that all are counted, the total number in their sum will not be affected by the manner or order in which they are counted. Let then each heap be separated into several heaps, and let these be again grouped together in any manner; then the sum may be found by counting each of these groups in succession. Hence in adding numbers, we may separate them into any parts we please, group the parts, and add the groups successively in any order we please.
It remains to be seen what is the most convenient method of separation, and of grouping; and what the order of adding the groups. To determine this, we consider, that the sum is to be presented in the form of a number of units, tens, &c. and therefore our object is to find these numbers. Also no denomination of figures can arise from, or be affected by, the addition of higher denominations, but may arise from, and be affected by, the addition of lower denominations. From these considerations it appears that the most convenient method of separation will be into units, tens, &c; and that the same must be the method of grouping. Also it appears that if we add the group of units first, we shall find the whole number of units; and if we then add to this sum the tens, hundreds, &c. in succession, we shall find the whole number of these denominations.
Let then the units be added, and the sum converted into tens and units; the units may be written down, as the units in the whole sum. Let next the tens be added to this partial sum, which is done (Prop. 2) by adding the numbers of tens in the several numbers to the tens in the partial sum. This sum being converted into tens and hundreds, the tens may be written down as the number of tens in the whole sum; and the hundreds being added to the hundreds in the several numbers, will give the nuinber of hundreds in the sum. In the same manner all the denominations may be found.
Prop. A.-To explain the Rule for Compound Addition.
As in Addition of numbers, we may separate the quantities into any parts, and add them in any order. And because the addition of higher denominations cannot affect the lower, but the addition of lower may affect the higher, therefore the quantities being separated into their several denominations, the addition is commenced by adding the lowest first, and proceeding in order to the highest; and this addition is performed (Cor. Prop.2) by adding the numbers of the several denominations in order.
Prop. 5.-To prove and explain the Rule for Subtraction of numbers.
If from a number of counters we are required to take away a given number, it can make no difference in the number that will remain, whether we take away the whole number at once, or in several portions, provided that the whole be taken away. Hence in subtracting one number from another we may separate the subtrahend into any parts we please, and subtract them in any order we please. But as the difference is to be expressed as a number of units, tens, &c. and as no denomination of figures can be affected by the subtraction of a higher, but may be affected by the subtraction of a lower; therefore it appears that it will be most convenient to separate the subtrahend into units, tens, &c. and to subtract first the units, then the