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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 10 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, &s. 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 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, fc. 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. we 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 che 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
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 thc 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
tens, and so on in order; for by so doing we shall certainly find the whole number of units, tens, &c. in the difference.
Let the units of the subtrahend be subtracted ; this is done (Prop. 3) by subtracting the units' figure of the subtrahend from that of the diminuend, if possible; or from that of the diminuend, increased by 10, the next higher figure of the diminuend being diminished by 1. The difference so obtained is the number of units in the whole difference: and the diminuend so altered is the first partial difference. We have now to subtract the tens of the subtrahend, which is done (Prop. 3) by subtracting the tens' figure from that of the partial difference, if possible; or (if not) from it when increased by 10, the next higher figure of the diminueud being diminished by 1. The difference so obtained is the number of tens in the whole difference; and the first partial difference so altered, is the second partial difference. In like manner the number of hundreds, &c. in the difference will be obtained.
Hence the Rule for Subtraction might be given thus :- When the upper figure is less than the lower, increase it by 10, and diminish the next higher figure of diminuend by 1. But whether we diminish the diminuend, or increase the subtrahend by any number, the final difference is the same, since in the one case we are subtracting in two parts, in the other in one. Hence (and because it is perhaps more convenient) the Rule is given to increase the next higher figure in the subtrahend by 1, whenever any figure of the diminuend is increased by 10.
Note. The increase of the figure of the diminuend by 10, and of the subtrahend by 1, may also be explained by considering that by so doing we are in fact increasing both by the same number. And if two numbers be both increased by the same number, their difference remains unaltered.
Cor. The Rule for Compound Subtraction may be proved and explained in precisely the same manner by substituting for units, tens, &c. the denominations of the given quantities.
Prop. 6.—The product of two numbers is the same, whichever
be the multiplier. Place in a line 12 dots: draw a line after every third, and every fourth dot. It is thus seen that the 12 dots are composed of four groups of 3 dots,
::::::::::| or of 3 groups of four dots. Hence 3 times 4 are equal to 4 times 3. The same may be proved of any numbers whatever, the multiplicand being abstract or concrete.
The same Prop. exhibited algebraically:-- If a and 6 be the two numbers, then a X b = 6 X a.
Prop. 7.— To prove that the product of one number by another
is equal to the sum of the products of each part of the mul
tiplicand and multiplier. Since Multiplication is only a short process of Addition, and since in adding numbers we are permitted to separate them into parts, and grouping the parts to add each group in succession, therefore in Multiplication we may do the same. The only difference between this process and Addition is, that the parts of each group being the same, the number in each is found by the aid of the Multiplication Table, instead of by direct Addition, and is called the product instead of the sum. In other respects the processes of Multiplication and Addition are precisely similar. The same Prop. exhibited algebraically ;-If A be the multiplicand, and
A = a + b + c + &c. and m be the multiplier ; then
A X m = ax m + 6 x m + cx mn •f. &c. Prop. 8.—T.
prove that the product of two numbers is equal to the sum of the products of the multiplicand by each part
of the multiplier. Since the product of two numbers is the same, whichever be the multiplier; therefore let the multiplicand be supposed to be the multiplier. Then the product of the two is equal to the sum of the products of the several parts of the present multiplicand by the present multiplier, i. e. is equal to the sum of the products of the several parts of the real multiplier by the real multiplicand; or is equal to the sum of the products of the multiplicand by the several parts of the multiplier.
The same Prop. exhibited algebraically :-If A be the multiplicand, M the multiplier, and M=a+b+c+ &c. then
AXM = M XA = (a + b +c+&c.) XA (Prop. 6.)
=a XA +b XA +cXA + &c. (Prop. 7.) = A Xa + A X6 + AXC + &c.
Prop. 9.-To prove the Rule for Multiplication by a com
posite number. The product of two numbers is equal to the sum of the products of the multiplicand by the parts of the multiplier. If all these parts, and therefore the products, be equal, the sum may be formed by the short process of Multiplication by their number. Now a composite number (supposed to consist of two factors) is composed by the sum of one number repeated as often as there are units in another. Therefore the product by a com