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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 В 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
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