Provided that the values of the several figures are remembered, the words written above them may be left out. And the values may easily be remembered by noticing that they increase from the right towards the left, so that each number has a value ten times greater than the same number one place further towards the right, and ten times less than the same number one place further towards the left. The above number would therefore be written 7864357. This, according to the usual practice, would be read seven millions, eight hundred and sixty-four thousand, three hundred and fiftyseven,' and comparing this reading with the expression of the number in words previously given, it will be at once seen that certain abbreviations are used. These abbreviations are as follows: firstly, the plural form of thousands and hundreds is not used; secondly, the words eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, are used in place of ten and 1, ten and 2, &c., as far as ten and 9; thirdly, the words twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, are used in place of two tens, three tens, &c., as far as nine tens, and after these words the conjunction 'and' is omitted. Taking as examples, numbers of three figures, 453, 218, 907 would be respectively read, four hundred and fifty-three, two hundred and eighteen, nine hundred and seven. If a number consists of four, five, or six figures, the figures after the third from the right should be read in the way just explained, and the word thousand added. Thus, 23896 should be read twenty-three thousand eight hundred and ninety-six ; 4702 is four thousand seven hundred and two; 250008 is two hundred and fifty thousand and eight. It should be observed that the conjunction 'and' is always used immediately before any number from 1 to 99 being mentioned, and that it is not used anywhere else. It has now been explained how any number of not more than six figures should be read. If a number consist of more than six figures, let it be divided by commas into periods of six figures, commencing from the right hand, and call the second period from the right millions, the third billions, the fourth trillions, and so on to quadrillions, quintillions, &c. As an example, the number of different ways in which the twelve black and twelve white draughtsmen might be arranged on the draught-board is 677794,282450,430456,394720; and this number is read, six hundred and seventy-seven thousand seven hundred and ninety-four trillions, two hundred and eighty-two thousand four hundred and fifty billions, four hundred and thirty thousand four hundred and fifty-six millions, three hundred and ninety-four thousand, seven hundred and twenty. As it is most important that the pupil should thoroughly understand the process of Notation, or writing down in figures any given number, and the process of Numeration, or reading numbers when expressed in figures, examples of both are here given. (Note A.) Ex. 1. Express in figures the numbers: 1. Twenty-nine. Eighty-six. Fifty-seven. 2. Three hundred and fifty-four. Five hundred and nineteen. Six hundred and eight. 3. Two thousand five hundred and eleven. Eight thousand seven hundred and ninety-six. Thirteen thousand and nine. 4. Twenty-four thousand nine hundred and two. Three hundred and eighty-nine thousand seven hundred and forty-six. 5. Fifteen millions, six hundred and forty-three thousand, five hundred and twenty-eight. Seventy-two millions one thousand and four. 6. Three thousand and two billions four hundred millions and seventeen. 11. 12. 48,275319. 675,000032. 5,179584,362814. 940,000000,320005. When numbers are used without reference to any particular objects, as five, twenty-seven, eight thousand and nineteen, they are called Abstract Numbers, but when they have such a reference, as for instance, nine men, two pounds, fifty-four sheep, they are called Concrete Numbers. Numbers may be subjected to four different operations in arithmetic, Addition, Subtraction, Multiplication, and Division, and all arithmetical processes, however complicated, can be nothing but combinations of these four simple operations. The object of Addition is to determine what number represents the sum of two or more numbers taken together. If, for example, three persons are in a room, and afterwards two more come in, it will be found that there are then five in the room, and from that we infer that the sum of the numbers 3 and 2 is 5. The foundation, therefore, of addition is the record of a certain number of facts established by experience, and it should always be taught to little children in this way. A child should find out for himself, for example, that 8 and 7 are 15, by putting down upon a slate 8 strokes, then afterwards putting down 7 more, and counting the total number. The number of primary facts to be thus recorded is 90, arising from each of the numbers 1, 2, 3, &c. as far as 10 being added to each of the numbers 1, 2, 3, &c. as far as 9. This series of results is sufficient to enable us to add together any two numbers, however large. For example, let the numbers be 7539 and 9645. Put down either of these under the other, and commence by adding the right hand figures, 5 and 9 are 14. It has been stated that 14 means 10 and 4, and here 7539 9645 17184 these two numbers are separated, the 4 being put down in the place of units, and 1 carried forward to the place of tens. This 1 carried forward, added to the 4 and the 3 in the place of tens, gives 8. The 6 and 5 in the place of hundreds give 11, that is 1 in the place of hundreds, and 1 carried forward to the place of thousands. Lastly, 1 and 9 are 10, and 10 and 7 are 17. Although this is the first and the most elementary example of any process in arithmetic, it yet affords an opportunity of explaining what constitute good and bad habits of working. It is very common and most objectionable for a child to accompany the addition with a monotonous repetition, or rather a chant, of the following words :Five and nine are fourteen, put down four and carry one; one and four are five, and three are eight, six and five are eleven, put down one and carry one; one and nine are ten, and seven are seventeen;' while the numbers to be carried are successively put down and smeared out, somewhere below on the slate or paper. There is much here that is perfectly useless. It would be far better to say nothing aloud, and to say to oneself,' as it is called, or register in the memory, by allowing the mind to dwell upon them, not processes, but only results. Thus, criticising the words given above; 'five and nine are fourteen' is more than is required to be remembered, the result, fourteen, being all that is wanted; 'put down four and carry one' is something to be done, and not to be talked about, and the 'carrying' is not made easier by the putting down and then smearing out a figure below, or, if made easier, it is at the expense of accustoming the child not to rely upon his memory at all. All that should be registered in the memory is the series of results, fourteen, five, eight, eleven, ten, seventeen.' This may appear at first sight a trivial matter, but attention to it greatly facilitates the acquisition of a power of correct and rapid calculation. When there are more than two numbers to be added together, the method is precisely the same, but larger 8723 5169 2874 3496 2178 4053 26493' After adding several numbers together, it is best to try the addition again, beginning at the top and adding downwards. If the results agree, the work is nearly certain to be correct. In Subtraction a smaller number is taken away from a larger, and the remainder is determined. Supposing that five persons are in a room, and two go away, it would be found that there would remain three. Hence we deduce the fact that two from five leaves three. Now, there are two ways of arriving at facts in subtraction; either they may be obtained independently, as, for example, by a child. putting down upon paper or a slate a certain number of marks, then rubbing out a given part of them, and counting the remainder; or they may be obtained by deducing them from the facts of Addition. The latter is practically the best. Thus, from the fact that 8 and 7 are 15, we infer that 7 from 15 leaves 8, and that 8 from 15 leaves 7. The 90 elementary facts of addition furnish us with sufficient inferences of this kind to enable us to work out any question in Subtraction. Let it be required then to subtract 53 from 85. Here, putting the 53 under the 85, we say 3 from 5 leaves 2, 5 from 8 leaves 3. Next, to subtract 243 from 527. Here, 3 from 7 leaves 4, and 32 4 from 2 gives no possible result, so that some alteration must be made in the arrangement of the numbers before the subtraction can be performed. Let the 243 number in the place of tens in the 527 be posed to be increased by the value of 1 taken away from sup 85 53 527 284 |