above, four digits must be used; for ten thousand, five digits; &c. (b). Numeration is the converse of notation, and is the art of expressing in words a number given in figures. The explanation given above will sufficiently indicate the method of doing this. It will be noticed that whatever the number of digits in a number, the last digit on the right always represents units; the last but one, tens; the next, hundreds; &c. The following table may be useful: 3 4 5. 9 Tens. Units. ∞ Thousands. Tens of thousands. • Millions. Tens of millions. Hundreds of thousands of millions. Thousands of millions. Hundreds of millions. + Billions. Ex. (1) Three hundred and twenty-five thousand, two hundred and fifty-seven, would be written in figures as 325257. Ex. (2) Twenty-eight millions seven hundred and fortyfour thousand one hundred and sixty-nine, would be 28744169. Ex. (3) Ten thousand three hundred and eighty, would be 10380. Ex. (4) Five hundred thousand three hundred and one. Here there are no tens of thousands, no units of thousands, and no tens; each of these places must be supplied by a o; and we shall have 500301. Ex. (5) Seventy billions, one million, and twenty-five, would be 70,000,001,000,025. The commas are here inserted for the sake of clearness, after the billions, thousands of millions, millions, and thousands. Ex. (6) 527434 when expressed in words, will be five hundred and twenty-seven thousand, four hundred and thirty-four. Ex. (7) 47012603 will be forty-seven millions, twelve thousand, six hundred and three. Ex. (8) 90,000,100,030,050 will be ninety billions, one hundred millions, thirty thousand, and fifty. The commas inserted here divide the digits into sets of 3, beginning from the right hand, and correspond with those in Ex. (5). (c). The consideration of any number expressed in figures will show that the value of a digit in a number depends on two things; first, the value attached to it independently of its position, as five to the figure 5, seven to 7, &c. and secondly, the value it acquires through the place it holds in a number; as in 327, the 2, being followed by another digit, means 20, the 3 means 300. The former of these is called the intrinsic value of the digit, the latter the local value. The common notation being obtained from the Arabs is known as the Arabic notation. 3.-A number used in reference to some particular unit, as 5 books, 21 horses, 13 minutes, is called a concrete number. But if deprived of its reference to any particular unit, as 5, 21, 13, it is said to be an abstract number. I used as an abstract number, is often spoken of as unity. Express in figures: Exercise 1. (1) Two thousand three hundred and seventy-five. (2) Four hundred thousand five hundred and thirty-one. (3) Eighty thousand nine hundred and ninety. Express in words :(4) 23716. (7) 904287643. (5) 829017. (6) 8014325. (8) 2010304000. (9) 12000004. Write down in figures: (10) Fifty thousand one hundred and three. (11) Two billions, thirty millions, one hundred thousand and forty-nine. (12) A million and sixty. (13) Thirteen thousand and one. (14) Two thousand four hundred and six millions, three hundred and ninety thousand seven hundred and seventy. (15) Ten millions, ten thousand and ten. (16) Twelve millions, one hundred and twenty thousand and twelve. Write down in words: (17) 97200102. Write in figures: (18) 326810012004. (19) Forty millions, ten thousand, seven hundred and six. (20) Eleven hundred and ten thousand and five. ADDITION. 4.-ADDITION is the operation of finding a number equal to two or more numbers taken together. The result is called the sum of the numbers. In the addition of numbers less than ro, the results, which at first are obtained by counting on from one number through as many numerals as are represented by the other, must be remembered and made perfectly familiar to the mind. When the numbers consist of more than one figure, they should be arranged in horizontal rows, one below the other, the units forming the vertical column, the tens another, &c.; then the units must be first added, then the tens, &c., as in the first of the following examples. If the result for any column exceed 9, the number of 10's in it must be carried on to the next column, the remaining units being set down. For instance, in the third column (from the right) of the second example below, the result is 29, of which the 9 is set down and the 2 carried on to the next column. The reason of this rule will be obvious, if it is remembered that every 10 tens becomes a hundred, which is expressed by 1 in the third column; every 10 hundreds becomes a thousand, i. e. 1 in the fourth column, &c. (7) Add together 27561, 2487, 46935, 367, and 2894. (8) Add together 341, 92768, 1004, 21238, 76, and 7428. (9) A traveller goes 132 miles by steamer, 93 by railway, 17 by coach, and 8 by carriage, how long was his journey? (10) If three towns contain respectively 21426, 45957, and 86025 inhabitants, what is their combined population? (11) An estate contains 746 oak trees, 957 elms, 83 beeches, 137 ash, 24 chesnut, 71 fruit trees, and 123 of other sorts: how many are there altogether? (12) Four estates are worth respectively 12245 pounds, 6274 pounds, 21290 pounds, and 11970 pounds, find the value of the whole. SUBTRACTION. 5.-SUBTRACTION is the operation of finding the number which results when a smaller number is taken from a larger. The result is called the remainder, or the difference between the two numbers. Subtraction is therefore the reverse of Addition, and amounts really to finding what number added to the smaller will produce the larger: and it is performed by means of this relation. Thus if 5 is to be subtracted from 7, we consider that 2 must be added to 5 to make 7, and therefore 2 is the difference required. The arrangement of numbers consisting of more than one digit is similar to that in Addition, the smaller being placed below the larger. The subtraction is then to be begun from the units. If the figure in the upper line in any column be less than that in the lower, 10 must be added to it before subtracting, and to counterbalance this, I must be added to the next figure to the left on the lower line. Thus in the second column of Ex. (2) below, 4 is less than 9; so 9 is subtracted from 14, leaving 5; and 1 being added to the 3, gives 4 to be subtracted from 6. So in the fourth column, 8 must be subtracted from 10; and then in the fifth 6 from 12. This addition of 10 is simply the removing (borrowing as it is generally called) of 1 from the next higher column, for use in the column we are considering; the local value making the I in any place equal to a 10 in the next lower place. Then since this 1, as well as the next lower figure, has to be taken from the next upper figure, the two are put together and then subtracted. It will be clear from what was said above, that the remainder, when added to the smaller number, will produce the larger. The number which is to be subtracted is sometimes called the subtrahend; and that which is to be diminished (i. e. from which the subtraction is made), the minuend. |