to this is 100, which represents ten tens of units, or one hundred of units, together with no tens of units, together with no units; or, as it is briefly read, one hundred. By pursuing the same system in higher numbers the figure occupying the fourth place from the right hand will represent so many tens of hundreds of units, or thousands of units; the figure in the fifth place will represent so many tens of thousands of units; and so on. 205 represents two hundreds of units, together with no tens of units, together with five units; or, as it is briefly read, two hundred and five. 5473 represents five thousands of units, together with four hundreds of units, together with seven tens of units, together with three units; or, as it is briefly read, five thousand, four hundred and seventy-three. 7040730 represents seven millions of units, together with no hundreds of thousands of units, together with four tens of thousands of units, together with no thousands of units, together with seven hundreds of units, together with three tens of units, together with no units; or, as it is briefly read, seven millions, forty thousand, seven hundred and thirty. 107834265 represents one hundred of millions of units, together with no tens of millions of units, together with seven millions of units, together with eight hundreds of thousands of units, together with three tens of thousands of units, together with four thousands of units, together with two hundreds of units, together with six tens of units, together with five units; or, as it is briefly read, one hundred and seven millions, eight hundred and thirty-four thousand, two hundred and sixty-five. 8. NOTATION is the art of expressing any number by figures which is already given in words. NUMERATION is the converse of Notation, being the art of expressing any number in words which is already given in figures. 9. The method above explained of denoting numbers by means of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and combinations of them, was brought into Europe by the Arabs, and it is therefore often called the ARABIC NOTATION. It was derived by the Arabs from the Hindoos. This method of notation is now in common use, not only in this country, but throughout Europe. Ex. 1. Exercises in Notation and Numeration. Express the following numbers in figures: (1) Sixty-three; eighty-one; ninety-nine; forty; thirteen. (2) Two hundred; three hundred and three; seven hundred and sixty-four; eight hundred and eighty-eight. (3) Four thousand; one thousand, four hundred and seventy-one ; six thousand, nine hundred and thirty; nine thousand and nine. (4) Twenty-seven thousand, five hundred and four; thirty-three thousand; nine thousand and sixteen. (5) One hundred thousand; six hundred and seventy-six thousand and fifty; two hundred and two thousand, five hundred and ninety*three. (6) Seven millions, three thousand; eleven millions, one hundred and eight thousand, one hundred and six; fifty-four millions, fifty-four thousand and eighty-eight; six hundred and thirteen millions, twenty thousand, three hundred and three. (7) Two billions; nine billions, three hundred thousand and twentyone; ninety-four billions, ninety millions, ninety-four thousand, nine hundred and four. Write down in words at full length the following numbers: (1) 43; 60; 88; 97; 59; 12; 21; 19. (2) 256; 401; 500; 999; 365; 578; 837. ADDITION. 10. ADDITION is the method of finding a number, which is equal to two or more numbers taken together. The number found by adding two or more numbers together is called the SUM or AMOUNT of the several numbers so added. 11. There are two kinds of Addition, SIMPLE and COMPOUND. It is Simple Addition, when the numbers to be taken together are all abstract numbers; or when they are all concrete numbers of the same denomination, as all pence, all days, all pints. It is Compound Addition, when the numbers to be taken together are concrete numbers of the same kind, but of different denominations of that kind; as pounds, shillings, and pence; or years, months, and days; or gallons, quarts, and pints. 12. The sign+, PLUS, placed between two or more numbers, signifies that the numbers are to be added together: thus 2+5+7 signifies that 2, 5, and 7 are to be added together, and denotes their sum. The sign =, EQUAL, placed between two numbers, signifies that the numbers are equal to one another. The sign VINCULUM, placed over numbers, and the sign () or { }, called a BRACKET, enclosing numbers within it, are used to denote that all numbers under the vinculum, or within the bracket, are equally affected by all numbers not under the vinculum or within the bracket: thus 2+3 or (2+3) or (2+3), each signify, that whatsoever is outside the vinculum or bracket which affects 2 in any way, must also affect 3 in the same way, and conversely. The sign.. signifies 'therefore.' SIMPLE ADDITION. 13. RULE. Write down the given numbers under each other, so that units may come under units, tens under tens, hundreds under hundreds, and so on; then draw a straight line under the lowest line. Find the sum of the column of units; if it be under ten, write it down under the column of units, below the line just drawn; if it exceed ten, then write down the last figure of the sum under the column of units, and carry to the next column the remaining figure or figures; treat each succeeding column in the same way, and write down the full sum of the extreme left-hand column. The entire sum so marked down will be the sum or amount of the separate numbers. 14. Add together 5469, 743, and 27. Proceeding by the Rule given above, we obtain 5469 743 27 6239 The reason for the Rule will appear from the following considerations. When we take the sum of 7 units and 3 units and 9 units, we get 10 6 units and 9 units, or 19 units; we therefore place the 9 units under the column of units and carry on the 1 ten units to the next column, viz. the column of tens. Now the sum of 1 ten, 2 tens, 4 tens, and 6 tens, is 10 tens and 3 tens, or 13 tens; we therefore place the 3 tens under the column of tens and carry on the 1 hundred units to the next column, viz. the column of hundreds. Again, the sum of 1 hundred, 7 hundreds, and 4 hundreds, is 10 hundreds and 2 hundreds, or 12 hundreds; we therefore place the 2 hundreds under the column of hundreds, and carry on the 1 thousand units to the next column, viz. the column of thousands. Again, the sum of 1 thousand and 5 thousands, is 6 thousands; we therefore place the 6 under the column of thousands, and the entire sum is 6239. 15. The above example might have been worked thus, putting down at full length the local value of all the figures. Now adding the columns, we get the sum =5000+1100+120+19 = 5000+1000+100+100+20+10+9, (since 1100=1000+100, 120=100+20, and 19=10+9) =6000+200+30+9, (collecting the thousands together, the hundreds together, and so on) =6239. Note. The truth of all results in Addition may be proved by adding the columns first upwards as in the above example, and then adding them downwards; if the results be the same, the operation in each case will in all probability have been performed correctly. (17) Add together 7384, 326, 6780, and 57; also 6740, 9745, 5769, 8031, 6543, 2002, and 9999; also 89, 4500, 423, 2024, 5408, 60546, and 9401. (18) Add together 83746, 2478, 692577, 456, and 7; also 935473, 262, 13897, 598453, 25, 3734, 724008, and 649768. (19) Find the sum of 4738685, 237869513, 148794343978, 865, 4647, and 250; also of 68539582, 78602045, 370489000, 7055591234, 276, 9123456789, and 5000: also of 888929944, 73600, 27978462, 333, 5875396006, 4827532, 486684836, 80032148379, 12345, 1112858673, and 53800000835. (20) Add together one thousand, four hundred and eighty-three; seven hundred and ninety-six; thirty-nine; forty thousand, seven hundred and forty-four; five thousand, eight hundred and sixty; fifty thousand and seven. (21) Add together the following numbers: fifteen thousand, seven hundred and ninety-six ; four hundred and nine; two hundred and thirtyfour thousand and fifty; four millions, three thousand and seventy-six ; forty thousand and thirty-six ; ten thousand, nine hundred and one. (22) Add together the following numbers: twenty-two millions, six hundred thousand, five hundred and three; five hundred and sixty-three |