POUND SUBTRACTION; and the distinctions between these may be easily understood by referring to the distinctions already explained (see pp. 14 and 15) between SIMPLE, FRACTIONAL, and COMPOUND ADDITION, with which they are in exact correspondence. The two given quantities, of which one is to be subtracted from the other, must be of the same kind. Thus, while it is possible to subtract 3 shillings and 2 pence from £4-that is, money from money; or to subtract 7 feet from 4 yards—that is, length from length; it is not possible to subtract 3 shillings and 2 pence from 4 hours. The sign-, called minus, when set between two numbers, denotes that the one which follows it, is to be taken from the one which precedes it. Thus, 16-9-7, which is read 16 minus 9 (that is, 16 less by 9) equal to 7, denotes that if 9 be taken from 16, the remainder is 7. RULE FOR SIMPLE SUBTRACTION. (1.) Place the less number below the greater,* with units under units, tens under tens, &c., as in addition. (2.) Beginning with the units, take, if possible, each figure in the lower line, from the figure above it, and set down the remainder. (3.) But if any figure in the lower line be greater than the figure above it, add ten to the upper; then subtract as before, and carry one to the next figure in the lower line. Methods of Proof. 1. Add the remainder and the less of the given numbers together: if the sum be equal to the greater, the work is correct. 2. Subtract the number found, from the greater of the given numbers: if the remainder be equal to the less, the work is correct. Exam. 1. From 7854 take 4513. Set the numbers as in the margin, and proceed thus:-3 from 4, and 1 remains; 1 from 5, and 4 remain; 5 from 8, and 3 remain; 4 from 7, and 3 remain: the remainder, therefore, is 3341. To prove the work, to the less of the given numbers add the remainder, and the sum is 7854, the greater; or, as in the second method, subtract the remainder from the greater number, and the result is 4513, the less. 7854 4513 3341, remainder. 7854, proof. 7854 4513 3341, remainder. 4513, proof. *Though this is the usual method of placing the numbers, the greater is sometimes placed with advantage below the less. In this case, the words upper and lower must be interchanged throughout the rule. So likewise must above and below. 3712 1831 Exam. 2. Required the difference of 3712 and 1831. In this example proceed thus:-1 from 2, and 1 remains: 3 from 11, and 8 remain: carry 1 to 8; then 9 from 17, and 8 remain : carry 1; and then 2 from 3, and 1 remains. The difference, therefore, is 1881; and the operation would be proved in the same manner as before. When we thus add 10, it is commonly said that we borrow 10. Reason of the Rule. 1881, remainder. The rule for subtraction depends on the principle, that the differences of the several parts of two numbers are, when taken together, equal to the difference of the numbers themselves. The reason of placing units under units, tens under tens, &c., is, that figures may be subtracted from others of the corresponding local value with more facility. By carrying one to the lower figure, we increase the lower line as much as we increased the upper; and thus the difference is the same as if neither had been increased. Thus, in the second of the preceding examples, when in the tens' place we subtract 3 from 11, we add 10 to the 1 in the upper line; then the lower line is increased by the same value, by adding 1 to the 8; because, by the nature of notation, 1 in the third column is equivalent to 10 in the second. Thus, therefore, both the given numbers are equally increased, and consequently the difference must be the same as if they had received no increase. 57 26 As a further illustration of subtraction, let it be required to find the difference of 83 and 57. Here, as 7 cannot be taken from 3, we may consider 83 as equal to 70 and 13; and subtracting 7 from 13, and 5 from 7, we find the difference to be 26. In this simple and natural method, the values of the given numbers undergo no change; and, with only one exception, it might be employed with as much facility as the common method, the next figure in the upper line being always diminished by a unit, when one would be carried to the figure below it in the common method. The exception is the case in which the next figure in the upper line is 0. In this case the common method is considerably preferable; and, as in practice, that method is in no case inferior, it is universally preferred. 1. 45079-32048 4. 90401-58270 5. 623417-32686 6. 8463192-177825 7. 4444444-1234567 Exercises. 8. 915161718-151617189 9. 202122223-192021222 10. 357912468-24680135 11. 7503046571-34992884 12. 376995145-49490718 13. 153425178-53845248 14. 10000000010001001 15. From a piece of linen 85 yards long, a piece 57 yards long is cut off: find how much remains. Observe this is a case of sub tracting a quantity from a quantity, and the result is to be a quantity, not merely a number. 16. Subtract three thousand from three millions. 17 From a cistern which contained 251 gallons of water, 87 gallons have been drawn off: how much remains in it if none has gone away except those 87 gallons? 18. Required the difference between three, and three hundred thousand. 19. La Place, the celebrated French mathematician and philosopher, was born in 1749, and died in 1827: how long did he live? 20. Mont Blanc, the highest mountain in Europe, is 15,680 feet high; and the height of Chimborazo, the highest in America, is 21,400 feet: how much is the latter higher than the former? 21. The following are the years of the Christian era in which the undermentioned events happened: required the number of years from each till the year 1875. Commencement of the Hegira, or era of the flight of Mohammed, 622; The Arabian or modern notation in arithmetic, introduced from Arabia into Europe by the Saracens, 991; First Crusade, 1096; Magna Charta signed by King John, 1215; Linen first made in England, 1253; Termination of the Crusades, 1291; Gunpowder first used in Europe, 1330; Algebra introduced into Europe from Arabia, 1412; Printing invented, 1440; University of Glasgow founded, 1450; Constantinople taken by the Turks, 1453; America discovered by Columbus, 1492; Copernicus died, 1543; Spanish Armada destroyed, 1588; Telescopes invented, 1590; University of Dublin founded, 1591; Decimal fractions invented, 1602; Logarithms published by Napier, 1614; Barometer invented, 1643; Air Pump invented, 1654; Newtonian Philosophy published, 1686; Union of Great Britain and Ireland, 1801; Battle of Trafalgar, 1805; Battle of Waterloo, 1815; Catholic Emancipation Bill passed by the British Parliament, 1829; Negro Emancipation in the British Colonies, 1838; Corn Laws repealed, 1846; Close of the Crimean War, 1855; Battle of Solferino, 1859; Close of the American War, 1865; Atlantic Cable successfully laid, 1866. SIMPLE MULTIPLICATION. THE object of MULTIPLICATION, in the cases to which the term most obviously and properly applies, is to find the sum of a whole number or a fractional numerical expression or a quantity repeated a whole number of times but the name multiplication is extended further to include also cases in which a repetition of the thing is made a whole number of times and a fraction of it is superadded; and also to include cases in which the thing is not repeated at all, nor taken even once, but only a fraction of it is taken. In SIMPLE MULTIPLICATION, integers, or whole numbers, called also proper numbers, alone are dealt with; the treatment of fractional numerical expressions and of compound numerical expressions not falling within the scope of simple multiplication. We may thus notice that when quantities are to be multiplied any number of times, they must be expressed simply in one denomination, and must be without fractions, if the process is to be one in simple multiplication. The process is called COMPOUND MULTIPLICATION when any quantity expressed in more denominations than oneas, for instance, a quantity of money expressed in pounds, shillings, and pence-is to be conceived as repeated or multiplied a certain number of times, and the amount due to the repetition is to be found. The number or quantity to be repeated is called the MULTIPLICAND; the number which shows how often the multiplicand is to be repeated is termed the MULTIPLIER; and the number or quantity found is called the PRODUCT. Both the multiplicand and multiplier are sometimes called FACTORS, from their making or producing the product.* It will readily be observed that when the multiplier is a whole number, the process of multiplication comes to be merely an abridged method of performing addition in the case in which the numbers or quantities to be added together are equal to one another. An important principle which holds good for numbers in general, whether whole or fractional, may now be established for whole numbers. It is that the product of two numbers is the same whichever of them is taken as multiplier: that, for instance, 3 times 5 are the same as 5 times 3; that is, that 3 times 5 objects of any kind are the same in number as 5 times 3 objects of any kind. To illustrate this let the objects be represented by dots, and let three rows each containing five dots be placed as in the margin. We have thus 15 objects represented, which may be regarded either as 3 times 5 objects, when we take the three horizontal rows; or as 5 times 3 objects, when we take the five vertical *It is plain, from the definition of multiplication, that at least one of the two factors that is to say, that the multiplier, at least-must be a mere number, whole or fractional, not a quantity of any kind of thing; not an expression for money, weight, measure, or any other object. Thus, while we may multiply 5 shillings by 6-that is, repeat 5 shillings 6 times-it would evidently be absurd to speak of multiplying 5 shillings by 6 shillings-that is, of repeating 5 shillings 6 shillings times. Further remarks on this subject will be found in the article on Compound Multiplication. rows, each of which contains three dots; and a like illustration may be given in every case when whole numbers are dealt with. Fractional numbers and their multiplication will be treated of farther on in this book, but not at this early stage in what is called simple multiplication. The sign, called the sign of multiplication, placed between two numbers, denotes the multiplication of those numbers together: that is to say, the multiplication of either of them by the other. Thus, 20 × 12=240, which, for brevity, may be read, 20 into 12 equal to 240, denotes that the product of 20 and 12 is 240. When the sign of multiplication × is placed between a number and an expression for a quantity, it denotes the multiplication of the quantity by the number. Thus, 2 × £2 - 3 - 4, or £2-3-4 × 2, would denote £4-6-8. * Though the part of the multiplication table given in the text is enough for the pupil to commit to memory at first; yet, after he has made some proficiency in arithmetic, he may find it advantageous to commit what follows, as it will enable him, in many cases, to shorten his work in a considerable degree. The |