The sign X (into) indicates that one number is to be multiplied into another; thus, 4X3 is 12. The sign • (by) indicates that the nurnber on the left hand is to be divided by the number on the right hand; thus, 12-3 is 4. The sign=(equal to) indicates that the number before it, is equal to the number after it; for example, 4+2=6. 6-254. 5 X 3=15. 153=5. III. ADDITION. ADDITION is the operation by which two or more numbers are united in one number, called their sum. It is the first and most simple operation in arithmetic, effecting the first and most simple combination of quantities. The priinary mode of forming numbers, by joining one unit to another, and, this sum to another, and so on, exhibits the principle of addition. When numbers, which are to be added, consist of units of several degrees, such as tens, hundreds, &c., it is found convenient to add together the units of each degree by themselves; and since ten units of any degree make one unit of the next higher degree, the number of tens in the sum of each degree of units is carried to the next higher degree, and added thereto. * RULE FOR ADDITION. Write the numbers, units under units, tens under tens, &c. Add each column separately, beginning with the column of units. When the sum of any column is not more than 9, write it under the column: when the sum is more than 9, write only the units figure under the column, and carry the tens to the next column. Finally, write down the whole sum of the left hand column. 1. What is the sum of 370+90264+1470+40060? 2. What is the sum of 4000+570+99+54 +273+ 69073+4000+01998+752? 3. What is the sum of 243 +5021 +7628 +927 +64 +5823+742+796 + 5009 + 325 +7426 +31156 + 987+6954+2748 ? 4. What is the sum of two thousand and seven, forty four million five hundred and sixty-one, one hundred mil. lion, six billion twenty-eight thousand and eleven? IV. SUBTRACTION. SUBTRACTION is the operation by which one number is taken from another. The number from which another is to be taken is called the minuend, and the number to be taken is called the subtrahend. The number resulting from the operation shows the remainder of the minuend, after the subtrahend has been taken out; it also shows the difference between the minuend and subtrahend, or the excess of the former above the latter. The subtrahend and remainder may be considered the two parts into which the minuend is separated by the operation; and in this view, subtraction is the opposite of addition, in as much as addition unites several quantities in one sum, and subtraction separates à quantity into two parts. Subtraction is performed by taking the units of each degree in the subtrahend, from those of corresponding degree in the minuend, and severally denoting the remainders. When the units of any degree in the subtrahend exceed those of the same degree in the minuend, we mentally join one unit of the next higher degree to the deficient place in the minuend, and consider the units of the higher degree to be one less than they are denoted: this process is the reverse of carrying in addition. One other method may be adopted in this case; viz. Increase both the minuend and subtrahend, py mentally adding ten to the deficient place in the former, and, one to the next higher degree of units in the latter. This method is justified by the self-evident truth, that, if two unequal quantities be equally increased, their difference is not thereby altered. RULE FOR SUBTRACTION. Write the smaller number under the greater, placing units under units, &c. Begin with the units, and subtract each figure in the lower number from the figure over it. When a figure in the upper number is smaller than the figure under it, consider the upper figure to be 10 more than it is, and the next upper figure on the left hand, to be 1 less than it is. PROOF. Add together the remainder and the smaller number: their sum will be equal to the greater number, if the work be right. 1. What is the difference between 70240 and 69418? 2. How much is the excess of the number 482724 above the number 194750? 3. Suppose 479021 to be a minuend, and 38456 the subtrahend; how much is the remainder? 4. 905106392-904623724=? 5. Subtract fifty-one thousand from one hundred billion, eighteen thousand, five hundred and one. V. MULTIPLICATION. MULTIPLICATION is the operation by which a number is produced, equal to as many times one given number, as there are units in another given number. It is an abridged method of finding the sum of several equal quantities, by repeating one of those quantities. The number to be multiplied or repeated is called the multiplicand; it may be viewed as one of several equal quantities, whose sum is to be produced by the operation. The number to multiply by is called the multiplier; it indicates how many such quantities as the multiplicand are to be united, or, how many times the multiplicand is to be repeated. The number resulting from the operation is called the product. The multiplicand and multiplier, considered as concurring to form the product, are called factors of the product. Either factor may be used as the multiplier of the other; that is, the multiplicand and multiplier may change places, and the product will be still the same. For example, 4 X3=12. 3X4=12. When a product arises from more than two factors, he numbers may be denoted thus, 6 X3 X5=90; but, n forming the product, a distinct operation is necessary to pring in each factor, after the first two. The numbers, 6, 3, 5, would, therefore, be multiplied into each other thus, 6 X3=18; 13 X5=90. Factors may be arranged in any succession whatever, since the mere order in which they are brought into the operation cannot affect their final product. For example, 5 X3 X4=60. 4X3 X5=60. 3 X5 X4460 The products of small numbers may be committed to memory; but when the product of factors consisting of several figures is required, it is necessary to multiply each figure in the multiplicand by each figure in the multiplier, and denote the several products in such order that they shall represent their respective values. When siniple units are employed as the multiplier, the product of each figure in the multiplicand is of the same degree as the figure multiplied; that is, units multiplying units give units, units multiplying tens give tens, units multi plying hundreds give hundreds, &c. When tens are employed as the multiplier, the product of each figure in the multiplicand is one degree higher than the figure multiplied; that is, tens multiplying units give tens, tens multiplying tens give hundreds, tens multiplying hundreds give thousands, &c. When hundreds are employed as the multiplier, the product of each figure in the multiplicand is two degrees higher than the figure multiplied; and so on. RULE FOR MULTIPLICATION. Write the multiplier under the multiplicand, placing units under units, &c. When there is but one figure in the multiplier, begin rith the units, multiply each figure in the multiplicand separately, and place each product under the figure in the multiplicand from which it arose; observing to carry the tens to the left as in addition. When there is more than one figure in the multiplier, multiply by each figure separately, and write its product in a separate line, placing the right hand figure of each line under the figure by which you multiply; and finally, add together the several products. The sum will be the whole product. Abbreviations of the above rule may frequently be adopted, as follows. When there are ciphers standing between other figures, in the multiplier, they may be disregarded. When ciphers stand on the right of either factor, or both, they may be disregarded till the multiplication is performed, and then annexed to the product. When either factor is 10, 100, 1000, &c., merely place the ciphers in this factor on the right hand of the other factor, and it becomes the product. When the multiplier is a number that can be produced by multiplying two smaller numbers together, multiply the multiplicand first by one of the smaller numbers, and the product thence arising by the other. 1. Suppose 479265 to be a multiplicand, and 9236 the multiplier; how much is the product ? 2. Suppose 26537 to be one factor, and 873643 another; how much is their product ? 3. Suppose the numbers 725, 38046 and 91, to be factors; how much is the product ? 4. What is the product of 62392 X 4003 ? 5. What is the product of 248000 X 9400 ? What is the product of 24 X 300 X13X10002? 7. Multiply one hundred five million, by one thousand. For the purpose of determining whether any error has happened in the process of multiplication, the following method of trial, which depends on the peculiar property of the number 9, and which is called casting out the nines, may be practised. Add together the figures of the product, horizontally, |