10 11 12 13 14 15 16 17 18 19 11 12 13 14 15 16 17 18 19 20 14 15 16 17 18 19 20 21 13 14 15 16 117 18 19 20 21 14 15 16 17 18 19 20 21 22 15 16 17 18 19 20 21 22 23 24 16 17 18 19 20 21 22 23 24 25 117 18 19 20 21 22 23 24 25 26 18 19 20 21 22 23 24 25 26 27 23 24 25 26 27 28 20 21 92 21 22 22 23 24 25 26 27 28 29 29 30 The square made by the ten Arabic characters forms an Addition and Subtraction Table. Beginning with the first line, thus, Zero and zero are zero; zero and 1 are 1; zero and 2 are 2; zero and 3 are 3; zero and 4 are 4; zero and 5 are 5; zero and 6 are 6, etc. The second line, one and zero are one ; 1 and 1 are 2: 1 and 2 are 3 ; 1 and 3 are 4; 1 and 4 are 5, etc. 2 and zero are 2 ; 2 and 1 are 3 ; 2 and 2 are 4; 2 and 3 are 5; 3 and 4 are 7, etc. 3 and 0 are 3; 3 and 1 are 4; 3 and 2 are 5; 3 and 3 are 6; 3 and 4 are 7, ; etc. Continue this, taking the first figure of the 1st column and adding it to each successive figure in the first line ; the adding of zero is only nominal, as it makes no increase. It also becomes a subtraction table, the figures of the first column being the subtrahend, and those of the first line the remainders. Take zero from 1 and 1 remains ; 0 from 2, 2 remain, etc. It may be thus expressed: 1–0=1; 2 – 0 = 2; 3 -0= 3; 4 – 0 = 4; 5 – 0 = 5; which is read, 1 minus zero equals 1, etc. Second line: take 1 from 2, 1 remains; 1 from 3, 2 remain; or, 2 – 1=1; 3-1= 2; 4-1= 3; 5 -1= 4, etc. Third line; 3 – 2 = 1; 4-2= 2; 5 - 2 = 3; 6-2 = 4; 1—2 = 5, eta In addition, we add two numbers at a time, never more, and in the first square we have the addition of every two units that can come together; so also in subtraction. = = = In the second square, the units correspond with the first square, and have an additional ten. In the third square, the units again are repeated, and another additional ten. As a column of tens, hundreds, and every higher or lower order is added and subtracted in the same way, the above table develops every principle of addition and subtraction. 20 Add the column of units. 1 3 One and 2 are 3 ; 3 and 3 are 6; 2 5 6 and 4 are 10; 10 and 5 are 15; 15 3 6 and 6 are 21 ; 21 and 7 are 28; or as 4 4 is customary to begin at the bottom 5 2 of the column, 7 and 6 are 13; 13 6 영 and 5 are 18; 18 and 4 are 22;.22 my 9 and 3 are 25; 25 and 2 are 27; 27 28 36 and 1 are 28. ; REM.-Although many numbers may be added together, in performing the operation only two at a time are added. Add the following numbers jointly and separately; thus, 35 30 and 5 24 20 4 43 40 3 52 50 2 21 67 60 ry 200 221 200 21 = 221 The sum of the column of units is 21; that is, 1 unit and 2 tens; the sum of the column of tens is 20; that is, 20 tens or 2 hundred; and the two sums united make 66 221; precisely the same as if the column of units is first added, and the units of the sum placed under the column of units, and the tens added with the colump of tens; and then the tens of the sum of the tens column placed under the column of tens, and the hundreds in place of hundreds. COR.-As the relation of each successive order is the same, hence for every ten of any order, the 1, or lefthand figure, belongs to the next order; and the process is the same in the addition of every column; that is, one l is carried to the next column for every ten in the addi. tion of each column. Minuend, 9820110 6354789 Minuend, 9820110 3465321 Cor. 1.- The minuend is always equal to the sum of the subtrahend and remainder, and is therefore greater than either. COR. 2.–Arithmetic is based upon the postulate contained in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, which is addition, etc.; and the application of Axiom 6 (page 9) to this postulate proves the principle of subtraction; thus, 19 +6= 25, then 25 – 6 must be equal to 19. REM. 1. - When the figure of the subtrahend is larger than the one above it of the same order of the minuend, 1 of the next order of the minuend must be united to the figure of the minuend and then the subtraction be performed; then in order to make up for this addition to the minuend, 1 mnst be added to the next order of the subtrahend, and then perform the subtraction ; this process is called carrying and requires all the attention of the student. REM. 2.-I prefer few examples, but these may be often repeated, and if thought necessary the teacher can give others in which the columns are longer. REM. 3.- Each order may be regarded as units, and the sum may reach one, two, or more hundred of its order. QUESTIONS, 1. When the sum of the column of units is 157, where do you place the 7, and what do you do with the 15 ? 2. When the sum of the column of tens and of the tens carried from the column of units is 246, what do you do with the 6, and what with the 24 ? |