This is true also of each of the following lines, which, as well as the former line, may be extended to any number of places in either direction. 2222.22 &c. 4444.44 &c. 3333-33 &c. 5555.55 &c. 7777·77 &c. 9999.99 &c. of 20. 20 of 200 200 of 2000 and so on. = = tenfold 02 02 = And, 20 tenfold 2 200. = tenfold 002 2000 tenfold 200- " The other lines may be proved in the same way. Ten, therefore, may be called the governing number of our method of arithmetical notation, which, for this reason, is called the Decimal System of Notation. The word "decimal" is derived from the Latin word "decem," ten. PRINCIPLE IV. In the decimal system of arithmetical notation, each digit has one tenth of the value it would have in the next place to the left, and tenfold the value it would have in the next place to the right. From this principle we derive the two rules for carrying, the use of which rules will be shewn hereafter. First rule for carrying. In decimal arithmetic, every ten of any place may be carried as one to the next place to the left. Second rule for carrying. In decimal arithmetic, every one of any place may be carried as ten to the next place to the right. Decimal Arithmetic is not the Arithmetic of Decimal Fractions only, but also that of Integers, when they are expressed by the Decimal System of Notation. ELEMENTARY PROCESSES OF ARITHMETIC. Numbers can be changed in only two ways. They can be increased, and they can be diminished. The increasing of a number, by adding or joining one or more other numbers to it, is called Addition. When equal numbers are added together, that kind of addition is called Multiplication. The diminishing of a number, by taking a smaller number out of it, is called Subtraction. The distribution of a number into its parts is called Division. Addition is the uniting of two or more numbers into one. Multiplication is the addition or repetition of equal numbers. Subtraction is the taking of a smaller number out of a larger one. Division is the distribution of a number into its parts. ADDITION. Addition is the increasing of a number by uniting one or more other numbers with it. Arithmetical Symbol. The sign of addition is a cross, composed of two equal lines, one vertical and the other horizontal, intersecting each other in the middle, thus: +; it is called, in reading, "plus," which is the Latin word for "more.' Thus, the expression 6+2+1 is read "six plus two plus one;" that is to say, "six and two more and one more;" and it means that 6 and 2 and 1 are to be added together, or collected into one number, namely, 9. Sum. The sum of numbers is found by adding them together. 21 Find the sum of 3. & 9. & 5. & 7. & 0. 13. & 6. & 14. of 8 & 10 & 10 & 10 & 10. & 10 & 5. & 2. 22. of 9. & 12. 23. 24. of 14 & 13 EXAMPLES IN CONCRETE NUMBERS. 35. Add 9. apples, 3. apples, 7 apples, and 12 apples. 36. Add 13 pence, 11 pence, 6. pence, and 2 pence. 37. Add of 8 pence, of 18. pence, and of 12. pence. 38. Henry had 10 shillings, Amelia had half as much as Henry, and Julia had 3 shillings more than Amelia; how much had all three? 39. The tail of Tom's kite was 30 feet long, the kite was of the length of the tail, and the string was 110 feet long; what was the length of kite, tail, and string together? 40 What is the joint weight of 7 pounds of coffee, 4 pounds of tea, 9 pounds of cheese, 2 pounds of currants, and 8 pounds of soap? 41 Little Mary's market-basket weighs 5 pounds when empty, what weight had she to carry when her basket contained 5. pounds of mutton, 6 pounds of flour, 2 pounds of treacle in a jug weighing 1 pound, and half a six-pound Dutch cheese? ADDITION OF NUMBERS TOO LARGE TO BE ADDED MENTALLY. First. Arrange the numbers carefully in a vertical column, so that the decimal point in each line shall stand directly under the decimal point in the line above; units being under units, tens under tens, hundreds under hundreds, tenths under tenths, hundredths under hundredths, &c. Draw a horizontal line under the column of numbers, and place a decimal point below it, in a line with those above it. Second. Find the sum of the digits in the right-hand place. Third. Carry one for every ten of this sum to the next place to the left. Set down the rest of the sum under its own place. Fourth. Proceed to the next place to the left. its digits and the number carried. for every ten, and set down the rest. Add Carry to the left, one *NOTE TO TEACHER. The pupil is supposed to find these fractions of the Integers by trial with strokes, or peas, or other units. Fifth. Work every column similarly. What is carried from the last column must be set on the left of the sum.* EXAMPLE WORKED OUT. INTEGERS ONLY. Find the sum of 86143. 9459. 574 and 74298. These numbers, when arranged, stand thus: 86143. 9459. 574. 74298. Then, sum of right-hand column = (8+4+9+ 3) units; 24 units, 2 tens and 4 units. " " Set down 4 (units) and carry 2 (tens) to tens' column. Sum of tens' column with 2 tens carried(2+9+7+5+4) = tens; 27 tens, 2 hundreds and 7 tens. = Set down 7 (tens) and carry 2 (hundreds) to hundreds' column. Sum of hundreds' column with 2 hundreds carried (2+2+5+ 4 + 1) hundreds; 14 hundreds, = 1 thousand and 4 hundreds. - Set down 4 (hundreds) and carry 1 (thousand) to thousands' Sum of thousands' column with 1 thousand carried == 20 thousands, 2 ten-thousands. Set down 0 (thousands) and carry 2 (ten-thousands) to tenthousands' column. Sum of ten-thousands' column with 2 (ten-thousands) carried (278) ten-thousands; = 17 ten-thousands, thousands. = 1 hundred-thousand and 7 ten Set down 7 (ten-thousands) and carry 1 (hundred-thousand) As there are no digits in hundred-thousands' place to be added, NOTE TO TEACHER. The word "sum" should never be employed in any other than this its true sense. A slovenly practice exists of designating as "sums" arithmetical operations of every kind. We hear of "sums in addition," sums in division," and even of "subtraction sums"! This practice should be abandoned, and instead of the loose phrase, "doing a sum," the pupil should be trained to use the correct expression," working an example." Nomenclature furnishes potent aid in the pursuit of any science, but to do so it must be both clearly understood and accurately applied. The example is now completed, and stands thus: EXAMPLES FOR PRACTICE, INTEGERS ONLY; the num bers being already arranged. Find the sum of each of the following sets of numbers. Ex. 1. 314. Method of Proof. After working an example in addition of Integers, the accuracy of the work may be tested by taking these two steps. First. Omit one of the numbers, and find the sum of the others. Second. Add this last-found sum to the number omitted. The number thus found is evidently the sum of all the numbers, and should be the same as that found by the first addition. |