PROGRESSIVE ARITHMETIC FOR ELEMENTARY SCHOOLS, PART I. ADAPTED TO STANDARD I. OF THE NEW CODE, 1871. Simple Addition and Subtraction of Numbers of not more than Four Figures, and the Multiplication Table to Multiplication by Six. ABOUT THE FIGURES. Arithmetic teaches us how to calculate or reckon by means of numbers. To express numbers we make use of certain signs or marks, called figures or digits. The figures or digits used in Arithmetic are ten in number, and are printed thus : I 2 3 4 5 6 7 8 9 Nought, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. In writing the figures must not be made like printed figures, but should be written thus : О 1 2 3 4 5 6 7 8 9 Nought, One, Two, Three, Four, Five, Six, Seven, Eight, Nine. The number nine or any less number is expressed by only one figure, but all numbers greater than nine must be expressed by two or more figures taken together. The ten figures or digits used in Arithmetic are, however, sufficient to express any number, however great it may be. NOTATION AND NUMERATION. I. OF ONE AND TWO FIGURES. The art of writing a number in figures is called Notation. The art of reading, or expressing in words, a number written in figures, is called Numeration. The sign o, which is called nought, and sometimes cipher, or zero, signifies nothing by itself. Its use when standing together with other figures, will be shewn hereafter. Any figure standing by itself, with the exception of o, expresses so many units or ones: thus : I expresses only one 2 3 6 expresses six ones nine ones 4 The first step in learning Arithmetic is to understand Notation and Numeration of one figure, that is, to know how to write single figures or units in words, and to express the word-names of units in figures. Ex. 1.-Numeration of One Figure. numbers in words : Write the following lines of I-I, 2, 3, 4, 5, 6, 7, 8, 9, 7-3, 0, 7, 4, 2, 8, 1, 9, 5, 6 2—2, 1, 3, 5, 4, 7, 6, 0, 9, 8 8-8, 6, 5, 0, 3, 2, 7, 1, 9, 4 3-5, 3, 7, 9, 0, 1, 2, 8, 4, 9-4, 6, 2, 8, 0, 9, 7, 5, 3, I 4-0, 1, 3, 9, 2, 6, 8, 7, 5, 4 10-9, 0, 8, 1, 7, 2, 6, 3, 4, 5 5-7, 6, 0, 2, 8, 5, 1, 3, 4, 9 11-6, 2, 5, 4, 3, 0, 7, 8, 1, 9 -6, 5, 3, 9, 0, 2, 7, 4, 1, 812-9, 8, 7, 6, 5, 4, 3, 2, I, O Ex. 2.-Notation of One Figure. Express the following words in figures :I-One, two, three, four, five, six, seven, eight, nine, nought. 2-Nought, two, four, six, eight, one, three, five, seven, nine. 3-Six, five, seven, four, eight, three, nine, two, nought, one. 4-Four, nine, seven, six, nought, three, one, five, eight, two. 5-Nine, one, three, two, eight, nought, seven, four, five, six. 6-Five, nought, two, seven, four, nine, three, six, one, eight, We can now take another step, which brings us to Numeration and Notation of Two Figures. The next number to nine is ten. This cannot be expressed in writing by a single figure. It requires two figures taken together to express it, and it is written thus,-10. The figures I and o thus placed together signify ten units, or one ten. From this we learn that the figure o, which stands for nought, or is of no value when standing singly, increases the value of I, or one, ten times, when placed to the right of it. We learnt that a figure standing by itself expresses so many units or ones. We now learn further that a figure standing to the right of a line of two or more figures, expresses so many units or ones, while a figure standing in the second place from the right expresses so many tens of units, called for the sake of shortness, tens. So, as 10 expresses one ten, or ten units,— 20 exp. two tens or twenty units. | 60 exp. three tens or thirty units. 40 four tens or forty units. 50 five tens or fifty units. six tens or sixty units. seven tens or seventy units. eight tens or eighty units. 90 nine tens or ninety units. وو By placing the units 1, 2, 3, 4, 5, 6, 7, 8, 9, in the place of the o or nought, in 10, we get the numbers which come in succession between ten, or 10, and 20, or twenty, namely : 11-eleven, or one ten and one. 12-twelve, or one ten and two. 13-thirteen, or one ten and three. 14-fourteen, or one ten and four. 15-fifteen, or one ten and five. 16-sixteen, or one ten and six. seven. 18-eighteen, or one ten and eight. 19-nineteen, or one ten and nine. In the same way by placing the units 1, 2, 3, 4, 5, 6, 7, 8, 9 in the place of the o or nought, in 20, 30, 40, 50, 60, 70, 80, and 90, we get the numbers which come in succession between 20 and 30, 30 and 40, 40 and 50, 50 and 60, 60 and 70, 70 and 80, 80 and 90, and thus learn to express any number of two figures from 20 to 99 in figures and words, as follows: |