A THEORETICAL AND PRACTICAL ARITHMETIC. INTRODUCTION. SECTION I. 1. ARITHMETIC is the science of numbers. 2. Numbers express how many units, or parts of a unit, there are in any quantity. 3. Quantity is any thing that can be increased, diminished, or measured. 4. The least whole number employed to express or measure quantity of the same kind is called a unit. 5. A number expressing a particular kind of a unit is called a concrete number; as, 1 dollar, 2 books, &c. 6. When a number does not express any particular kind of a unit, it is called an abstract number; as, 1, 4, 7, &c. OBS. 1. Concrete numbers are also called denominate or compound numbers. OBS. 2. Whole numbers are sometimes called integers. 7. In the computation of numbers, ten characters are employed, called figures; thus: 1, one; 2, two; 3, What is arithmetic ? What are numbers? What is quantity? What is a unit? What is a concrete number? What is an abstract number? three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; 0, cipher. The first nine figures are called significant, because they have a given value assigned them. The cipher has no representative value, but is used where no number is to be expressed. OBS. 1. By the aid of these ten characters any possible or conceivable quantity may be expressed. OBS. 2. The first nine figures are sometimes called digits, from the Latin word digitus, signifying a finger. 8. The various operations of arithmetic are performed by NUMERATION, ADDITION, MULTIPLICATION, SUBTRACTION, and DIVISION. Addition and multiplication are employed to show how numbers may be increased; subtraction and division how they may be diminished. NUMERATION. SECTION II. 9. NUMERATION is the art of expressing any number whatever by figures. Figures are arranged in different orders or places, and have different values assigned them, according to the place they occupy. The first place, which is always at the right, represents units; the second, tens ; the third, hundreds; the fourth, thousands; the fifth, tens of thousands; the sixth, hundreds of thousands; the seventh, millions, &c. Thus the figure 1 represents a unit, a ten, a hundred, a thousand, &c., according to the place it occupies. In all places in which no number is to be expressed, ciphers must be written. Which are significant figures? How are the various operations of arithmetic performed? What is numeration? How are figures arranged? What does the first place represent? The second? The third? The fourth? The fifth? &c. When must ciphers be used? Thus, if any figure be written in the fourth place, or the place of thousands, a cipher must be written in the place of units, and in the place of tens, and in the place of hundreds; as, 7000. It is evident that the same figure represents in the second place a value ten times greater than in the first place, and in the third place ten times greater than in the second, and a hundred times greater than in the first; and by each removal of any figure to the next place on the left its representative value is increased ten times. 10. The different places in which figures are arranged may be divided into periods of three figures each. NUMERATION TABLE. Hundreds of quintillions. Hundreds of trillions. 6 4 9 8 7 6 5 4 8 7 1 9 8 7 6 5 4 3 6 0 1 Hundreds of billions. ∞ Hundreds of millions. Millions. Hundreds of thousands. Tens of thousands. co Thousands. • Hundreds. - Tens. Units. The number indicated in the above table is read thus: six hundred and forty-nine quintillions; eight What effect is produced on the representative value of a figure by changing its place? B hundred and seventy-six quadrillions; five hundred and forty-eight trillions; seven hundred and nineteen billions; eight hundred and seventy-six millions; five hundred and forty-three thousand; six hundred and 1. Twenty-four. 2. Two hundred and four. 3. Two hundred and forty. 4. Two thousand and four. 5. Two thousand and forty. 6. Forty-six thousand, five hundred and twenty. 7. Four hundred and six thousand, five hundred and two. 8. Eight hundred thousand, one hundred and one. 9. One million, one thousand, one hundred and one. 10. Ten millions, ten thousand, and ten. 11. One hundred millions, one hundred thousand, and one hundred. 12. Two millions, six hundred and ten thousand, four hundred and forty-six. 13. Sixty-four millions, nine hundred and ten. 14. Two hundred and forty millions, three thousand. 15. Five hundred and sixty-seven billions, three hundred and forty-eight millions, seven hundred and twenty thousand, six hundred and forty. 16. Fourteen trillions, six billions, three hundred and forty millions, and twenty-two. ADDITION. SECTION III. 13. ADDITION is the process of finding the sum of two or more numbers of the same kind. OBS. The sum expresses the total value of the several numbers, or as many units as there are in all of them. 14. RULE. Write the numbers under each other, units under units, tens under tens, hundreds under hundreds, &c. First, add the column of units, and write under this column the right hand figure of the sum, and add the remaining figure or figures to the next column. Add all the columns in the same manner, and under the last write the whole sum contained in it. PROOF. Beginning at the top of the column of units, add each column downwards; and if the result be the same as the first, the work is supposed to be right. 15. Two signs are often employed in addition; the one, +, called plus, which signifies added to, or and, and the other,=, called the sign of equality, which signifies equal to, or are: thus, 4+6= 10 is read, four and six are ten. 1. Add together the following numbers: — 2472 51856 27692 70347 81850 234217 The sum of the first column is 17; the 7 is written What is addition? What does the sum express? What signs are used in addition? Recite the rule. |