PRACTICAL ARITHMETIC, DEFINITIONS. 1. Arithmetic is that branch of mathematics which treats of quantity and its relations, when expressed by number. 2. Mathematics is the science of quantity, and embraces arithmetic, algebra, geometry, etc. 3. Quantity is anything which may be measured; as, distance, weight, time. 4. A number is either a single unit, or a collection of units; as, 1, 2, 5, 95; 4 men, 7 dollars, 9 houses. 5. A unit is a single thing; as, 1, 1 man, 1 book. 6. A number is abstract or concrete. 7. An abstract number is a collection of abstract units; as, 4, 7. 8. An abstract unit is simply one; as, 1. 9. A concrete unit denotes the kind of thing expressed; as, 1 man, 3 cents, 5 apples. FUNDAMENTAL RULES. 10. The fundamental rules of arithmetic are Notation and Numeration, Addition, Subtraction, Multiplication, Division. 11. All the operations of arithmetic are based upon one or more of these fundamental rules. NOTATION AND NUMERATION. 12. Notation is the art of writing numbers. 13. In arithmetic numbers are expressed by means of certain characters. These characters are either figures or letters. 14. Two systems of notation are sometimes used: the Arabic, and the Roman. 15. In the Arabic system numbers are expressed by means of the following characters, called digits : one. two. three. four. five. six. seven. eight. nine. naught. 1 2 3 4 5 6 7 8 9 0 16. The first nine digits are significant figures, because they express value when standing alone. 17. Naught expresses no value when standing alone; but when used in connection with significant figures it performs an important office in the Arabic system. 18. The Arabic system of notation is based on a scale of TEN; that is, ten units of one order make one unit of the next higher order throughout this system. 19. The orders increase in value from units' place toward the left, each order consisting of one figure. Illustration. In the number 729, The figure 9 belongs to the order of units; The figure 7 belongs to the order of hundreds. 20. A period consists of three orders, except the left hand period, which often contains only two orders, or even one. 21. Each significant figure may have a simple value, or a local value. 22. The simple value of a figure is the value it expresses when standing alone, or in units' place; as, 7, 9, 4. 23. The local value of a figure is the value it expresses when used in connection with other figures to express a number. This number is read, Seven hundred and forty-two sextillion, thirty-six quintillion, one hundred and seventy-five quadrillion, eight hundred and nine trillion, two hundred and thirteen billion, four hundred and fifty-six million, three hundred and forty-five thousand, seven hundred and eighty-nine. 24 a. The value expressed by any number depends upon two things: 1st. Upon the number of units expressed by each digit. 2d. Upon the position of each digit with reference to units' place. EXERCISES. 1. Write five units of the first, or units order. 2. Write eight units of the second, or tens order. 3. Write five units of the second and eight units of the first order. 4. Write six units of the third, or order of hundreds. 5. Write eight hundred, five tens, and six units. 6. Write fifty-eight; eighty-five. 7. Write four hundred and seventy-five. 8. Write four hundred and five. 9. Write forty-five. RULE FOR WRITING NUMBERS. RULE.Write the figure expressing the number of units in the place expressing the given order. cant orders with ciphers. Fill va 1. Four in the second order expresses how many units of the first order? 2. Write four units of the second order. 3. Five units of the third expresses how many units of the first? Of the second? 4. Write five units of the third order. Of the second order. Of the first. 5. Write five units of the first, second, and third, as one number. 6. Write seven units of the third, and two of the first, as one number. In this case, what character Why? fills the second order? 7. Name the eight periods given in the scale. 8. Name each order in each period. 9. Name the several orders consecutively, from the order of units to the order of sextillions. 10. How many orders are required to express hundred-thousands? 11. How many orders to express millions? How many periods? 12. How many orders to express tens of millions? Tens of billions? 13. How many significant figures are required to express eleven hundred million, seven thousand and twenty-three? What orders do they occupy? What orders are filled with ciphers? 14. Which orders are filled with ciphers, and which with significant figures, in writing forty-nine hundred thousand and seventy-seven? 15. Write thirty-two million, seventy-seven thousand and twenty-two. |