REVIEW. 112. 1. What are the fundamental operations in this section? Ans. Addition and Subtraction. 2. What relation have Multiplication and Division to these? (83, 101) 3. When two or more numbers are given, how do you find their sum? 4. What is the method of performing the operation?(81) 5. When the given numbers are all equal, what shorter method is there of finding their sum? (83) 6. How is Multiplication performed?(88) 7. What are the given numbers employed in Multiplication called? (87) 8. What is the result of the operation called?(87) 9. How would you find the difference between two numbers?(94) 10. By what names would you call the two numbers?(98) 11. What is the difference called? 12. If the minuend and subtrahend were given, how would you find the remainder? 13. If the minuend and remainder were given, how would you find the subtrahend? 14. If the subtrahend and remainder were given, how would you find the minuend? 15. If the sum of two numbers, and one of them were given, how would you find the other? 16. If the greater of two numbers and their difference be given, how would you find the less? 17. If the less of two numbers and their difference be given, how would you find the greater? 18. How would you find how many times one number is contained in another? 19. By what name would you call the number divided?[105] 20. What would you call the other number? 21. By what name would you call the result of the operation? 22. Where there is a part of the dividend left after performing the operation, what is it called? 23. How can you denote the division of this remainder?[108] 24. If the divisor and dividend were given, how would you find the quotient? 25. If the dividend and quotient were given, how would you find the divisor? 26. If the divisor and quotient were given, how would you find the dividend? 27. If the multiplicand and multiplier were given, how would you find the product? 28. If the multiplicand and product were given, how would you find the multiplier? 29. If the multiplier and product were given, how would you find the multiplicand? 30. When the price of an article is given, how do you find the price of a number of articles of the same kind?[83] 31. Does the proof of an arithmetical operation demonstrate its correctness?[82] What then is ite use? NOTE. The definitions of such of the following terms as have not been already explained, may be found in a dictionary. What is Arithmetic? What is a Science? Number? Notation? Numeration? Quantity? Question? Rule? Answer? Proof? Principle? Illustration? Explanation? DECIMALS and fedERAL MONEY. DECIMALS. 113. The method of forming numbers, and of expressing them by figures, has been fully explained in the articles on Numeration. (71, 72, 73) But it frequently happens that we have occasion to express quantities, which are less than the one fixed upon for unity. Should we make the foot, for instance, our unit measure, we should often have occasion to express distances which are parts of a foot. This has ordinarily been done by dividing the foot into 12 equal parts, called inches, and each of these again into 3 equal parts, called barley corns. (38) But divisions of this nature, which are not conformable to the general law of Notation, (73) necessarily embarrass calculations, and also encumber books and the memories of pupils, with a great number of irregular and perplexing tables. Now, if the foot, instead of being divided into 12 parts, be divided into 10 parts, or tenths of a foot, and each of these again into 10 parts, which would be tenths of tenths or hundredths of a foot, and so on to any extent found necessary, making the parts 10 times smaller at each division; then in recomposing the larger divisions from the smaller, 10 of the smaller would be required to make one of the next larger, and so on, precisely as in whole numbers. Hence, figures expressing tenths, hundredths, thousandths, &c. may be written towards the right from the place of units, in the same manner that tens, hundreds, thousands, &c. are ranged towards the left; and as the law of increase towards the left, and of decrease towards the right, is the same, those figures which express parts of a unit may obviously be managed precisely in the same manner as those which denote integers, or whole numbers. But to prevent confusion, it is customary to separate the figures expressing parts from the integers by a point, called a separatrix. The points used for this purpose are the period and the comma, the former of which is adopted in this work; thus to express 12 feet and 3 tenths of a foot, we write 12.3 ft. for 8 feet and 46 hundredths, 8.46 feet. DEFINITIONS. 114. Numbers which diminish in value, from the place of units towards the right hand, in a ten fold proportion, (as described in the preceding article,) are called Decimals. Numbers which are made up of integers and decimals, are called mixed numbers. NUMERATION OF DECIMALS. 115. It must be obvious from the two preceding articles, that the figures in decimals, as in whole numbers, have a local value, called the name of the place,(74) which depends upon their distance from the separatrix, or the place of unity, each removal of a figure one place towards the right diminishing its value ten times. (73) The names of the places, both of integers and decimals, are expressed in the following co Billions. -100 Millions. 100 Thousands. Hundreds. Hundredths. 10 Millionths. From this table it will be seen, that the names of the places, each way from that of units are the same, excepting the termination th, or ths, which is added to the name of the last, or right hand place, in the enunciation of decimals. 116. Ciphers on the right of decimals do not alter their value; for while each additional cipher indicates a division into parts ten times smaller than the preceding, it makes the decimal express 10 times as many parts, (113) Thus 5 tenths denotes 5 parts of a unit, which is divided into 10 parts; 50 hundredths denotes 50 parts of a unit, which is divided into 100 parts, and so on: but as 5 is half of 10, and 50 half of 100, the value of each is the same, namely, one half a unit. On the contrary, each cipher placed at the left hand diminishes the value of a decimal 10 times, by removing each significant figure one place towards the right.(115) In the decimals, 0.5, 0.05, 0.005, the second is only 1 tenth part as much as the second; and they are read, 5 tenths, 5 hundredths, and 5 thousandths. ADDITION OF DECIMALS. ANALYSIS. 117. 1. What is the sum of 4 tenths of a foot, 75 hundredths of a foot, and 9 hundredths of a foot? 0.4 0.09 We first write 0.4; then as .75 is 0.7 and 0.05, we write 0.7 under 0.4, and place the 5 at the right hand in the place of hundredths; and lastly, we write 9 under the 5 in the place of hundredths. We then add the hundredths, and find them to be 0.14, equal to one 1 tenth and 4 hundredths; we therefore reserve the 0.1, to be united with the tenths, and write the 4 under the column of hundredths. We then say, 1 to 0 is 1, and 7 are 8, and 4 are 12; but 12 tenths of a foot are equal to 1 foot and 2 tenths; we therefore write 2 in the place of tenths, and place the I foot on the left of the separatrix in the place of units. Thus we find the sum of 0.4, 0.75, and 0.09 of a foot, to be 1.24 ft. Ans. 1.24 ft. RULE. 118. Write down the whole numbers, if any, as in Simple Addition, and place the decimals on the right in such manner that tenths shall stand under tenths, hundredths under hundredths, and so on, and draw a line below. Begin at the right hand, and add up all the columns, writing down and carrying as in Simple Addition. Place the decimal point directly under those in the numbers added. ANALYSIS. 119. 1. How much butter in 3 boxes, each containing 4 pounds and 75 hundredths of a pound? The method of solving this question By Addition. by Addition, must be sufficiently obvi- By Multiplication. 4.75 4.75 4.75 Ans. 14.25 lb. ous,[117] In doing it by Multiplica- 4.75 are hundredths of a pound, the product Ans. 14.25 lb. is obviously hundredths; but 0.15 are 0.1 and 0.05, we therefore write 5 in the place of hunPredths, and reserve the 1 to be joined with the tenths. We then say, 3 times 7 are 21, which are so many tenths, because the 7 are tenths, and to these we join the 1 tenth reserved, making 22 tenths; but 22 tenths of a pond are equal to 2 pounds and 2 tenths of a pound. We therefore write the 2 tenths in the place of tens, and reserve the 2 lbs. to be united with the pounds. Lastly, we say, 3 times 4 lbs. are 12 lbs. to which we join the 2 lbs. reserved, making 14 pounds, which we write as whole numbers on the left hand of the separatrix. From this example it appears, that when one of the factors contains decimals, there will be an equal number of decimal places in the product. 120. 2. If a person travel 4.3 miles per hour, how far will he travel in 2.5 hours? 4.3 2.5 2.15 Having written the numbers as at the left hand, we say 5 times 3 are 15. Now as the 3, which is multiplied, is tenths, it is evident, that if the 5, by which it is multiplied, were units, the product, 15, would be tenths,(119 But since the 5 is only tenths of units, the product, 15, can be only 10ths of 10ths, or 100ths of units; but as 0.15 are 0.1 and 0.05, we write 5 in the place of hundredths, reserving the 1 to be joined with the tenths. We then say 5 times 4 are 20, which are tenths, because the 5 is tenths; joining the 0.1 reserved, we have 21 tenths, equal to 2.1 miles; we therefore write I in the place of tenths, and 2 in the place of Aas.10.75 miles. |