Note to the Teacher. Changing from words to figures when the pupil reads the words is of little importance. Writing numbers in figures which one hears spoken is very important. Dictation exercises should be used freely. NOTATION AND NUMERATION OF DECIMALS 17. Parts of numbers expressed in tenths, hundredths, thousandths, etc., are called decimals. 18. The principles of the Arabic system are used in writing and reading decimals. In writing decimals, the decimal point (.) is used to tell where the units end in the number written. Units' place is the first place at the left of the decimal point. The location of units' place is the key to decimals as it is to whole numbers, and to the entire Arabic system. In 365.24 the decimal point shows us that 5 is in units' place and stands as the key to the number. Each figure at its left has its value multiplied by ten for each step or place it is moved from the key position of units. Each figure at its right has its value divided by ten for each step or place it is removed from the key position of units. The decimal part, therefore, like the integral part, is fixed by a law of place value of the figures, by a law of place values, etc. Each step of a figure to the left multiplies its value by ten, each step of a figure to the right divides its value by ten. In the number given, 365.24, the 2 has one tenth of its unit value, as it is one place to the right of units' place; the 4 has one hundredth of its unit value, as it is two places to the right of units' place. The 6 has ten times its unit value, as it is one place to the left of units' place; the 3 has one hundred times its unit value, as it is two places to the left of units' place. This law is so important, as it is the key to all number expressions, that you should study it until you understand it thoroughly. STUDY OF NUMBER EXPRESSIONS 19. Express fully what is meant by the following numbers: 1. 645.32. (a) The 5 is in units' place and expresses 5 units. (b) The 4 is one step to the left of units' place and expresses 10 times 4 units or 40 units. (c) The 6 is two steps to the left of units' place and expresses 100 times 6 units or 600 units. (d) The 3 is one step to the right of units' place and expresses of 3 units or units. 3 10 (e) The 2 is two steps to the right of units' place and expresses 10 of 2 units or 180 units. 2. 300.07. (a) The 3 is two steps to the left of units' place and expresses 100 times 3 units or 300 units. (b) The 7 is two places to the right of units' place and expresses 10 of 7 units or 1 units. 3. 45.18. 4. 8650.132. 5. 7006.008. Note. While the decimal notation is not usually taken at this tage of the pupil's progress, it may well be studied here, if the units' lace and not the decimal point is used as the key to number expressions. Careful study of the law of place values at this time will save much loss of time later, and will enable the pupil to associate the fundamental operations of integers and decimals. A little more careful study of the fundamental operations will thus enable the pupil to omit the entire chapter usually devoted to decimals. 20. The following chart shows the principles which govern the Arabic system of numbers, both integral and decimal. 9 8 7 6 5 4 3 2 1 Key 1 2 3 4 5 6 21. To read the decimal part, we read the num bers in that part as if they were integers, then give them the group name of the place value of the last or right-hand figure. In reading aloud, we read the decimal point "and" to show its location. 1. Read 3415.016. 3 thousand, 415 and 16 thousandths. 2. Read 42,678.143. 3. Read 30,250.0015. 4. Read 84,132,143.0875. 5. Read 360,204.00045. READING AND WRITING NUMBERS WITH 22. Read the following numbers; then close your book and see if you can write them when you hear them read: NOTATION AND NUMERATION OF U. S. MONEY 23. 1. The sign $ is used in place of the word "dollars," and is written before the number, thus $6, for six dollars. 2. Cents are hundredths of dollars; therefore, the last figure of cents must be in hundredths' place, the second place at the right of units. 3. The decimal point is used after dollars, which are the units numbered when the dollar sign is used. Three dollars and five cents is written $3.05. Ten dollars and forty-five cents is written, $10.45. Eight cents is written $.08, or, when written alone, 8. 24. Read the following; then close your book and see if you can write them when you hear them read: 1. $80.15, $65.04, $750.10, $753.25, $7850.25, $60.03. 2. $150.04, $250.43, $601.15, $78.02, $6701.15, $100.05. 3. $243.50, $6780.25, $783.15, $.54, $.05, $.09. 4. $25,145.15, $524,785.10, $67,620.50, $130,345.25, $1.06. 5. $60.02, $600.15, $707.07, $606.06, $5000.05, $5200.52. ROMAN NOTATION 25. The Roman notation is little used except to number clock dials, for dates on monuments, public buildings, etc., and to number the chapters in books. It is never used in computation. 1. The first nine numbers are: I II III IV V VI VII VIII IX 6 7 8 9 |