NOTE 1.-Observe that a figure always stands for units. If it occupies the first place, it stands for primary units; if it occupies the second place, it stands for tens (that is, units of tens); the third place, for hundreds; the first decimal place, for tenths; the second decimal place, for hundredths, etc. Thus, a figure 5 always stands for fivefive primary units, five thousand, five hundredths, five tenths, according to the place it occupies. NOTE 2.—In reading integral numbers the primary unit should be, and usually is, most prominent in consciousness. Thus, the number 275 is made up of 2 hundreds, 7 tens, and 5 primary units; but 2 hundreds equal two hundred (200) primary units, and seven tens equal seventy (70) primary units; these (200 + 70 + 5) we almost unconsciously combine in our thought, and that which is ordinarily present in consciousness is 275 primary units. So in the number 125,246, there are units of six orders, which we reduce in thought to primary units and say, one hundred twenty-five thousand two hundred forty-six primary units. NOTE 3.-In reading decimals, too, the primary unit should be prominent in consciousness. Thus, .256 is made up of 2 tenths, 5 hundredths, and 6 thousandths; but 2 tenths equal 200 thousandths, and 5 hundredths equal 50 thousandths; these (200+ 50+ 6) we combine in our thought, and that which should be present in consciousness is 256 thousandths of a primary unit. Write in figures: 10. EXERCISE. 1. Two hundred fifty-four thousand one hundred sixty. 2. One hundred seventy-five and two hundred six thousandths. 3. Eighty-four and three hundred twenty-five thousandths. 4. One hundred ninety-seven and twenty-seven hundredths. 5. Seven thousand four hundred twenty-four and six tenths. 6. Twenty-four thousand six hundred fifty-one. 7. One hundred thirty-five thousand two hundred fifty. (a) Find the sum of the seven numbers. 11. EXERCISE. Read in two ways as suggested in the following: 324.61. (1) 3 hundreds, 2 tens, 4 primary units, 6 tenths, 1 hundredth. (2) Three hundred twenty-four and sixty-one hundredths. Use the word and in place of the decimal point only. 12. EXERCISE. Observe that any number may be read by giving the name of the units denoted by the right-hand figure, to the entire number: thus, 146 is 146 primary units; 21.8 is 218 tenths; 3.25 is 325 hundredths. 1. 27 = 2 tens + 7 primary units = 27 primary units. 2. 2.7 2 primary units + 7 tenths = 3. .27 4. .027 = = tenths. hundredths. = 2 hundredths + 7 thousandths = thou 6. 5.247 5 primary units + 2 tenths + 4 hundredths + · = NOTE. Exercise 12 and Exercise 13 are important as a preparation for the clear understanding of division of decimals. Do not omit them nor permit the work to be done carelessly. 13. EXERCISE. Observe that any part of a number may be read by giving the name of the units denoted by the last figure of the part, to the entire part; thus, 24.65 is 246 tenths and 5 hundredths; 14.275 is 1427 hundredths and 5 thousandths. In a similar manner read each of the following: Observe that in reading a mixed decimal in the usual way, we divide it into two parts and give the name of the units denoted by the last figure of each part to each part; thus, 2346.158 is read 2346 (primary units) and 158 thousandths. Read the following in the usual manner. Do no not use the word and in reading the numbers in the second column: 1. Two hundred and eight thousandths. NOTE 1.-The names of the periods above tredecillions are: quatuordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion and vigintillion. 16. Note the number of decimal places in each of the following expressions: 1. .44 tenths. (1 decimal place.) 2. .27 27 hundredths. (2 decimal places.) = = 3. .346 346 thousandths. (3 decimal places.) = 5. .07286 7286 hundred thousandths. = 6. .000896 896 millionths. (6 decimal places.) = 7. .000,468,275 = 8. .000,000,000,462 = billionths. (9 decimal places.) trillionths. = 9. .000,000,000,000,527 = 10. .000,000,000,000,000,004 = 11. .000,000,000,000,000,000,037 12. .000,000,000,000,000,346,275 quadrillionths. = = 13. .000,000,000,000,002,427,836 = 14. In any number of thousandths there are decimal places. 15. In any number of millionths there are places. decimal 16. In any number of billionths there are places. decimal Algebra-Notation. 17. Letters are used to represent numbers; thus, the letter a, b, or c may represent a number to which any value may be given. 18. Known numbers, or those that may be known without solving a problem, when not expressed by figures, are usually represented by the first letters of the alphabet; as, a, b, c, d. ILLUSTRATIONS. (a) To find the perimeter of a square when its side is given. Hence the rule: To find the perimeter of a square, multiply the number denoting the length of its side by 4. (b) To find the perimeter of an oblong when its length and breadth are given. Then 2a + 2b, or (a + b) × 2 = the perimeter. Hence the rule: To find the perimeter of an oblong, multiply the sum of the numbers denoting its length and breadth by 2. 19. Unknown numbers, or those which are to be found by the solution of a problem, are usually represented by the last letters of the alphabet; as, x, y, z. (a) There are two three times the first. Let Then ILLUSTRATION. numbers whose sum is 48, and the second is What are the numbers? * That is, the number of units in one side. The letter stands for the number. |