« ΠροηγούμενηΣυνέχεια »
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. .4 = 4 tenths. (1 decimal place.) 2. .27 = 27 hundredths. (2 decimal places.) 3. .316 = 346 thousandths. (3 decimal places.) 4. .2758 = 2758 ten-thousandths. 5. .07286 = 7286 hundred thousandths. 6. .000896 = 896 millionths. (6 decimal places.) 7. .000,468,275 = billionths. (9 decimal places.) 8. ,000,000,000,462 = trillionths. 9. .000,000,000,000,527 = quadrillionths. 10. .000,000,000,000,000,001 = quintillionths. 11. .000,000,000,000,000,000,037 12. .000,000,000,000,000,346,275 = 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 decimal places.
16. In any number of billionths there are decimal places.
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.
(a) To find the perimeter of a square when its side is given.
Let a = one side. *
Then 4a the perimeter. 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.
Let a = the length.
the breadth. Then 2a + 2b, or (a + b) X 2= the perimeter. Hence the rule: To firid 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, 2.
ILLUSTRATION. (a) There are two numbers whose sum is 48, and the second is three times the first. What are the numbers? Let
x = the first number. Then
3x = the second number, and x + 3x = 48.
4x = 48.
* That is, the number of units in one side. The letter stands for the number.
20. The sign of multiplication is usually omitted between two letters representing numbers, and between figures and letters; thus, a x b, is usually written ab; b 4, is written 4b. 6 ab, means, 6 times a times b, or 6 x a x b.
21. EXERCISE. Find the numerical value of each of the following expressions, if a = 8,6 = 5, and c= 2: 1. a + b + c =
5. 2 ab 2. a + b
6. 3 abc 3. 2 a + b + c =
7. 2 ab + 50 = 4. a + b - 20 =
8. ab + bc
(a) Find the sum of the eight results.
Find the numerical value of each of the following expressions if a = 20, 6= 4, and c= 2:
1. 3(a + b)
5. a = b = 2. 2(a - b) =
6. (a + b) = (= 3. 4(a + b + c) =
7. (a + b) + 3 C =
8. (a +26)
+ 20 =
24. A geometrical line has length, but neither breadth nor thickness.
NOTE.—Lines drawn upon paper or upon the blackboard are not geometrical lines, since they have breadth and thickness. They represent geometrical lines.
25. A straight line is the shortest distance from one point to another point.
26. A curved line changes its direction at every point.
27. A broken line is not straight, but is made up of straight lines.
1. The line AB is a
5. The line EK is a
of equal lines. 7. The perimeter of a regular pentagont is a line made of equal lines. 8. The circumferences of a circle is a
line, every point in which is equally distant from a point called the center of the circle.
9. The diameter of a circle is a line. 10. Imagine a straight line drawn upon the surface of a stovepipe. Can you draw a straight line upon the surface of a sphere? * See Book I., p. 53. + See Book I., p. 63.
I See Book II., p. 256.
28. MISCELLANEOUS REVIEW. 1. If a equals one side* of a regular pentagon, the perimeter of the pentagon is
2. If b equals the perimeter of a square, the side of the square equals b +
3. If a equals a straight line connecting two points and 6 equals a curved line connecting the same points, then a is
† than b. 4. Find the difference between two hundred seven thousandths, and two hundred and seven thousandths. I
5. How many zeros in 1 million expressed by figures ? 1 billion ? 1 trillion ?
6. How many decimal places in any number of millionths ? billionths? trillionths?
7. How many decimal places in 5 thousandths ? in 25 thousandths ? in 275 thousandths ? in 4346 thousandths ?
8. A figure in the second integral place represents units how many times as great as those represented by a figure in the second decimal place?
9. If a = 6, b = 2, and d 8, what is the numerical value of the following ? 12 a + 3b - 5 d.
10. John had a certain amount of money and James had 5 times as much ; together they had 354 dollars. dollars had each ?
the number of dollars John had. Then 5 x= the number of dollars James had, and x + 5 4: 354 dollars.
6 x = 354 dollars.
5 x =
* The expression a equals one side" means that a equals the number of units in one side. Remember that in this kind of notation the letters employed stand for numbers.
+ Longer or shorter? #See p. 15, Exercise 14.