16. Note the number of decimal places in each of the following expressions :

1. .4 = 4 tenths. (1 decimal place.)

2. .2727 hundredths. (2 decimal places.)

=

3. .346 346 thousandths. (3 decimal places.)
4. .27582758 ten-thousandths.

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.

11. In any number of millionths there are places.

decimal

12. In any number of billionths there are

decimal

places.

13. In any number of hundredths there are places.

decimal

14. In any number of ten-thousandths there are decimal places.

15. In any number of hundred thousandths 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.

ILLUSTRATIONS.

(a) To find the perimeter of a square when its side is given. Let a one side.*

Hence the rule:

number denoting the

(b) To find the

breadth are given.

Then 4 a = the perimeter.

To find the perimeter of a square, multiply the length of its side by 4.

perimeter of an oblong when its length and

Let a = the length.

Then 2a

Let b = the breadth. 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.

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 x 4, is written 4 b. 6 ab, means, 6 times a times b, or 6 × a × b.

21. EXERCISE.

Find the numerical value of each of the following expressions, if a = 8, b = 5, and c = : 2:

Find the numerical value of each of the following expressions if a = 20, b = 5, and c = 2:

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.

7. The perimeter of a regular pentagon is a line made up of equal lines.

8. The circumference of a circle is a

line, every point in which is equally distant from a point called the center of the circle.

9. Imagine a straight line drawn upon the surface of a stovepipe. Can you draw a straight line upon the surface of a sphere?

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 b 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.

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 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. How many dollars had each ?

*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.