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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. .346 = 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. .000,000,000,000,000,004 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.

ILLUSTRATIONS. (a) To find the perimeter of a square when its side is given.

Let a = one side.*

Then 4 a 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.

Let 6 the breadth. Then 2 a + 2b, or (a + b) x 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, 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 3 x= the second number and

x + 3x= 48.

4x=48.

x= 12.

3 x= 36.
The numbers are 12 and 36.

* That is, the number of units in one side. The letter stands for the number.

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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. .346 = 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. .000,000,000,000,000,004 = 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.

ILLUSTRATIONS. (a) To find the perimeter of a square when its side is given.

Let a = one side.*

Then 4 a 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.

the length.

Let b = the breadth. Then 2 a + 2b, or (a + b) x 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, 2.

Let a =

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 3 x= the second number and x + 3x = 48.

4 x = 48.

x = 12.

3 x= 36.
The numbers are 12 and 36.

That is, the number of units in one side. The letter stands for the number.

20. The sign of the 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 xa x b.

21. EXERCISE. Find the numerical value of each of the following expressions, if a= 8, b= 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 +56=
4. a+b-2c=

8. ab + bc
(a) Find the sum of the eight results.

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22. EXERCISE. Find the numerical value of each of the following expressions, if a= 20, b= 4, and c = 2: 1. 3(a + b)=*

5. a+b= 2. 2(a - b)=

6. (a + b)+c= 3. 4(a+b+c)=

7. (a+b): 3c= 4. 2(a+b-c)=

8. (a + 2b): 2c= (b) Find the sum of the eight results.

.

23. EXERCISE.
1. If 2x = 20, 2. If 5 x = 40, 3. If 6 x = 72,
x=?

=?
4. If 2x + 3 x= 60, 5. If 3x +4x=56,
x = ?

X = ?

X =

X = ?

* This means, 3 times the sum of a and b.
+ This means, the sum of a and b, divided by c.

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