Division is also denoted by placing the divisor under the dividend, with a horizontal line between them like a vulgar fraction. Thus, is the same as 6÷2.

6. A number placed above another number, a little to the right, is called an exponent. Thus, 62, 73, in these expressions, 2 and 3 are exponents of 6 and 7 respectively.

7. An exponent placed over a quantity denotes that the quantity is to be used as a factor as many times as there are units in the exponent. Thus, 24=2×2×2×2=16.

S. When the exponent is two, the result is called the second power of the quantity over which it is placed. Thus, 72=7X7-49=the second power of 7.

9. When the exponent is three, the result is called the third power of the quantity over which it is placed. Thus, 43=4X4×4=64 the third power of 4.

The higher powers are denoted in the same way.

10. The symbol, √, denotes that the square root of the quantity over which it is placed is to be taken. Thus,√4=2,

which is read, the square root of 4 equals 2.

3

11. The symbol, ✔, denotes in a similar manner the cube root of the number over which it is placed. Thus, 764,

=4 which is read, the cube root of 64 equals 4. The roots of higher dimensions are denoted in a similar way.

12. The symbol, ..., is equivalent to the phrase therefore, or consequently. Thus, 62=36, and 4×9=36 ... 62=4×9, which is read, the square of 6 equals 36, and the product of 4 and 9 equals 36, therefore the square of 6 equals the pro

duct of 4 and 9.

13. The parenthesis, (), when it encloses several quantities, requires these quantities to be regarded as one single quantity. Thus, (5+2)×7=49, which is read, the sum of 5 and 2 multiplied by 7=49.

EXAMPLES.

ILLUSTRATING THE FOREGOING DEFINITIONS AND SYMBOLS.

3. The expression, 11+5-2=2x7=28÷÷2=14, when translated into common language, becomes the sum of 11 and 5 diminished by 2 equals the product of 2 and 7, equals 28 divided by 2, equals 14.

2. The expression,

(42-8)

2

+3=4x5=20, is equivalent

to the following: Forty two diminished by 8, and the remainder divided by 2, and the quotient increased by 3, equals 4 multiplied into 5, equals 20.

3. The expression,√144=3×4=36÷3=12, is the same as the square root of 144 equals 3 multiplied into 4, equals 36 divided by 3, equals 12.

4. Translate the expression, (10+3)x7=182÷2=91, into common language.

5. Translate the expression, (√16+7)×4=√64×11, into common language.

6. Translate the expression, (√49—√64)×3—20—11, into common language.

"Five

7. What expression is equivalent to the following? times nine divided by three, and that quotient multiplied by seven, equals the square of ten increased by five?"

EXAMPLES ILLUSTRATING DEFINITIONS AND SYMBOLS. 13

8. What expression is equivalent to the following ?"Three times twenty-one, increased by five times seven, and diminished by three times the square of four, is equal to twice the square of five?"

9. What expression is equivalent to the following? "The cube root of sixty-four, increased by two, and the sum multiplied by ten, is equal to the square of eight diminished by four?"

MULTIPLICATION OF COMPOUND EXPRESSIONS. 4. Let it be required to multiply 3+2 by 4+5.

We must repeat 3+2 as many times as there are units in

4+5.

First, repeating 3+2 as many times as there are units in 4, we get (3+2)×4=12+8, for first partial product:

Secondly, repeating 3+2 as many times as there are units in 5, we get (3+2)×5=15+10, for second partial product : Hence, 3+2 repeated as many times as there are units in 4+5, becomes (3+2)×(4+5)=12+8+15+10.

2. Again, let it be required to multiply 7-3 by 4+2: Proceeding as in the last example, we find (7-3)×(4+2)= 28-12+14-6.

3. In a similar way we find that 4-3, multiplied by 3-2, gives (4-3)×(3—2)=12—9—8+6.

By carefully reviewing these examples, we deduce the following rule. To multiply together two compound expressions.

RULE.

Multiply each term of one of the factors, by each term of the other factor; observing that like signs produce plus, and unlike signs produce minus.

Examples.

4. What is the product of 8+3 by 6+4 ?

Ans. 48+18+32+12.

5. What is the product of 6-2 by 4+3?

Ans. 24-8+18-6.

6. What is the product of 11-3 by 13-7?

Ans. 143-39-77+21.

7. What is the product of 3+2-1 by 4-1+5?

Ans. 12+8-4-3-2+1+15+10−5.

8. What is the product of 1+2-3 by 4-5+6?

Ans. 4+8-12-5-10+15+6+12-18.

9. What is the product of 7-9 by 5-11?
10. What is the product of 21-3 by 9-2?
11. What is the product of 1+7+5 by 2+3?
12. What is the product of 9+8+7 by 6+5+4?

INTERESTING PROPERTIES OF NUMBERS,

PROPOSITION I.

5. Every number will divide by 9, when the sum of its digits is divisible by 9.

For, take any number as 78534; this number is, by the nature of decimal arithmetic, the same as 70000+8000+ 500+30 +4.

78534=9999×7+999×8+99×5+9×3+(7+8+

5+3+4.)

Now, since each expression, 9999 × 7, 999 × 8, 99 × 5, and 9 × 3, is divisible by 9, it follows that the first number, 78534, will be divisible by 9 when the sum of its digits (7+8+5+3 +4) is.

Hence it follows, that any number being diminished by the sum of its digits, will become divisible by 9.

Also any number divided by 9 will leave the same remainder as the sum of the digits when divided by 9.

NOTE. These singular properties of the digit 9, have been made use of by many authors for proving the work of the four fundamental rules of arithmetic.

PROPOSITION II.

6. Every number is either a prime number, or composed of prime factors.

For, all numbers which are not prime are composite, and can therefore be separated into two or more factors; and if these factors are not prime, they can again be separated into other factors, and thus the decomposition can be continued until all the factors are prime.

Hence, to resolve any composite number into its prime factors, we have this

RULE.

Divide the number by any prime number, which will divide it without any remainder; then divide the quotient in the same way; and so continue until a quotient is obtained which is a prime. Then will the successive divisors, together with the last quotient, form the prime factors required.