« ΠροηγούμενηΣυνέχεια »
What is the value of .6875 of a yard?
3=number of feet in a
(yard. 2.0625 12=number of inches in
(a foot. .7500 12=number of lines in
(an inch, 9.0000 The answer here is 2 feet 9 lines. What is the value of .084 of a furlong? Ans. 3
per. I yd. 2 ft. 11 in. What is the value of .683 of a degree ? Ans. 40
m. 58 sec. 48 thirds. What is the value of .0053 of a mile? Ans. I
per. 3 yds. 2 ft. 5 in. + What is the value of .036 of a day? Ans. 51
IN DECIMAL FRACTIONS.
Having reduced all the fractional parts in the given quantities to their corresponding decimals, and having stated the three known terms, so that the fourth, or required quantity, may be as much greater, or less than the third, as the second term is greater, or less than the first, then multiply the second and third terms together, and divide the product by the first term, and the quotient will be the answer;- in the same denomination with the third term.
If 3 acres 3 roods of land can be purchased for 93 dollars 60 cts. how much will 15 acres 1 rood cost at that rate ?
3 acs. 3 rds. = 3.75 acres.
15 acs. I rd. = 15.25 acres.
$ 93, 60 cts. = $ 93.60 Then 3.75: 15.25 :: 93 60:
If a clock gain 14 seconds in 5 days 6 hours, how much will it gain in 17 days 15 hours? Ans. 47 seconds.
If 187 dollars 85-cents gain 12 dollars 33 cents interest in a year, at what rate per cent is this interest? Ans. 6.56+
INVOLUTION AND EVOLUTION.
INVOLUTION is the method of raising any number, considered as the root, to any required power.
Any number, whether given, or assumed at . pleasure, may be called the root, or first power of this number; and its other powers are the products, that result from multiplying the number by itself, and the last product by the same number again; and so on to any number of multiplications.
The index, or exponent, is the number denoting the height, or degree of the power, being always greater by one, than the number of multiplications employed in producing the power. It is usually written above the root, as in the following EXAMPLE, where the method of involution is plainly exhibited.
Required the fifth power of 8 ) the root, or first first multiply by
then multiply the product 64 = 8z = square, or
by [second power. &c. 512 = 83 = cube, or
8 (third power. 4096=8, = biquadrate
8 for fourth power.
32768 = 89 = Answer.
EXAMPLES FOR EXERCISE,
What is the second power of 3.05? Ans. 9.3025 What is the third power of 85,3! Answer,
620650 477 What is the fourth power of .073? Answer,
090028398241 What is the eighth power of .09?. Answer, 1.00.00.00.0043046721
Note: When two, or more powers are multiplied together, their product is that power, whose index is the sum of the indices of the factors, or powers multiplied.
EVOLUTION is the method of extracting any required root from any given power.
Any number may be considered as a power of some other number; and the required root of any given power is that number, which, being multiplied into itself a particular number of times, produces the given power; thus, if 81 be the given number, or power, its square, or second root is 9; because 9 x 9 = 9=81; and 3 is its biquadrate, or fourth root, because 3x3x3x3=34 =81. Again, if 729 be the given power, and its cube root be required; the answer is 9, for 9x9x9=729 ; and if the sixth root of that number be required, it is found to be 3, for 3x3x3x3x3x3=729.
The required power of any given number, or root, can always be obtained exactly, by multiplying the number continually into itself; but there are many numbers, from which a proposed root can never be completely extracted;
yet by approximating with decimals, these roots may be found as exact as necessity requires. The roots that are found complete, are denominated rational roots, and those, which cannot be found completed, or which only approximate, are called surd, or irrational roots.
Roots are usually represented by these characters or exponents; V, or which signifies the square root; thus, ✓9, or 9} = 3 V or cube root; }
V 64, or 643=4 v, oro biquadrate root ; '16, or 169=2 &c.
Likewise 8} signifies the square root of 8 cubed; and, in general, the fractional indices imply, that the given numbers are to be raised to such powers as are denoted by their numerators, and that such roots are to be extracted from these powers, as are denoted by their denominators.
• The reason for separating the figures in the dividend into periods or portions of two places each, is, that the square of any single figure never consists of more than two places ; the square of a number of two figures, of not more than four places, &c. So that there will be as many figures in the root, as the given number has periods so divided.
And, the reason of the several steps in the operation appears by assuming any root and raising it actually to the second power, as in common Arithmetic; or from the Algebraic form of the square of any number of terms, whether two, three, or more. Thus,
Let 24 be squared, whose square is 576, put a for the ten's place, and b for the units, a+b=24, therefore a2 + 2 a 6462=576. The square root of the algebraic form is performed thus,
1st. Divisor a) a2 +2 ab + b2(a + b
24. Divisor 2 a76)2 a b + b2
2 ab + 62 First, I find the root (a) of the first term (az), and place it in the quo. tient, and its square under the first term, then I bring down the next term (2 ab) which is equal to twice the quotient (a) x by the next root (0) to be found, hence divide by twice the root (a) already found and it gives the next root, (6) which place in the quotient, as also to the right hand of the divisor, then í multiply both by (6) the root last found, and subtract said product from the dividend ; that is, (2 ab +62) which is equal to it, hence, The method of extracting the square root by, the Algebraic form ; but to. apply this to the Arithmetical form, which is thus,
176 I find the nearest root to the first period which is 2, then I place this in the quotient and its square under the first period as in the Algebraic form, then I bring down the next period to the remainder, this dividend is equal to double the root found multiplied by the next figure of the root, plus the square of the figure to be found; hence I double the quotient figure for a divisor and set the figure found in the quotient, as also to the right hand side of the divisor, (because the first root found is in the ten's place) the same as in the Algebraic form, in the same manner it can be demonstrated if there be three or terms