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fraction, as in the numerical examples already given, the integral frac

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&c. will terminate, and we shall obtain an expres

sion for the exact value of the given fraction by taking them all.

235. We will now explain the manner in which any approximating fraction may be found from those which precede it.

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By examining the third approximating fraction, we see, that its numerator is formed by multiplying the numerator of the preceding fraction by the denominator of the third integral fraction, and adding to the product the numerator of the first approximating fraction and that the denominator is formed by multiplying the denominator of the last fraction by the denominator of the third integral fraction, and adding to the product the denominator of the first approximating fraction.

We should infer, from analogy, that

neral. But to prove it rigorously, let

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be the three

this law of formation is ge. P Q R Q"R" approximating fractions for which the law is already established. Since c is the denominator of the last integral fraction, we have from what has already been proved

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Hence we see that the fourth approximating fraction is deduced from the two immediately preceding it, in the same way that the third was deduced from the first and second; and as any fraction may be deduced from the two immediately preceded in a similar manner, we conclude, that, the numerator of the nth approximating fraction is formed by multiplying the numerator of the preceding fraction by the denominator of the nth integral fraction, and adding to the product the numerator of the n -2 fraction; and the denominator is formed according to the same law, from the two preceding denomina

tors.

236. If we take the difference between any two of the consecu. tive approximating fractions, we shall find, after reducing them to a common denominator, that the difference of their numerators will be equal to ±1; and the denominator of this difference will be the product of the denominators of the fractions.

Taking, for example, the consecutive fractions

we have,

1

b

and

ab+1'

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and

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ab+1 (ab+1)c+a ̄ ̄(ab+1)((ab+1)c+a)

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To prove this property in a general manner, let

three consecutive approximating fractions. Then

P Q PQ'-P'Q

PQR

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But R-Qc+P and R'=Q'c+P' (Art. 235).

Substituting these values in the last equation, we have

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P Q

From which we see that the numerator of the difference

P Q

Q R'

is equal, with a contrary sign, to that of the difference

Q' R'

That is, the difference between the numerators of any two consecutive approximating fractions, when reduced to a common denominator, is the same with a contrary sign, as that which exists between the last numerator and the numerator of the fraction immediately following.

But we have already seen that the difference of the numerators of the 1st and 2d fractions is equal to +1; also that the difference between the numerators of the 2d and 3d fractions is equal to -1; hence the difference between the numerators of the 3d and 4th is equal to +1; and so on for the following fractions.

Since the odd approximating fractions are all greater than the true value of the continued fraction, and the even ones all less (Art. 232), it follows, that when a fraction of an even order is subtracted from one of an odd order, the difference should have a plus sign; and on the contrary, it ought to have a minus sign, when one of an odd order is subtracted from one of an even.

237. It has already been shown (Art. 232), that each of the approximating fractions corresponding to the odd numbers, exceeds the true value of the continued fraction; while each of those corresponding to the even numbers is less than it. Hence, the difference between any two consecutive fractions is greater than the difference between either of them and the true value of the continued fraction. Therefore, stopping at the n' fraction, the result will be true to within 1 divided by the denominator of the nt fraction, multipli. ed by the denominator of the fraction which follows. Thus, if Q and R are the denominators of consecutive fractions, and we stop at the fraction whose denominator is Q', the result will be true to 1 within But since a, b, c, d, &c. are entire numbers, the deQR" nominator R' will be greater than Q', and we shall have

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that is, the approximate result which is obtained, is true to within unity divided by the square of the denominator of the last approximating fraction that is employed.

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Here we have in the quotient the whole number 2, which may

either be set aside and added to the fractional part after its value shall have been found, or we may place 1 under it for a denominator and treat it as an approximating fraction.

Of Exponential Quantities.

Resolution of the Equation a2=b,

238. The object of this question is, to find the exponent of the power to which it is necessary to raise a given number a, in order to produce another given number b.

Suppose it is required to resolve the equation 2=64. By rais. ing 2 to its different powers, we find that 2o=64; hence x=6 will satisfy the conditions of the equation.

Again, let there be the equation 3*=243. The solution is x=5. In fact, so long as the second member b is a perfect power of the given number a, x will be an entire number which may be obtained by raising a to its successive powers, commencing at the first.

Suppose it were required to resolve the equation 2=6. By making x=2, and x=3, we find 22=4 and 23=8: from which we perceive that x has a value comprised between 2 and 3.

or

1

Suppose then, that x=2+1, in which case x'>1.

Substituting this value in the proposed equation, it becomes,

22+_6 or

x' power.

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=2, by changing the members, and raising both to the

To determine x', make successively r'=1 and 2; we find

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