Elements of arithmetic

There is a useful method of finding fractions which shall be nearly equal to a given fraction, and with which the computer ought to be acquainted. Proceed as in the rule for finding the greatest common measure of the numerator and denominator, and bring all the quotients into a line. Then write down,

Then take the third quotient, multiply the numerator and denominator of the second by it, and add to the products the preceding numerator and denominator. Form a third fraction with the results for a numerator and denominator. Then take the fourth quotient, and proceed with the third and second fractions in the same way; and so on till the quotients are exhausted. For example, let the fraction be ⁹¹³¹/₁₃₁₂₈.

• 9131)13128(1, 2
• 1137   3997(3, 1
• 551    586(1, 15
• 201      35(1, 2
• 26        9(1, 8
• 8        1

This is the process for finding the greatest common measure of 9131 and 13128 in its most compact form, and the quotients and fractions are:

It will be seen that we have thus a set of fractions ending with the original fraction itself, and formed by the above rule, as follows:

and so on. But we have done something more than merely reascend to the original fraction by means of the quotients. The set of fractions, ¹/₁, ²/₃, ⁷/₁₀, ⁹/₁₃, &c. are continually approaching in value to the original fraction, the first being too great, the second too small, the third too great, and so on alternately, but each one being nearer to the given fraction than any of those before it. Thus, ¹/₁ is too great, and ²/₃ is too small; but ²/₃ is not so much too small as ¹/₁ is too great. And again, ⁷/₁₀, though too great, is not so much too great as ²/₃ is too small.

Moreover, the difference of any of the fractions from the original fraction is never greater than a fraction having unity for its numerator and the product of the denominator and the next denominator for its denominator. Thus, ¹/₁ does not err by so much as ¹/₃, nor ²/₃ by so much as ¹/₃₀, nor ⁷/₁₀ by so much as ¹/₁₃₀, nor ⁹/₁₃ by so much as ¹/₂₉₉, &c.

Lastly, no fraction of a less numerator and denominator can come so near to the given fraction as any one of the fractions in the list. Thus, no fraction with a less numerator than 249, and a less denominator than 358, can come so near to

The reader may take any example for himself, and the test of the accuracy of the process is the ultimate return to the fraction begun with. Another test is as follows: The numerator of the difference of any two consecutive approximating fractions ought to be unity. Thus, in our instance, we have ¹⁶/₂₃ and ²⁴⁹/₃₅₈, which, with a common denominator, 23 × 358, have 5728 and 5727 for their numerators.

As another example, let us examine this question: The length of the year is 365·24224 days, which is called in common life 365¼ days. Take the fraction ²⁴²²⁴/₁₀₀₀₀₀, and proceed as in the rule.

• 24224)100000(4, 7, 1, 4, 9, 2
• 2496     3104
• 64       608
• 0         32

and ⁷⁵⁷/₃₁₂₅ is ·24224 in its lowest terms. Hence, it appears that the excess of the year over 365 days amounts to about 1 day in 4 years, which is not wrong by so much as 1 day in 116 years; more accurately, to 7 days in 29 years, which is not wrong by so much as 1 day in 957 years; more accurately still, to 8 days in 33 years, which is not wrong by so much as 1 day in 5313 years; and so on.

This method may be applied to finding fractions nearly equal to the square roots of integers, in the following manner:

•     
• √43 = 6 + ...

Set down the number whose square root is wanted, say 43. This square root is 6 and a fraction. Set down the integer 6 in the first and third row, and 1 in the second row always. Form the successive rows each from the one before, in the following manner:

and so on. In process of time the second column, 1, 7, 1, occurs again, after which the several columns are repeated in the same order. As a final process, take the set in the lowest line (excluding the first, 6), namely, 1, 1, 3, 1, 5, 1, 3, &c. and use them by the rule given at the beginning of this article, as follows:

Hence, 6¹⁶⁵/₂₉₆ is very near the square root of 43, not erring by so much as

If we try it, we shall find (⁶¹⁶⁵/₂₉₆) to be ¹⁹⁴¹/₂₉₆, the square of which is ³⁷⁶⁷⁴⁸¹/₈₇₆₁₆, or 43⁷/₈₇₆₁₆.

This rule is of use when it is frequently wanted to use one square root, and therefore desirable to ascertain whether any easy approximation exists by means of a common fraction. For example, √2 is often used.

Here it appears that

considering the ease of the operation, a fair approximation. In fact, ⁹⁹/₇₀ is 1·4142857 ... the truth being 1·4142135 ...

The following is an additional example: