Reduction

There are four types of reduction given:

(A) to reduce a vulgar faction to a decimal.

The method is to annex ciphers (0’s) to the numerator and divide by the denominator. The decimal must consist of as many figures as there are ciphers added to the numerator.

Examples:

(1) Reduce to a decimal.

The answer given without explanation is .078125.

(2) Reduce to a decimal.

Macdonald works this one out with longhand division.

 

318

 

22

70

 

 

66

3×22

 

40

 

 

22

1×22

 

180

 

 

176

8×22

 

     4

 

 

I have added the ciphers in red boldface. Three have been added so there are 3 figures in the decimal, i.e. Macdonald gives his result as .318. Notice that the final subtraction yields the number 4 so that if we add the cipher to get 40 we are back to the line under 66. Consequently, the full answer (not given by Macdonald is a repeating decimal .3181818 etc.

(3) Reduce to a decimal.

The answer given in the notebook is 0.03448275862068965517241379310. No details are given. It is a good assumption that Macdonald just copied out the number and did not work it through. It is a lot of work and cannot be checked with the usual calculator or a program such Excel, which does the calculation only to 16 decimal places, not 29. I did check the division using the programming language R using function mpfr in the library Rmpfr, which does calculation to any specified level of precision. The answer given is correct. In writing out the next set of 28 decimals, it is also a repeating decimal with blocks of 34482753448275862068965517241379310 repeating.

(B) to reduce a decimal to a vulgar fraction.

Place, for denominator, unity with as many ciphers after it as there are figures in the numerator & reduce the fraction to its lowest terms.

Examples:

(1) Reduce .875 to a vulgar fraction.

Macdonald gives no further calculation. However, from earlier material in the notebook, the greatest common divisor between 875 and 1000 is 125. Dividing numerator and denominator by 125 yields .

(2) Reduce 25.84 to a vulgar fraction.

(3) Reduce .265625 to a simple fraction.

As in the other two examples, Macdonald gives no further calculation. Here are the missing steps to get the greatest common divisor between the numerator and the denominator:

1000000 = 3 × 265625 + 203125

265625 = 1 × 203125 + 62500

203125 = 3 × 62500 + 15625

62500 = 4 × 15625

Therefore 15625 is the greatest common divisor. Dividing 256625 and 1000000 by 15625 yields the given answer.

(C) to reduce infinite decimals to vulgar factions.

These are repeating decimals where blocks of digits after the decimal place repeat ad infinitum. A standard notation is to draw a line over the repeating block of digits. For example, in his section on reducing fractions to decimals Macdonald had written as which I mentioned was a repeating decimal etc. This is written compactly as .

To obtain the fraction, write out the digits of the decimal until the numbers begin to repeat (318). Subtract the nonrepeating digits from this number to get the numerator in the fraction (318 – 3 = 315). The denominator is comprised of a string of 9’s where the length of the string equals the number of repeating digits, followed by a string of ciphers (0’s) where the length of this string equals the number of nonrepeating digits (990). Then reduce the resulting fraction to a simple fraction ().

Examples:

(1) Reduce to a vulgar fraction.

23562

   235

23327

Macdonald gives the result without further explanation or possible simplification. It turns out that 23327 and 99000 are relatively prime since their greatest common divisor is 1. This is verified with the algorithm to find the greatest common divisor.

99000 = 4 × 23327 + 5692

23327 = 4 × 5692 + 559

5692 = 10 × 559 + 102

559 = 5 × 102 + 49

102 = 2 × 49 + 4

49 = 12 × 4 + 1

4 = 4 × 1

(2) Reduce to a vulgar fraction.

740384615

             740

740383875

The answer is . This reduces to , which Macdonald provides without further demonstration. The reduction is obtained through the usual algorithm which shows that the greatest common divisor between numerator and denominator is 9615375.

(D) to reduce interminate decimals to a common denominator.

An interminate decimal is another word for a repeating decimal. To find the common denominator there are two cases: pure repeating decimals and leading digits followed by repeating decimals.

In the case of pure repeating decimals set out the numbers in their cycles until the first time the end of the cycles coincide. The common denominator is made up of the replications of the digit 9; the number of replications is the length of the common cycle.

Example:

Find the common denominator of and .

Set out the repeating decimals in their cycles separating each cycle by | .

974|974|

 56|56|56|

The end of the two cycles coincides after 6 digits. Therefore 999999 is the common denominator.

In the case of leading digits that are not repeated find the decimal fraction that contain the largest number of leading digits that are not repeated. Set out the repeating digits in their cycles beginning after nonrepeating digits until the first time that the cycles coincide. The common denominator is made up of replications of the digit 9 followed by replications of the digit 0. The number of 9’s is the length of the common cycle and the number of 0’s is largest number of leading digits that are not repeated.

Example:

Find the common denominator of , , and .

The longest length of no repeating digits is 2. Set out the all the numbers with repeating digits in their cycles after the first 2 digits.

.3

 

.04

 

.72

572|572|572|572|

.14

5624|5624|5624|

 The end of the cycles coincides after 12 digits. Therefore 99999999999900 is the common denominator.

 

 

Some, but not all, of this material is covered in Bonnycastle’s Introduction to Arithmetic: (A) is found on page 139 of the 1810 edition and (C) is found on page 148 of the same edition. The term “interminate” is not used by Bonnycastle or Pike.