Auxiliary Fractions
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[edit] Auxiliary Fractions: Converting a fraction to the equivalent decimal value by using the auxiliary fraction
[edit] Dividing by powers of ten
When we divide by a power of ten first we move the decimal point in both the numerator and the denominator to the left the same number of places as the number of zeros at the end of the denominator. Then we divide. For example, 1/800 = 0.01/8; 39/70 = 3.9/7; 3741/110000 = 0.3741/11; and 97654/90,000,000 = 0.0097654/9. We make the same adjustment in the decimal point when forming an auxiliary fraction.
[edit] Forming the Auxiliary Fraction: three types
For the purpose of dividing with an auxiliary fraction there are three types of fractions which are named by their denominator: (Type One A) denominators which end in nine or (Type One B) denominators which end in several nines, (Type Two) denominators which end in one, and (Type Three) denominators which end in the digits, 2, 3, 4, 5, 6, 7, and 8. If the denominator ends in zero(s), slide leftward to the first non-zero digit to identify the family of this fraction.
Type One A, Nines Family Fractions
When the fraction's denominator ends in nine we use the Ekādhika Purva. First we replace the denominator by its Ekādhika which means we drop the last digit (a nine) and increase the penultimate (next to the last) digit by one unit. In the numerator we slide the decimal point to the left by the number of terminal nines in the denominator.[1]
Nines family examples:
When F = 1/19, then the A.F. = 1/20 = 0.1/2;
When F = 4/29, then the A.F. = 4/30 = 0.4/3;
When F = 8/59, then the A.F. = 8/60 = 0.8/6;
Type One B, Nines Family Fractions
When the denominator ends in several nines, increase the denominator by one, then slide the decimal point to the left in both the numerator and the denominator the number of places as the number of nines. When we divide, we use digit-bundles in the dividend and generate the quotient in a set of digits whose number is the number of terminal nines in the denominator.
When F = 174/1299, then the A.F. = 1.74/13 (two nines means we use bundles of two digits);
When F = 1/6999, then the A.F. = 0.001/7 (three nines means we divide with bundles of three digits);
If the denominator ends in 1, 3, or 7, we may convert to an equivalent fraction of the nines family and proceed as above to form the auxilliary fraction.
When F = 36/121 = 324/1089, then the A.F. = 32.4/109;
When F = 53/93 = 159/279, then the A.F. = 15.9/28;
When F = 15/37 = 105/259, then the A.F. = 10.5/26.
Type Two, Ones Family Fractions (when the denominator ends in one)
When the fraction has a denominator ending in one, we form the auxilliary faction by dropping the one and decreasing the numerator by one. Then we slide the decimal point in both the numerator and the denominator to the left by the number of places as the number of terminal zeros in the new denominator.[2]
When F = 3/61, then the A.F = 2/60 = 0.2/6;
When F = 28/71, then the A.F. = 27/70 = 2.7/7;
When F = 1/81, then the A.F. = 0/80 = 0.0/8;
When F = 14/131, then the A.F. = 13/130 = 1.3/13;
When F = 1/301, then the A.F. = 0/300 = 0.00/3 (two terminal zeros means we divide with two-digit groups);
When F = 6163/8,001, then the A.F. = 6162/8000 = 6.162/8 (with 3-digit bundles);
When F = 2175/80,000,001, then the A.F. = 2174/80,000,000 = 0.0002174/8 (with digit-bundles of 7 dividend digits at-a-time).
We may convert threes family and sevens family fractions to the ones family also.
When F = 10/27 = 30/81, then the A.F. = 29/80 = 2.9/8;
When F = 5/67 = 15/201, then the A.F. = 14/200 = 0.14/2(with two-digit groups);
When F = 4/17 = 12/51, then the A.F. = 11/50 = 1.1/5.
Type Three, denominators ending in the other endings, 2, 3, 4, 5, 6, 7, and 8.
For these denominator endings use the nines family auxiliary fraction and count the number of units (above or below) that the ending is from the normal nine.
[edit] Dividing a Nines Family Fraction by using its Auxiliary Fraction
Given a fraction, we first write the auxiliary fraction. Next, we divide in the auxiliary fraction to generate one (or more) quotient digit(s) at-a-time. Then we write the quotient digit(s) and the remainder. The remainder at each step is prefixed to the just generated quotient digit for the next division.
This algorithm meets the Vedic ideal of mental math with one line notation.[3]
[edit] Type One A, Nines Family Fractions (when the denominator ends in 9)
Example One
Given a fraction, F = 1/169 we shall convert the fraction to a (repeating) decimal. First, we may estimate the quotient as about six thousandths. Then, set up the auxiliary fraction, AF = 0.1/17. The first dividend is 0.1. The working divisor is 17. Calculate one quotient digit at a time. Set down the remainder as a (sub-scripted) prefix to the quotient-digit just produced. Continue dividing to generate the quotient of the precision desired. Remember that the prefixed remainders are not parts of the quotient but only prefixes to the quotient-group in question and are dropped out of the answer. As the Swami wrote, the lower row is a mere scaffolding and goes out.[4]
As there are 168 remainders for the repeating decimal value of one one-hundred-sixty-ninth we may expect a maximum of 168 repeating digits. If the remainder were to be zero or 169, then the fraction would terminate as an exact decimal value. Here, we have a special case with 78 repeating digits. 169 = 132. 169-13=156. Even number, 156/2 = 78.
17 into 0.1 goes 0 rem 1. 17 into 10 goes 0 rem 10. 17 into 100 goes 5 rem 15. 17 into 155 goes 9 rem 2. (The third dividend, 155, is formed by the remainder 15 prefixed to Q-digit 5.)
F = 1/169 = .10. 10 100 155 29 121 27 101 165 129 107 56 53 23 61 103 16 150 79 114 126 77 ...
This all the notation that is needed. The calculating is mental using the multiples of 17: 17, 34, 51, 68, 85, 102, 119, 136, 153, 170.
F = 1/169 ≈ 0.00591715976331361...
Example Two
If the denominator is 73, we have a special case because (73)(137) = 10001. F = 1/73 = 137/10001 = (137)(9,999)/(99,999,999) = 0.0136,9863 repeating. Thus we may expect an eight-digit repeating decimal with complementary halves, i.e., the digits in the first half of the repeating decimal digit set are complements of nine for the digits of the second half.[5]
Example Three
When F = 3/73 = 9/219, then the A.F. = 9/22. The working divisor is 22. Prefix the remainder to the Q-digit.
F = 0.90 24 21 210 129 195 198 09 90 24 21 210...
F = 3/73 ≈ 0.04109589,0410... The fraction repeats after eight decimal places. Furthermore, the eight digits have complementary halves.
[edit] Type One B, Nines Family Fractions (when the denominator ends in several nines)
Example One
When F = 53/799, then the A.F. = 0.53/8
The working divisor is 8. As the denominator has two nines, we divide in bundles of two dividend digits, generating two quotient digits at-a-time. Prefix the remainder to the pair of quotient digits just generated.[6]
F = 53/799 = 0.506 263 732 491 361 145 18...
F = 53/799 ≈ 0.06633291614518...
[edit] Generating a Denominator with More Terminal Nines
An additional technique is available to find another A.F. with a smaller divisor. By a judicious choice of the multiplier one can produce an equivalent fraction with more nines in the denominator.[7]
When F = 1/7 = 7/49, then the A.F. = 0.7/5; We use a multiplier of 7 to produce an equivalent fraction, 7/49. Look at the denominator of the equivalent fraction, 49. Consider the 4. To build the 4 to a nine we need to add 5 in the tens place. The 5th multiple of 7 ends in 5, so we may use 5 tens or 57 as the multiplier.
F = 1/7 = 57/399, and the A.F. = 0.57/4 (dividing in bundles of two digits).
We may proceed likewise, until we have F = 1/7 = 142,857/999,999 giving a bundle of six nines and an A.F. = 0.142857/1. This means we have the repeating decimal digit set on sight because when we divide 0.142857 this bundle of six digits by one and we have no remainder and this bundle repeats!
[edit] Type Two, Ones Family Fractions
After forming the auxiliary fraction we divide in the first step, but here we prefix the remainder, not the the quotient digit, but to the Q-digit's complement from nineand using this as the next dividend. Then we carry on step-by-step, dividing to the precision desired.
As the numerator is reduced by one, when the fraction has a numerator of one, the A.F. will have a numerator of zero. Having a dividend of zero is not a problem because on the second step the complement of zero is nine, an adequate dividend.
Example One
F = 13/31, then the A.F. = 12/30 = 1.2/3
The working divisor is three. The lower row is the dividend.
F = 13/31 = 0. 4 1 9 3 5 4 8 3 8 7 0 9
Prefix R to complement of Q-digit: 05 28 10 16 14 25 11 26 21 02 29 20
F = 13/31 ≈ 0.419354838709...
[edit] Type Three (when the denominator ends in a 2, 3, 4, 5, 6, 7, or 8)
When the fraction has a denominator ending in a number not immediately above or below a multiple of ten or a power of ten we apply the Ānurūpya Sūtra, whereby, after prefixing each remainder to the quotient-digit in question, we have to add to (or subtract from) the dividend at each step, as many times the quotient-digit as the divisor (the denominator) is below (or above) the normal nine. The process can be wholly mental with practice![8]
Example One
If the denominator is 68, As 68=(4)(16), we may expect two fixed digits and 17 repeating digits. When F = 15/68, the A.F. = 1.5/7 (As D ends in 8, one below the normal ending, 9, we add the quotient-digit to the dividend at each step). The working divisor is 7. Divide to achieve the desired precision.
F = 15/68, A.F. = 1.5/7
A prefixed rem: F = 0.12 02 40 55 48 08 22 33 15 62 19 04 11 51
Plus the Q-digit: 2 2 0 5 8 8 2 3 5 2 9 4 1 1
7 into actual dividend: 14 04 40 60 56 16 24 36 20 64 28 08 12 52
F = 15/68 ≈ 0.22058823529411...
Example Two
When F = 163/275, A.F. = 16.3/28. (Since D ends in 5, four below the normal ending, 9, we add four times the Q-digit to the dividend at each step.) The working divisor is 28. Since D = 275 = (52)(11), we may expect two fixed digits, then a two-digit repeater. Remember that every factor of 2, 5, or 10 in the denominator generates one fixed decimal digit. Multiples of 28: 28, 56, 84, 112, 140, 168, 196, 224, 252, 280.
Prefix the rem: F = 0. 235 39 192 47 192 47
Plus four times the Q-digit: x 20 36 08 28 08 28
28 into the actual dividend: 163 255 75 200 75 200 75
F = 163/275 ≈ 0.592727...
[edit] References
- ^ Pages 255-256, Vedic Mathematics
- ^ Pages 259-262, Vedic Mathematics
- ^ Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas, Auxiliary Fractions, chapter XXVIII, pages 255-272.
- ^ Page 258, Vedic Mathematics
- ^ Page 227, Vedic Mathematics
- ^ Page 257, Vedic Mathematics
- ^ Page 259, Vedic Mathematics
- ^ Pages 262-263, Vedic Mathematics
- Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas, by Jagadguru_Swami_Sri_Bharati_Krishna_Tirthaji_Maharaja (1884-1960), Motilal Banarsidass Indological Publishers and Booksellers, Varnasi, India, 1965; reprinted in Delhi, India, 1975, 1978. 367 pages.
