Auxiliary fraction

In Swami Bharati Krishna Tirtha's Vedic mathematics, the auxiliary fraction method is used to convert a fraction to its equivalent decimal representation. The "auxiliary fraction" is not a true fraction, but is simply a mnemonic aid used in the calculation. The method is essentially the long division algorithm adapted for mental calculation. It is simplest when the fraction's denominator is one less than a multiple of 10, when it uses the identity

: $a=qb+r\; Rightarrow\; 10a=q(10b-1)+(10r+q).$

Variants of the method used when the denominator is not one less than a multiple of 10 become progressively more complex but still in the realm of mental math or with one line of notation.

Dividing by powers of ten

To divide by a power of ten first 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 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.

Forming the auxiliary fraction

The formation of the auxiliary fraction depends on the denominator. There are four cases:

*Type One A: denominators which end in a single nine
*Type One B: denominators which end in several nines
*Type Two: denominators which end in one
*Type Three: denominators which end in the digits, 2, 3, 4, 5, 6, 7, and 8.

If the denominator ends in zero(s), its first non-zero digit (from the right) identifies the family of the denominator.

Type One A

When the fraction's denominator ends in a single nine use the "Ekādhika Purva." First replace the denominator by its "Ekādhika" which means adding one to the denominator and dividing the result by 10. Divide the numerator by 10 also. [Pages 255-256, "Vedic Mathematics"] Hereafter, F is the actual fraction to be converted to a decimal value and A.F. is the auxiliary fraction used to remember the original dividend and the recurrent divisor.

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

When the denominator ends in several nines, increase the denominator by one, then divide both numerator and denominator by a power of 10 equal to the number of terminal nines in the denominator.

Examples:

*When F = 174/1299, then the A.F. = 174/1300 = 1.74/13;

*When F = 1/6999, then the A.F. = 1/7000 = 0.001/7;

If the denominator ends in 1, 3, or 7, multiply both denominator and numerator by 9, 3 or 7 respectively to convert to an equivalent fraction in which the denominator ends in 9.

Examples:

* 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

When the fraction has a denominator ending in one, form the auxiliary fraction by subtracting one from the denominator and from the numerator. Then divide both numerator and denominator by a power of 10 equal to the number of terminal zeros in the new denominator. [Pages 259-262, "Vedic Mathematics"]

Examples:

* 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;

* When F = 6163/8001, then the A.F. = 6162/8000 = 6.162/8;

If the denominator ends in 3 or 7, multiply both denominator and numerator by 7 or 3 respectively to convert to an equivalent fraction in which the denominator ends in 1.

Examples:

* When F = 2/3 = 14/21, then the A.F. = 13/20 = 1.3/2;

* 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;

* When F = 4/17 = 12/51, then the A.F. = 11/50 = 1.1/5.

Type Three

For denominators that do not end in 1 or 9, use the nines family auxiliary fraction and count the number of units (above or below) that the ending is from the normal nine.

Using the auxiliary fraction - Type One

Given a fraction whose denominator ends in 9, first write the auxiliary fraction. Next divide in the auxiliary fraction to generate one (or more) quotient digit(s) at-a-time. Then 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. So if the quotient and remainder from one step are "q" and "r", the dividend for the next step is 10"r"+"q".

This algorithm meets the Vedic ideal of mental math with one line notation. ["Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas", Auxiliary Fractions, chapter XXVIII, pages 255-272.]

Word examples - Type One A

Example One

To convert the fraction F = 1/169 to a (repeating) decimal:

First, estimate the quotient as about six thousandths. Then, set up the auxiliary fraction, AF = 0.1/17. The first dividend is 0.1 and 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 - the lower row is a mere scaffolding and goes out. [Page 258, "Vedic Mathematics"]

As there are 168 remainders for the repeating decimal value of one one-hundred-sixty-ninth, there can be at most 168 digits before the decimal expansion repeats. If the remainder were to be zero or 169, then the fraction would terminate as an exact decimal value. There are in fact 78 digits before the decimal expansion of 1/169 repeats, and 78 is half of 156, which is the totient of 169.

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 quotient from the previous step, 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, this is 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. [Page 227, "Vedic Mathematics"]

Example Three

When F = 3/73 = 9/219, then the A.F. = 9/22. The working divisor is 22.The dividend at each step is created by prefixing the remainder to the quotient from the previous step.

F = 0.₉0 ₂4 ₂1 ₂₁0 ₁₂9 ₁₉5 ₁₉8 ₀9 ₉0 ₂4 ₂1 ₂₁0...

F = 3/73 ≈ 0.041095890410... The fraction repeats after eight decimal places. Furthermore, the eight digits have complementary halves (see Midy's theorem).

Worked example - Type One B

When F = 53/799, then the A.F. = 0.53/8

The working divisor is 8. As the denominator has two nines, the division proceeds in steps of two dividend digits at a time, generating two quotient digits at each step. The remainder is prefixed to the pair of quotient digits from the previous step to create the dividend for the next step. [Page 257, "Vedic Mathematics"]

F = 53/799 = 0.₅06 ₂63 ₇32 ₄91 ₃61 ₁45 18...

F = 53/799 ≈ 0.06633291614518...

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. [Page 259, "Vedic Mathematics"]

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 this bundle of six digits (0.142857) by one and we have no remainder and this bundle repeats!

Using the auxiliary fraction - Type Two

After forming the auxiliary fraction, divide in the first step, but create the next dividend by prefixing the remainder not to the quotient digit, but to the quotient digit's complement from nine. So if the quotient and remainder from one step are "q" and "r", the dividend for the next step is 10"r" + (9 - "q").

When the fraction has a numerator of one, the auxiliary fraction 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.

Worked example - Type Two

F = 13/31, then the A.F. = 12/30 = 1.2/3

The working divisor is 3. 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 quotient: 05 28 10 16 14 25 11 26 21 02 29 20

F = 13/31 ≈ 0.419354838709...

Using the auxiliary fraction - Type Three

When the fraction has a denominator that does not end in 1 or 9 apply the "Ānurūpya" Sūtra, whereby, after prefixing each remainder to the quotient-digit in question, 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. [Pages 262-263, "Vedic Mathematics"]

Worked examples - Type Three

Example one

If the denominator is 68, as 68=(4)(17), we may expect two fixed digits and 16 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. Continue dividing 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...

References

* "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.

See also

*Fraction (mathematics)