Vedas and Mathematics
An interesting application of this formula is in computing squares of numbers ending in five. Consider:
35 × 35 = ((3 × 3) + 3)*10 + 25 = 1225 or
= (3 x 4) *10 +(5 x 5) = 1225
125 x 125 = (12 x 13)*10 + 25 = 15625
Explanation: The latter portion is multiplied by itself (5 by 5) and the previous portion is multiplication of first digit and the digit higher to the first digit resulting in the answer 1225
This is a simple application of (a+b)2 = a2 + 2ab + b2 ....Here 2b = 10
It can also be applied in multiplications when the last digit is not 5 but the sum of the last digits is the base (10) and the previous parts are the same. Consider:
37 × 33 = (3 × 4)*100 + (7 × 3) = 1221
29 × 21 = (2 × 3)*100 + (9 × 1) = 609
This uses (a + b)(a − b) = a2 − b2 twice combined with the previous result to produce: (10c + 5 + d)(10c + 5 − d) = (10c + 5)2 − d2 = 100c(c + 1) + 25 − d2 = 100c(c + 1) + (5 + d)(5 − d).
The Sutra is very useful in its application to convert fractions into their equivalent decimal form. Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.
Method 1: Using multiplication. The sutra "one more than the one before" provides a simple way of calculating values like 1/x9 (e.g 1/19, 1/29, etc). Let's take one 1/x9 and calculate e.g. 1/19. In this case, x=1. To convert 1/19 to decimals, since 19 is not divisible by 2 or 5, the fractional result is a purely circulating (recurring) decimal, i.e the digits would repeat themselves after some time. (If the denominator contains only factors 2 and 5 it is a purely non-circulating decimal i.e it is a perfect decimal, else it is a mixture of the two.).
So to get the decimals of 1/19, we start with the numerator digit i.e 1 in this case and Multiply this by "one more" (1+x), in this case: 1+1 = 2.
21 Multiplying 2 by 2, followed by multiplying 4 by 2
421 → 8421 Now, multiplying 8 by 2, sixteen
68421 1 ← carry
If the result of the multiplication is greater than 10, keep (value - 10) and keep the "1" as "carry over" which you'll add to the next digit.
multiplying 6 by 2 is 12 plus 1 carry gives 13
368421 1 ← carry
Continuing
7368421 → 47368421 → 947368421
Now we have 9 digits of the answer. There are a total of 18 digits (= denominator − numerator i.e 19-1 = 18) in the answer. The last 9 digits can be computed by complementing the lower half (with its complement from nine i.e number + complement = 9):
052631578
947368421
Thus the result is 1/19 = 0.052631578,947368421 repeating.
If you picked up 1/29, you'll have to do it till 28 digits (i.e. 29-1). You'll get the following
1/29 = 03448275862068,
96551724137931
Run this on your calculator and check the result!
I do not like this method a lot as it seems academic. Why would someone want to find all the numbers in the ratio, until he is doing some research. For all practial purposes 4-5 digits of the decimal (from the left) are enough. So the next technique is my favoured method.
Method 2: Using division. The earlier process can also be done using division instead of multiplication.
For A/X9, We divide A by (1+X). Incase of 1/19, we divide 1/(1+1), the answer is 0 (lets say N) with remainder 1 (lets say D)
0.0: D (in this case 1) carries forward and become (D*10 + N) i.e. 10. This is then divided by 2 for N = 5
0.05: Next 05 divided by 2 with answer as 2 and remainder 1
0.052: Next 12 (1 from earlier remiander and 2 from the answer) is divided by 2 for answer 6
0.0526: and so on.
So the asnwer of 1/19 = 0.0526.
Consider another example, 1/7, this same as 7/49 which has last digit of the denominator as 9. The previous digit to 9 in 49 is 4. So by the sutra we take one more i.e. 5. So we divide the numerator by 5, that is,
...7/5: 1 R=2; 21/5: 4 R=1; 14/5: 2 R=4, 42/5: 8 R=2 and so on...so the answer is 1/7 = 0.1428
Now try to convert 18/19 into decimals.
(Numerator 18 to be divided by x+1=2)...18/2:9 R=0 =>9/2:4 R:1 =>14/2: 7 R=0 =>7/2: 3 R=1 and so on.
So the answer for 18/19 is 0.9473...
Now try 18/29.Can you do it mentally? The answer is 0.6206...
Note this technique can be used for conversion of vulgar fractions ending in 1, 3, 7 including 9 such as 1 / 11, 1 / 21, 1 / 31 - - -- ,1 / 13, 1 / 23, - - - -, 1 / 7, 1 / 17, - - - - - by multiplying appropriate factors to the numerator and denominator to make them ed with 9.
1/11 = 9/99 --> 0.09090909
1/13 = 3/39 --> 0.07692307
Amazed at the speed at which you can calculate these sums MENTALLY!!!
People say there are so many small things that we have to remember, instead of one general method of division or multiplication. My answer is once you get used to it the effort in remembering such an useful method is far surpassed by the ease and speed.
Note: 1/17 = 7/119 --> This technique becomes a bit difficult when you try to solve with 1/x9, where x is greater than 9. Will discuss other techniques for these type of big fractions.
LET US NOW LEARN HOW TO CONVERT ANY FRACTION TO A DECIMAL