Merely 16 Sutras and 13 Sub-Sutra form the basis of Vedic Mathematics. These sutras can be used to solve all types of Mathematical Problems!!!
These sutras had been long lost in the cryptic Vedas where the mathematical solutions were hidden in Sanskrit Verses for centuries. Sri Bharati Krsna Tirthaji's "Vedic Mathematics" recently discovered these techniques again and given it to all of us to use.
The 16 sutras, or rules, and 14 sub-sutras can be used to solve complex mathematical problems mentally, leaving your fellow students and colleagues in awe.
The 16 Sutras of Vedic Mathematics:
Sutra 1 : By one more than the one before
Sutra 2 : All from 9 and the last from 10
Sutra 3 : Vertically and crosswise
Sutra 4 : Transpose and apply
Sutra 5 : If the Samuccaya is the same it is zero
Sutra 6 : If one is in ratio the other is zero
Sutra 7: By addition and by subtraction
Sutra 8: By the completion or non-completion
Sutra 9: Differential calculus
Sutra 10: By the deficiency
Sutra 11: Specific and general
Sutra 12: The remainders by the last digit
Sutra 13: The ultimate and twice the penultimate
Sutra 14: By one less than the one before
Sutra 15: The product of the sum
Sutra 16: All the multipliers
The 13 Sub-sutras:
SS 1: Proportionately
SS 2: The remainder remains constant
SS 3: The first by the first and the last by the last
SS 4: For 7 the multiplicand is 143
SS 5: By osculation
SS 6: Lessen by the deficiency
SS 7: Whatever the deficiency lessen by that amount and set up the square of the deficiency
SS 8: Last totaling 10
SS 9: Only the last terms
SS 10: The sum of the products
SS 11: By alternative elimination and retention
SS 12: By mere observation
SS 13: The product of the sum is the sum of the products
We will work through these Sutras in the next few pages and also will give you lot of practice to get used to the techniques.
Check out the blog on the link here to see the problems in the present method of teaching mathematics in schools.
Sutra 1: "By one more than the one before"
provides a simple way of multiplication of numbers with the same first digit and the sum of their last unit digits is 10.
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
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):
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,
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.
Interested in learning how to convert any decimal to fractions??
Sutra 2: "All from nine and the last from 10"
comes handy when we wish to subtract a number such as 578 from a number having zeros equivalent or more than the number of number being subtracted e.g. 1,000. Each figure in 578 is subtracted from nine and the last figure is subtracted from 10, yielding 422. The interesting thing is that this problem can be solved from left to right the same way we write and read text.
This sutra can be extended to solve problems such as 60,000 minus 987. We simply reduce the number left after taking out the 0s to account for the number being subtracted (i.e 60 left after removing the 3 left 0s) by 1 and then apply the sutra, to get the answer 59,013.
The concept gives rise to Sub-Sutra 7: "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square of that deficiency". Let me explain this further:
For instance, in computing 92 we go through the following steps:
Let us take 10 as our base. To get 9 to the base 10 we have to subtract 1 from 10.
Since decrease 9 further by 1 to 8. This is the left side of our answer.
On the right hand side put the square of the deficiency, which is 12. Hence the answer is 81.
Similarly, 82-->(8-2)*10+ 22 --> 64, 72 = 49.
For numbers above 10,we take the surplus. For example:
112 = (11+1)*10 + 12 = 121
132 = (13+3)*10 + 32 = 169
and so on.
This is based on the fact that a2= (a + b)(a − b) + b2 where a is the number whose square is to be found and b is the deficit (or surplus) from nearest factor of 10.
FOR MORE DETAILS ON EACH OF THE SUTRAS CLICK ON THE LINKS AT THE TOP OF THE PAGE...