The Magic Of Vedic Maths 6ti4v

THE MAGICS OF VEDIC MATHS

The system of Vedic Mathematics was rediscovered from ancient Sanskrit texts earlier this century. The system uses sixteen word-formulae which relate to the way in which we use our mind. Vertically and Crosswise is one of these formulae. Its application in multiplying numbers is fairly well known now but in fact its range of application is very great- as shown in this article. MULTIPLICATION If you are not aware of its use in multiplication here is an example. Suppose we want to multiply 33 by 44:

Multiplying vertically on the right we get 3×4 = 12, so we put down 2 and carry 1 (written 12 above). Then we multiply crosswise and add the two results: 3×4 + 3×4 = 24. Adding the carried 1 gives 25 so we put 5 and carry 2 (25). Finally we multiply vertically on the left, get 3×4 = 12 and add the carried 2 to get 14 which we put down. The simple pattern used makes the method easy to and it is very satisfying to get the answer in one line. It is also easy to see why it works: the three steps find the number of units, number of tens and number of hundreds in the answer. This multiplication can also be carried out from left to right, and this has many advantages. Let us find 33 × 44 from left to right:

Vertically on the left, 3×4 = 12, put 1 and carry 2 to the right (12 above). Crosswise we get 3×4 + 3×4 = 24 (as before), add the carried 2, as 20, to get 44 and put

TARUN GEHLOT (B.E, CIVIL HONOURS)

down 44. Finally, vertically on the right 3×4 = 12, add the carried 4, as 40, to get 52which we put down. We always add a zero to the carried figure as shown because the first product here, for example, is really 30×40 = 1200 and the 200 is 20 tens. So when we are gathering up the tens we add on 20 more. This does not seem so strange when you realise that a similar thing occurs when calculating from right to left: when we started the first calculation above with 3×4 = 12 the 1 in 12 was counted as 1 in the next column even though its value is 10. Although the first method above is useful for mental multiplication the second method is better because we write and pronounce numbers from left to right and so it is easier to get our answers the same way. This method can be extended to products of numbers of any size. Another advantage of calculating from left to right is that we may only want the first one, two or three figures of an answer, but working from the right we must do the whole sum and get the most significant figure last. In the Vedic system all operations can be carried out from left to right (right to left is not excluded though) and this means we can combine operations: add two products for example. We can extend this further to the calculation of sines, cosines, tangents and their inverses and the solution of polynomial and transcendental equations (Nicholas et al, 1999). The same vertical and crosswise method can be used for algebraic multiplication's. For example (2x + 5)(3x + 1):

Either method will do. From the left we have DIVISION The above left to right method can be simply reversed to give us a one line division method. Suppose we want to divide 1452 by 44. This means we want to find a number which, when multiplied by 44 gives 1452, or in other words we want a and b in the multiplication sum:

Since we know that the vertical product on the left must for the 14 on the left of 1452, or most of it, we see that a must be 3.

This s for 1200 of the 1400 and so there is a remainder of 200. A subscript 2 is therefore placed as shown. Next we look at the crosswise step: this must for the 25 (25), or most of it. One TARUN GEHLOT (B.E, CIVIL HONOURS)

crosswise step gives: 3×4 = 12 and this can be taken from the 25 to leave 13 for the other crosswise step, b×4. Clearly b is 3 and there is a remainder of 1:

We now have 12 in the last place and this is exactly ed for by the last, vertical, product on the right. So the answer is exactly 33. It is not possible in this short article to describe all the variations but the method is easily extended for a) dealing with remainders, b) dividing any two numbers, c) continuing the division (if there is a remainder) to any number of figures, d) dividing polynomial expressions. The multiplication method described here simplifies when the numbers being multiplied are the same, i.e. for squaring numbers. And this squaring method can also be easily reversed to provide one line square roots: easy to do, easy to understand. ADDITION AND SUBTRACTION OF FRACTIONS The usual method using common denominators is cumbersome and difficult to learn. By contrast the Vedic method allows the answer to be written straight down.

We multiply crosswise and add to get the numerator of the answer and we multiply the denominators to get the denominator of the answer. This looks like "horizontally and crosswise" rather than "vertically and crosswise" but fractions can also be written: 2/3 + 4/7, in which case we have:

in which we see "vertically and crosswise". Subtraction is similar, we cross-multiply and subtract:

When the denominators are not relatively prime we may divide out the common factor and cross-multiply with these reduced figures (see Williams & Gaskell 1997). EQUATION OF A LINE ING TWO POINTS Find the equation of the line ing (5, 3) and (2, 7). By conventional methods we need to know or look up the appropriate formula:

We substitute the four values, simplify, remove the fraction, open the brackets and rearrange the equation to finally get 3y = -4x + 29. TARUN GEHLOT (B.E, CIVIL HONOURS)

Or, by the one-line Vedic method:

By vertically and crosswise: we subtract vertically in the first column to get the y-coefficient, 5 - 2 = 3, we subtract vertically in the second column to get the x-coefficient, 3 - 7 =-4, and we cross-multiply and subtract to get the absolute term, 5×7 - 3×2 =29. We can also solve all sorts of problems in coordinate geometry, transformations, trigonometry etc. and there are more advanced applications in 3-dimensional work, trigonometrical equations, differential equationS, complex numbers, simple harmonic motion and so on. In addition to the general methods described above the Vedic system offers many special methods which can be used when certain conditions are satisfied. These are often extremely effective and powerful. The final example is a special method. MULTIPLYING NUMBERS NEAR A BASE To multiply, say, 88 by 98 we observe that these numbers are close to the base of 100 and once again we obtain the answer by one line mental arithmetic:

We see that 88 is 12 below 100 and 98 is 2 below, as shown. Cross-subtracting we get 88-2 = 86 (or 98-12 = 86) for the first part of the answer, and multiplying vertically we get 12×2 = 24 for the second part. So 88 × 98 = 8624.

Vertically and Crosswise has a huge range of applications- and re it is just one of sixteen formulae used in Vedic Mathematics! The Vedic system is extremely coherent and unified, the methods are so easy they really amount to mental arithmetic Fasten up your calculations using Ancient Indian principles of Vedic Math’s

TARUN GEHLOT (B.E, CIVIL HONOURS)

Trick 1 : Multiply any two numbers from 11 to 20 in your head. Take 15 x 13 for example..Place the larger no. first in your mind and then do something like this Take the larger no on the top and the second digit of the smaller no. in the bottom. 15 3 The rest is quite simple. Add 15+3 = 18 . Then multiply 18 x 10 = 180 ... Now multiply the second digit of both the no.s (ie; 5 x 3 = 15) Now add 180 + 15 Here is the answer 180 + 15 = 195 . Think over it. This is a simple trick. It will help you a lot. Trick 2 : Multiply any two digit number with 11. This trick is much simpler than the previous one and it is more useful too. Let the number be 27 . Therefore 27 x 11 Divide the number as 2 _ 7 Add 2+ 7 = 9 Thus the answer is 2 9 7 Wasn't this one e simple. But there is one complication. If you take a number like 57 Thus _57 x 11 Divide the number as 5 _ 7 Add 5 + 7 = 12 Now add 1 to 5 and place 2 in the middle so the answer is 5+1_2 _7 = _627 Thus the answer is 627 Trick 3 : Multiply any number fro from m 1 to 10 by 9 To multiply by 9,try this: (1) Spread your two hands out and place them on a desk or table in front of you. (2) To multiply by 3, fold down the 3rd finger from the left. To multiply by 4, it would be the 4th finger and so on. (3) the answer er is 27 ... READ it from the two fingers on the left of the folded TARUN GEHLOT (B.E, CIVIL HONOURS)

down finger and the 7 fingers on the right of it.This one was really cool wasn't it Trick 4 : Square a two digit number ending in five This one is as easy as the previous ones but you have to pay a little more attention to this one . Read carefully :Let the number be 35 35 x 35 Multiply the last digits of both the numbers ; thus ___ 5 x 5 = 25 now add 1 to 3 thus 3 + 1 = 4 multiply 4 x 3 = 12 thus answer 1225 You will have to think over this one carefully.As 5 has to come in the end so the last two digits o the answer will be 25 . Add 1 to the first digit and multiply it by the original first digit . Now this answer forms the digits before the 25. Thus we get an answer . Trick 5 : Square any two digit number Suppose the number is 47 . Look for the nearest multiple of 10 . ie; in this case 50 . We will reach 50 if we add 3 to 47. So multiply (47+3) x (47-3) = 50 x 44 = 2200 This is the 1st interim answer. We had added 3 to reach the nearest multiple of 10 that is 50 thus 3x 3 = 9 This is the second interim answer. The final answer is 2200 + 9 = 2209 ... Practice This one on paper first. Trick 6 : Multiply any number by 11 . Trick number 2 tells you how to multiply a two digit number by 11 but what if you have a number like 12345678 . Well that is very easy if you our trick as given below . Read it carefully. Let the number be 12345678 __ thus 12345678 x 11 Write down the number as 012345678 ( Add a 0 in the beginning) Now starting from the units digit write down the numbers after adding the number to the right So the answer will be 135802458 This one is simple if you think over it properly all you got to do is to add the number on the right . If you are getting a carry over then add that to the number on the left. So I will tell you how I got the answer . Read carefully. The number was 12345678 ___ I put a 0 before the number ____ so the new number 012345678 Now I wrote ___ 012345678 Then for the answer 8+0=8 7 + 8 = 15 (1 gets carry carried over) 6+1+7 = 14 ( 1 gets carried over) TARUN GEHLOT (B.E, CIVIL HONOURS)

5 + 1 + 6 = 12 ( 1 gets carried over) 4 + 1 + 5 = 10 ( 1 gets carried over) 3+1+4=8 2+3=5 1+2=3 0+1=1 Thus the answer = 135802458 Trick 7 : Square a 2 Digit Number, for this example 37: Look for the nearest 10 boundary In this case up 3 from 37 to 40. Since you went UP 3 to 40 go DOWN 3 from 37 to 34. Now mentally multiply 34x40 The way I do it is 34x10=340; Double it mentally to 680 Double it again mentally to 1360 This 1360 is the FIRST interim answer. 37 is "3" away from the 10 boundary 40. Square this "3" distance from 10 boundary. 3x3=9 which is the SECOND interim answer? Add the two interim answers to get the final answer. Answer: 1360 + 9 = 1369 with practice this can easily be done in your head. I hope you enjoyed this article Please don’t forget to rate this!

TARUN GEHLOT (B.E, CIVIL HONOURS)