Vedic Mathematics

Vedic MathematicsVadiraja Veda- IntroductionThe Sanskrit word Veda is derived from the root Vid, meaning to know without limit.The word Veda covers all Veda-sakhas known to humanity.The Veda is a repository of all knowledge, fathomless, ever revealing as it is delved deeper.Vedas, Vedangas and UpangasFourteen-fold Vedic Knowledge:1) 4 Vedas (Rg, Yajur, Sama and Atharva) -originally oral in nature.2) 6 Vedangas (phonetics, grammar, etymology, metronomy(chandas), Astronomy and Astrology, and Kalpa)4) 4 Upavedas (Analysis, Logic, Puranic Literature) and Darma Sastra

Upavedas4 Upavedas (Ayurveda, Gandharvaveda, Dhanurveda and Sthapatyaveda)Vedas refer to a body of knowledge that reveals different means and ends available to the human being.

Vedas, Vedangas and UpangasFourteen-fold Vedic Knowledge:1) 4 Vedas (Rg, Yajur, Sama and Atharva) -originally oral in nature.2) 6 Vedangas (phonetics, grammar, etymology, metronomy(chandas), Astronomy and Astrology, and Kalpa)4) 4 Upavedas (Analysis, Logic, Puranic Literature) and Darma Sastra

Upavedas4 Upavedas (Ayurveda, Gandharvaveda, Dhanurveda and Sthapatyaveda)Vedas refer to a body of knowledge that reveals different means and ends available to the human being.

Reference: http://www.cs.rpi.edu/~moorthy/vmVedic mathematics - IntroductionSwami Bharati Krishna Tirtha (1884-1960), former Jagadguru Sankaracharya of Puri made a set of 16 Sutras (aphorisms) and 13 Sub - Sutras (corollaries) from the Atharva Veda.According to him, there has been considerable literature on Mathematics in the Veda-sakhas. Unfortunately most of it has been lost to humanity as of now. This is evident from the fact that while, by the time of Patanjali, about 25 centuries ago, 1131 Veda-sakhas were known to the Vedic scholars, only about ten Veda-sakhas are presently in the knowledge of the Vedic scholars in the country.

Swamy BharathiKrishna TirtajiHe was gifted with extraordinary ability and talent.At the age of 16 he was awarded as the SARASWATHI.At the age of 20 he had 7 MA degrees. They were in the fields of numeracy, language, psychology and other fields.He had that extraordinary talent to extract maths involved in vedas and define the use of it.To recollect the technique easily, he gave 16 basic sutras or aphorisms or rhyming statements. With which one can easily remember and solve any given problem quickly. (mostly in mind).He came out with 14 volumes of books on Vedic Maths. Which covered all the aspect of problem solving.Unfortunately these books were stolen/lost from his ashram. (Mighty loss to India)In his late ages, with his disciple he came out with a book called as VEDIC MATHS.

Vedic mathematics - AdvantagesSaves time in hand calculations. Vedic Mathematics system also provides us with a set of checking procedures for independent crosschecking of whatever we do. Vedic Mathematics provides with possibility of solving the same problem in different alternative ways. Mathematics, a dreadful subject is converted into a playful and blissful subjectVedic Mathematics enriches our knowledge and understanding of mathematics. Vedic Mathematics come to us as a boon to all competitions.Vedic Mathematics being most natural way of working can be learnt and mastered with very little efforts and in a very short time. The element of choice and flexibility at each stage keeps the mind lively and alert and develops clarity of mind and intuition. Holistic development of the human brain automatically takes place through Vedic Mathematics multidimensional thinking.Vedic Mathematics systematically utilizes both halves of the human brain and therefore the chances of development of intuition part of the personality are increased.

Vedic mathematics - AdvantagesHigh speed VLSI arithmetic architectures can be derived from Vedic MathsDue to its parallel and regular structure the Vedic algorithms can be easily laid out on silicon chip .

Basic terms and lawsBase NumbersBase numbers will be used in many cases during vedic math calculations. Let us see what it is. Base numbers of 10 includes 101, 102, 103, 104etc..,

Basic terms and lawsComplement of a NumberUsually to find the complement of any number, we will subtract the value from thenext highest base value. But in Vedic maths you can find complement of any values without using the base number.For example,425Here, subtract each digit of the value fromnineand the last digit fromten.9 - 4=59 - 2=710 - 5=5Complement of425 is 575.Basic terms and lawsComplement of a NumberExample 1:Let us find, what is the complement of 8977.9 - 8=19 - 9=09 - 7=210 - 7= 3So, the complement of8977 is 1023.Multiplication

The Vedic Method requires a subtraction (crosswise), anda single digit multiplication (vertically)Multiplication

Beejank and Vedic check Beejank - It means conversion of a number to single digit by getting the sum of digits of the number. If the sum of the digits turns out to be of more than 1 digit, then keep continuing this process till a single digit number is got (similar to Numerology!!!)Beejank of 61 =6+1=7Beejank of 187=1+8+ -> 16 1+6 =7Vedic check (Gunit Samuchyayah, sutra 15) The operation carried out with the numbers have the same effect as that carried out on their beejanks. It can be used to quickly identify errors in a mathematical calculation.

Beejank and Vedic check Check for addition 26 + 15 = 41Beejanks (LHS)8+6=14 =5Beejank (RHS)4+1=5

Check for Subtraction 26 - 15 = 11Beejanks (LHS)8-6=2Beejank (RHS)1+1=2

Check for Multiplication 91 x 96 = 8736Beejanks (LHS)1*6=6Beejank (RHS)24=>6

Check for Square 12 x 12 = 144Beejanks (LHS)3*3=9Beejank (RHS)144=>9

Beejank - DivisionDividend =(divisor *quotient)+ reminderBeejanks of Dividend =( Beejanks of divisor * Beejanks of quotient) + Beejanks of reminder

e.g. `134/6=22 reminder=2lhs = beejank of 134=8rhs = (beejank of 6*beejank of 22)+beejank of 2 = beejank of (6*4)+2= 6+2=8lhs=rhsif you get negative number add 9Beejank - DivisionDividend =(divisor *quotient)+ reminderBeejanks of Dividend =( Beejanks of divisor * Beejanks of quotient) + Beejanks of reminder

e.g. 4969/41=121 reminder=8lhs = 28=>10=>1rhs = >(5*4)+ 8 = > 2+8=> 10=>1lhs=rhsif you get negative number add 9Beejank - Simplification

Multiplication by 9, 99,999,9999, etc a) The left hand side digit (digits) is ( are) obtained by applying the ekanyunenapurvena i.e. by deduction 1 from the left side digit (digits) .e.g. ( i ) 7 x 9; 7 1 = 6 ( L.H.S. digit )b) The right hand side digit is the complement or difference between themultiplier and the left hand side digit (digits) . i.e. 7 X 9 R.H.S is 9 - 6 = 3.c) The two numbers give the answer; i.e. 7 X 9 = 63.Example 1: 8 x 9 Step ( a ) gives 8 1 = 7 ( L.H.S. Digit )Step ( b ) gives 9 7 = 2 ( R.H.S. Digit )Step ( c ) gives the answer 72Example 2: 15 x 99 Step ( a ) : 15 1 = 14Step ( b ) : 99 14 = 85 ( or 100 15 )Step ( c ) : 15 x 99 = 1485Example 3: 24 x 99

Multiplication by 9, 99,999,9999, etc Answer :

Example 4: 356 x 999

AnswerNote the process : The multiplicand has to be reduced by 1 to obtain the LHSand the rightside is mechanically obtained by the subtraction of the L.H.S fromthe multiplier which is practically a direct application of Nikhilam Sutra.

Multiplication by 9, 99,999,9999, etc

Multiplication by 9, 99,999,9999, etc We have dealt the cases wheni) the multiplicand and multiplier both have the same number of digitsii) the multiplier has more number of digits than the multiplicand.In both the cases the same rule applies. But what happens when the multiplier has lesser digits?i.e. for problems like 42 X 9, 124 X 9, 26325 X 99 etc.,For this let us have a re-look in to the process for proper understanding.Multiplication table of 9. a b2 x 9 = 1 83 x 9 = 2 74 x 9 = 3 6- - - - - - - - - -8 x 9 = 7 29 x 9 = 8 110 x 9 = 9 0Observe the left hand side of the answer is always one less than the multiplicand (here multiplier is 9) as read from Column (a) and the right hand side of the answer is the complement of the left hand side digit from 9 as read from Column (b).Multiplication by 9, 99,999,9999, etc Multiplication table when both multiplicand and multiplier are of 2 digits. a b11 x 99 = 10 89 = (111) / 99 (111) = 108912 x 99 = 11 88 = (121) / 99 (121) = 118813 x 99 = 12 87 = (131) / 99 (131) = 1287-------------------------------------------------18 x 99 = 17 82 ----------------------------19 x 99 = 18 8120 x 99 = 19 80 = (201) / 99 (201) = 1980The rule mentioned in the case of above table also holds good here.Further we can state that the rule applies to all cases, where the multiplicand and the multiplier have the same number of digits.Example: 124 X 009123 //// 9-123=1141230-114 = 1116Multiplicationnumbers with same first digit and sum of last digits equals 10This multiplication Vedic shortcut is applicable when,Rule 1:The first digit of both the numbers are same for 2 digit numbers,First two digits of both the numbers are same for 3 digit numbers and so on.Rule 2:Sum of the last digit of the numbers equals 10.Example:Step 1:Take the first digit and add it with 1. Multiply the resultant value with the first digit.For example.,56 x 54 =>(5 + 1) x 5 = 6 x 5 =30Step 2:Multiply the last digits of both the numbers.6 x 4 =24Step 3:Write down the Step 1 result followed by the Step 2 result. The result is56 x 54 = 3024Multiplication numbers with same first digit and sum of last digits equals 10 With further examples,Example 1:93 x 97Step 1:(9 + 1) x 9 = 10 x 9 =90Step 2:3 x 7 =21Step 3:Join the values,93 x 97 = 9021.Example 2:66 x 64Step 1:(6 + 1) x 6 = 7 x 6 =42Step 2:6 x 4 =24.Step 3:Join the values,36 x 34 = 4224. Example 3:366 x 364Step 1:(36 + 1) x 36 = 37 x 36 =1332Step 2:6 x 4 =24.Step 3:Join the values,366 x 364 = 133224.Multiplication numbers with same last digit and sum of first digits equals 10

MULTIPLICATION: LEFT TO RIGHT237*2We multiply each of the figures in 237 by 2 starting at the left.The answers we get are 4, 6, 14.Since the 14 has two figures the 1 must be carried leftwards to the 6.So 4, 6,14 = 474.Again we build up the answer mentally from the left: first 4, then 4,6=46,then 4, 6,14 = 474.Example 2: 236 7 =First we have 14,then 1 4,2 1 = 161,then 161,4 2 = 1652Example 3: 1 0 5 9 x 7=Example 4: l48 6 3 1 x 8=Example 5: 7468 X 5 =Example 6: 9999 x 9 =Multiplication Special Multiplication

MULTIPLICATION: Vertically and CrosswiseMultiply 3-Digit Numbers

MULTIPLICATION: Vertically and CrosswiseMultiply 3-Digit Numbers

MULTIPLICATION: Vertically and CrosswiseMultiply 3-Digit Numbers

SquaresSquares of numbers ending with 5452 = (4+1)x4 followed by 25 i.e. 5x4=20 so answer is 2025.Ask if anyone can prove this by algebra (Homework)Similarly when sum of last digit =10 46x44 = (4+1)x4 followed by (6x4) i.e. 2024

SquaresSquares of numbers ending with 5

Squares - of numbers close to the base of powers of 10 (10, 100, 1000, etc)Eg 1: 92 Here base is 10.The answer is separated in to two parts by a/Note that deficit is 10 - 9 = 1Multiply the deficit by itself or square it12 = 1. As the deficiency is 1, subtract it from the number i.e., 91 = 8.Now put 8 on the left and 1 on the right side of the vertical line or slashi.e., 8/1.Hence 81 is answer.Eg. 2: 962 Here base is 100.Since deficit is 100-96=4 and square of it is 16 and the deficiencysubtracted from the number 96 gives 96-4 = 92, we get the answer 92 / 16Thus the answer is 9216.Squares - of numbers close to the base of powers of 10 (10, 100, 1000, etc)Eg. 3: 9942 Base is 1000Deficit is 1000 - 994 = 6. Square of it is 36.Deficiency subtracted from 994 gives 994 - 6 = 988Answer is 988 / 036 [since base is 1000]Eg. 4: 99882 Base is 10,000.Deficit = 10000 - 9988 = 12.Square of deficit = 122 = 144.Deficiency subtracted from number = 9988 - 12 = 9976.Answer is 9976 / 0144 [since base is 10,000].Exercise: Find the value of 882 Squares - Squaring Numbers Near 50

Squares - General squaringThe Duplex, D, of a number1 digit D(n) = n2e.g. D(5) = 252 digits D(n) = twice the product of the digitse.g. D(26) = 2(2)(6) = 243 digits D(n) = twice the product of the outer digits +the square of the middle digit

Squares - General squaring

Squares - General squaring

Powers - Meru prasthar, calculating powers of numbers squares, cubes, x4, x5 etc.

Powers - Meru prasthar, calculating powers of numbers squares, cubes, x4, x5 etc.Example 2:96396=(100-4) Base is 100. Check meru prashtar, for cube the pattern is 1 3 3 1963 = 1x(-04)0 | 3x(-04)1 | 3x(-04)2 | 1x(-04)3=1 |12 |48|64= 0884736

Example 3:9983998=(1000-2) Base is 1000. Check meru prashtar, for cube the pattern is 1 3 3 19983 = 1x(-002)0 | 3x(-002)1 | 3x(-002)2 | 1x(-002)3=1 |006 |012|008= 0994011992

SubtractionUsing Vedic Maths, you can easily subtract two large numbers without using the concept of borrowing numbers. All you have to remember is the base values. Proceed with the steps given below to do subtraction using the vedic math shortcuts.Lets take an example,9 9 5 3 8 -7 8 6 2 9Step 1:Subtract the right most digit of both values. Here 9 is greater than 8, so subtract 9 from the nearest base value.10 - 9 = 1.Add the resultant value with 8,8 + 1 = 9Last digit of the answer is9.Step 2:If you have taken the complement previously, carry a dot to the preceding value. This dot means one3 - 2 + 1 = 3 - 3 = 0Step 3:5 is less than 6, so the nearest base is 10.10 - 6 = 4Now add the 4 with 5,4 + 5 =9Step 4:As we have taken complement, add one with the next value,8 + 1 = 9,9 - 9 = 0Step 5:Now take the left most digit of both the numbers and subtract them.9 - 7 = 2. Answer is,99538 -78629 = 20909Vinuculum Number (bar number)All From 9 and the Last From 10

Vinuculum Number (bar number)

Vinuculum Number (bar number)

Vinuculum Number (bar number)

Vinuculum Number (bar number)

Fun with mathsMagic squareMagic Squares are square grids, where numbers are written in equal number of cells, along rows and columns; so that when these numbers are added along rows ,columns or diagonals we get the same sum.Magic squares have fascinated humanity throughout the ages, and have been around for over 4,120 years. They are found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity and prevention of diseases.Fun with mathsMagic squareAs far back as 550AD, Varahamihira used a 4 by 4 magic square to decribe a perfume recipe, but the earliest known Indian writings about an order 3 square comes from 900AD, as a medical treatment!The 33 magic square has been a part of rituals in India since Vedic times, and still is today. The Ganesh yantra is a 33 magic square. There is a well-known 10th-century 44 magic square on display in the Parshvanath Jain temple in Khajuraho, India.The Kubera-Kolam is a floor painting (Rangoli) which is in the form of a magic square of order three. Kubera Kolam has a belief that when drawn the house will never be short of money.

Fun with mathsMagic square odd Rows and columnHere we learn to create magic squares of 3x3 and 5x5 dimension using the Paravarthya Method.Steps:Take consecutive numbers and put the first number in the central square of the top row. Start moving towards the next row above and next column on the right. If you land outside the magic square on top, transpose the number to the cell at the bottom in the same column. If you land outside the magic square on the right side, transpose the number to the cell at the leftmost column in the same row. If a cell is already occupied or if you reach a corner of the magic square, move to the cell below that.

816357492Fun with mathsMagic square ObservationsTotal of each row will be as per the following formula: X For example, for a 5x5 magic square, the order is 5 and for a 3x3 magic square it is 3. In the below example, the sum of each row/column is 25x3=75. The number in the middle square happens to be the number which lies exactly in the middle of the series. For example, 25 is the number in the middle of 21-29 series. Exercise: Do a magic square of 3x3 using 11 to 19.Fun with mathsMagic square 5x5 magic square using 1-25:

Exercise: Do a magic square of 7x7 using 1 to 49.Fun with mathsMagic square - 4x4 magic square We can also talk about 4x4 magic square (slightly different approach) if required. The Magic Square of 4x4 is determined by examining the sum of the diagonals of the Natural Square of 4x4. It is the same as 4(42 + 1)/2 if the series starts from 1. We begin by creating a 4 4 square matrix and then we draw two diagonal lines to get a figure as follows.

We then start at the upper left corner to put the number 1,2,3,...,14,15,16 into the cells. However, we do not put a number in any cell where the diagonal line appears. We start with 1, but that cell has a diagonal line in it, so we go to the next cell which is blank and enter a 2, then we put a 3 in the next cell. The last cell in the first row has a diagonal line, so we do not write in the 4. We go to the next row and enter 5 in the first cell, which is blank, the next two cells have a diagonal line, so we skip 6 and 7. We continue this pattern until we get to the last cell in the last row. Our square will look like this:

Fun with mathsMagic square - 4x4 magic square Now we begin in the lower right-hand corner and work our way back using the numbers 1,4,6,7,10,11,13, and 16. We put these number in the cells which originally had the diagonal lines starting with 1 in the lower right-hand corner. Our finished product looks like this:

We see that in our finished square every row, column and diagonal sums to the magic number 34, which is 4(42 + 1)/2. Exercise: Do a magic square of 4x4 using 11 to 26.

Fun with mathsMagic square - of order 4nWe can use almost the same process as we used to generate a fourth-order magic square to create any 4n 4n magic square. For example to create an 8 8 magic square. Again we begin with a square matrix of size 8 8 and draw all the diagonals in the four 4 4 block that make up the matrix to get.

We proceed as we did in the 4 4 case, but this time we will be using all the positive integers from 1 to 64. We start in the upper left-hand corner to put numbers in the cells. As before, we will only put the cell number in the cell if the cell is blank; that is, does not have a diagonal line in it. We continue this process until we reach the lower right-hand corner. Our square would then look like this:

Fun with mathsMagic square - of order 4nNow we begin in the lower right-hand corner and work our way back using the numbers 1,4,5,8,10,11,... and 64. We put these number in the cells which originally had the diagonal lines starting with 1 in the lower right-hand corner. Our finished product looks like this:

The magic number is 260 = 8(82 + 1)/2. You can check all the rows, columns and diagonals to see that they each have a sum of 260. ReferencesReferences:Vedic Mathematics book written by Jagadguru Swami Sri Bharati Krishna Teerthaji Maharaj (Pub: Motilal Banarasidas) http://www.indiadivine.org/content/files/file/40-vedic-mathematics-by-shankaracharya-bharati-krishna-tirtha-pdf/ www.vedamu.org/Veda/1795$Vedic_Mathematics_Methods.pdfhttp://tutorials.vedicmaths.org/Books/Natural%20Calculator/Contents.php Magical World Of Mathematics: Vedic Mathematics - by V G Unkalkar