The Main Sutras

By one more than the one before.

All from 9 and the last from 10.

Vertically and Cross-wise

Transpose and Apply

If the Samuccaya is the Same it is Zero

If One is in Ratio the Other is Zero

By Addition and by Subtraction

By the Completion or Non-Completion

Differential Calculus

By the Deficiency

Specific and General

The Remainders by the Last Digit

The Ultimate and Twice the Penultimate

By One Less than the One Before

The Product of the Sum

All the Multipliers

 The Sub Sutras 

Proportionately

The Remainder Remains Constant

The First by the First and the Last by the Last

For 7 the Multiplicand is 143

By Osculation

Lessen by the Deficiency

Whatever the Deficiency lessen by that amount and set up the Square of the Deficiency

Last Totalling 10

Only the Last Terms

The Sum of the Products

By Alternative Elimination and Retention

By Mere Observation

The Product of the Sum is the Sum of the Products

On the Flag

Try a Sutra

Mark Gaskell introduces an alternative system of calculation based on Vedic philosophy

At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.

Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago.

Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.

The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.

1,000   - 564  =  436

1,000     -    5                         6                       4

 
                subtract       subtract          subtract
                 from             from                 from
                    9                         9                        10
                    ¯                         ¯                         ¯ 
                4                    3                   6

The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use.

The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply 32 by 44. We multiply vertically 2x4=8. Then we multiply crosswise and add the two results: 3x4+4x2=20, so put down 0 and carry 2. Finally we multiply vertically 3x4=12 and add the carried 2 =14. Result: 1,408.

             32       x       44         =       1,408

    
  A              3            2              Starting from the right  
                                x               multiply vertically

                   4           4               2 x 4 = 8

  B              3           2               Multiply crosswise
                                   3 x 4 = 12 and 2 x 4 = 8
                    4           4               Add them together 
                    _______
                     0          8               3 x 4 + 2 x 4 = 20
                  2                             Put down 0 and carry 2

  C                3          2               Finally multiply vertically
                 x                     3 x 4  =  12 and add the
                      4          4                carried over 2 = 14

         _______________
         14       0       8
                     2

We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.

All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either "special", in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.

Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.

We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.

  96    x    92     =       8,832

  A    96               4        (4 is the difference from base)     

        92               8       (8 is the difference from base)
         _____________

B    96               4      Subtract crosswise from the left
             
       92               8      96 - 8 = 88 or 92 - 4 = 88
       ______________
        88

C   96                4        Multiply vertically
                             x       4 x 8 = 32
      92                8
    ____________
      88               32