Vedic maths tricks based on the 16 sutras or vedic maths formulae
Ekadhikena Purvena The Sutra (formula) Ekādhikena Pūrvena means: “By one more than the previous one”.
Let us deal with one simple example. Apply this sutra to the square of numbers ending in 5 say 25^2
In the number 25, the last digit is 5 and the ‘previous’ digit is 2. Hence, applying rule ‘one more than the previous one’, we get, 2+1=3. The Sutra, in this context, asks ‘to multiply the previous digit 2 by one more than itself, that is, by 3. This becomes the L.H.S (left hand side) of the result, that is,
2 X 3 = 6. The R.H.S (right hand side) of the result is kept as it is 5^2, that is, 25.
The answer is 625. You can try this for 35^2
Previous one = 3 one more = 3+1=4
Hence 4 x 3 = 12 keeping the left hand side as it is and finding its square we get 5^2= 25
Answer = 1225
Nikhilam navatascaramam Dasatah meaning : “all from 9 and the last from 10”
Corollary: Sisyate sesasamjnah
The above sutra when applied to multiply numbers, which are nearer to bases say 10, 100, 1000 i.e., to the powers of 10 (eg: 96 x 98 or 102 x 104).
The procedure of multiplication using the Nikhilam involves minimum number of steps, space, time saving and only mental calculation. The numbers taken can be either less or more than the base considered.
Case (i) : when both the numbers are lower than the base.
Solve 97 X 94. These 2 numbers are close to base 100.
Now following the rules, the working is as follows:
97-100 = -3
94-100 = -6
Deviations = 3 and 6
It is written down like this for easy understanding and calculation
LHS is product of deviations 6 x 3 = 18
RHS is cross subtraction of one of the multipliers with the deviation = 97-6 or 94-3 = answer for both will be same = 91
Case ( ii) : When both the numbers are higher than the base.
The method and rules follow as they are. The only difference is the positive deviation. Instead of cross – subtract, we follow cross – add.104 X 102. Base is 100.
Note: We are considering 04×02=08 and appending ’08’ and not just ‘4×2=8’. This is done because, we need to consider two digits in deviation as it the base 100 has two zeros. If the deviation is near 1000 then we need to consider 3 digits in the deviation (eg, 004 and not just 4).
Case ( iii ): One number is more and the other is less than the base.
In this situation one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted. To have a clear representation and understanding a vinculum is used. It proceeds into normalization. 13 X 7. Base is 10
Urdhva – tiryagbhyam is the general formula applicable to all cases of multiplication and also in the division of a large number by another large number. It means “Vertically and cross wise.”
Paravartya Yojayet ‘Paravartya – Yojayet’ means ‘transpose and apply’
Sunyam Samya Samuccaye The Sutra ‘Sunyam Samyasamuccaye’ says the ‘Samuccaya is the same, that Samuccaya is Zero.’ i.e., it should be equated to zero. The term ‘Samuccaya’ has several meanings under different contexts.
Anurupye – Sunyamanyat The Sutra Anurupye Sunyamanyat says : ‘If one is in ratio, the other one is zero’. We use this Sutra in solving a special type of simultaneous simple equations in which the coefficients of ‘one’ variable are in the same ratio to each other as the independent terms are to each other. In such a context the Sutra says the ‘other’ variable is zero from which we get two simple equations in the first variable (already considered) and of course give the same value for the variable.
Sankalana – Vyavakalanabhyam This Sutra means ‘by addition and by subtraction’. It can be applied in solving a special type of simultaneous equations where the x – coefficients and the y – coefficients are found interchanged.
Puranapuranabhyam The Sutra can be taken as Purana – Apuranabhyam which means by the completion or non – completion. Purana is well known in the present system. We can see its application in solving the roots for general form of quadratic equation.
Calana – Kalanabhyam In the book on Vedic Mathematics Sri Bharati Krishna Tirthaji mentioned the Sutra ‘Calana – Kalanabhyam’ at only two places. The Sutra means ‘Sequential motion’.
Ekanyunena Purvena The Sutra Ekanyunena purvena comes as a Sub-sutra to Nikhilam which gives the meaning ‘One less than the previous’ or ‘One less than the one before’.
Anurupyena The upa-Sutra ‘anurupyena’ means ‘proportionality’. This Sutra is highly useful to find products of two numbers when both of them are near the Common bases i.e powers of base 10 .
Adyamadyenantya – mantyena The Sutra ‘ adyamadyenantya-mantyena’ means ‘the first by the first and the last by the last’.
Yavadunam Tavadunikrtya Varganca Yojayet The meaning of the Sutra is ‘what ever the deficiency subtract that deficit from the number and write along side the square of that deficit’. This Sutra can be applicable to obtain squares of numbers close to bases of powers of 10.
Antyayor Dasakepi The Sutra signifies numbers of which the last digits added up give 10. i.e. the Sutra works in multiplication of numbers for example: 25 and 25, 47 and 43, 62 and 68, 116 and 114. Note that in each case the sum of the last digit of first number to the last digit of second number is 10. Further the portion of digits or numbers left wards to the last digits remain the same. At that instant use Ekadhikena on left hand side digits. Multiplication of the last digits gives the right hand part of the answer.
Antyayoreva ‘Atyayoreva’ means ‘only the last terms’. This is useful in solving simple equations of the following type. The type of equations are those whose numerator and denominator on the L.H.S. bearing the independent terms stand in the same ratio to each other as the entire numerator and the entire denominator of the R.H.S. stand to each other.
Lopana Sthapanabhyam Lopana sthapanabhyam means ‘by alternate elimination and retention’. Consider the case of factorization of quadratic equation of type ax2 +by2 + cz2 + dxy + eyz + fzx This is a homogeneous equation of second degree in three variables x, y, z. The sub-sutra removes the difficulty and makes the factorization simple.