The document describes the Urdhva Tiryak method of division from Vedic mathematics. It provides examples to illustrate the method. For algebraic divisions, the divisor terms are divided in descending order to obtain the quotient terms. The examples show finding the quotient and remainder when dividing polynomials using this method. In the recap, it notes this method is well-suited for algebraic expressions but more difficult for arithmetic calculations, and each of the Vedic division methods only apply to certain cases rather than providing a single general division algorithm.

1. Urdhva Tirayak Sutra
Vedic Mathematics
Composed by A V Prakasam for the benefit of his daughters -
from the book ‘Vedic Mathematics’
By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja
Argumental Division – Urdhva Tirayak Sutra

2. Division - Urdhva Tiryak Sutra(Contd.)
Argumental Division – Urdhva Tiryak Sutra
In addition to the Nikhilam method and the Paravartya method, which are of use
only in certain special cases, there is a third method of division called the
‘Urdhva Tiryak’. The following examples will explain and illustrate it;
Example 1. x2+2x+1 / x+1
i. The first term of the dividend is X2. The first term of the divisor is x.
Therefore the first term of the quotient should be x.
ii. The second term of dividend is 2x. Therefore we should get 2x if the
divisor and quotient are multiplied. This is possible only if the second
term of the quotient is 1
iii. Now the independent term in the dividend is 1 and if the independent
terms in the divisor and quotient are multiplied, we get 1.
iv. Therefore, the quotient is x+1 and the reminder is 0.

3. Division - Urdhva Tiryak Sutra(Contd.)
Example 2: 12x2-8x-32 / x-2
i. 12x2 divided by x gives us 12x.Therefore the first term of the
quotient is 12x
ii. 12x multiplied by -2 (of the divisor) gives us -24x.
iii. But we want -8x in the dividend.
iv. Therefore the second term of the quotient should be +16, because
16 multiplied by x gives 16x and we already have -24x from the
first term of the quotient and -24x+16x = -8x.
v. Therefore the quotient now is 12x+16.
vi. 16 multiplied by -2, of the divisor, gives -32 which is the last term
of the dividend. Therefore reminder is 0.

4. Division - Urdhva Tiryak Sutra(Contd.)
Example 3. x3+7x2+6x+5 / x-2
i. x3 divided by x gives x2. Therefore the first term of the quotient is x2
ii. X2 multiplied by -2 (of the divisor) is -2x2.
iii. But we have 7x2 in the dividend. Therefore the second term of the
quotient should be +9x (9x * x -2x2=7x2)
iv. 9x multiplied by -2 is -18x. But we have 6x in the divisor.
v. Therefore the third term of the quotient should be +24 (24x-18x=6x)
vi The quotient till now is x2+9x+24.
vii. 24 multiplied by -2 gives -48. The last term of the divisor is 5
viii. Therefore the reminder has to be +53 ( 53-48=5).
All the three examples given above may be a bit confusing in the initial stages.
But when once the method is understood, division of this nature becomes easy
and the procedure becomes simple. The examples in the next slide will give the
necessary practice.

5. Division - Urdhva Tiryak Sutra(Contd.)
Examples for Practice
Sl.No. Dividend / divisor Quotient Reminder
1. 16x2+8x+1 / 4x+ 1 4x+1 0
2. X4-4x2+12x-9 / x2-2x+3 X2-2x-3 0
3. X4+4x3+6x2+4x+1 / x2+2x+1 X2+2x+1 0
4. 12x4+41x3+81x2+79x+42 / 4x2+7x+6 0
3x2+5x+7
5. 2x3+9x2+18x+20 / x2+2x+4 2x+5 0
6. 6x4+13x3+39x2+37x+45 / 6x2+25x+143 548x+1332
X2-2x-9
7. 16x4+36x2+6x+86 / 4x2-6x+9 6x+5
4x2+6x+9
8. 2x5-9x4+5x3+16x2-16x+36 / X3-3x2-21/2x+53/4 33/4x+301/4
2x2-3x+1

6. Recap
Recapitulation and Conclusions
The three methods of division expounded are free from the following handicaps which the
traditional system suffers from;
(i) Multiplication of large numbers by trial numbers
(ii) Subtraction of large numbers from large numbers
(iii) Length, cumbersome and clumsy processes
(iv) Risk of errors being committed.
The Vedic system, although superior to the processes now in vogue, also suffer in some cases
from these disadvantages. Further all the three methods are suitable only for some
particular type or types of cases and none of them is suitable for general application to all
cases as explained below:
(i) Algebrac divisions - ‘Nikhuilam’ method is generally unsuitable.
‘Paravartya’ process suits them better
(ii) Arithmetical Computations – ‘Nikhilam is useful only when the
divisor digits are large i.e. 6,7,8,9. Not at
all useful helpful when divisor digits are
1.2.3.4.5. Only ‘Paravartya’ can be useful
in the latter cases.
(iii) ,’Urhva Tiryak’ sutra’s utility for Algebrac expressions is plin enough, but difficult in
respect of Arithmetic calculations
End of Urdhva Tityak Sutra

7. Recap
Recapitulation and Conclusions
The three methods of division expounded are free from the following handicaps which the
traditional system suffers from;
(i) Multiplication of large numbers by trial numbers
(ii) Subtraction of large numbers from large numbers
(iii) Length, cumbersome and clumsy processes
(iv) Risk of errors being committed.
The Vedic system, although superior to the processes now in vogue, also suffer in some cases
from these disadvantages. Further all the three methods are suitable only for some
particular type or types of cases and none of them is suitable for general application to all
cases as explained below:
(i) Algebrac divisions - ‘Nikhuilam’ method is generally unsuitable.
‘Paravartya’ process suits them better
(ii) Arithmetical Computations – ‘Nikhilam is useful only when the
divisor digits are large i.e. 6,7,8,9. Not at
all useful helpful when divisor digits are
1.2.3.4.5. Only ‘Paravartya’ can be useful
in the latter cases.
(iii) ,’Urhva Tiryak’ sutra’s utility for Algebrac expressions is plin enough, but difficult in
respect of Arithmetic calculations
End of Urdhva Tityak Sutra