1. Urdhva-Tirayk Sutra
Multiplication
Urdhva-Tirayak Sutra
Principle : (ax2+bx+c)*(dx2+ex+f)=
x4*ad+x3(ac+bd)+x2(af+bc+cd)+x(bf+ce)+cf
Explanation: - Any number can be written as the sum of digits multiplied by a power of 10
e.g. 8 = 100*8
18= 101*1+100*8
218 = 102*2+101*1+100*8 etc.
- Replace x by 10 in the above equation, and a, b,c etc by digits.
- The left hand side of the equation represents the multiplication of two
numbers (First one is the multiplicand and the second one, multiplier)
- The right side of the equation is the result (Product)
- The coefficient of x4 is the first digit of the product, the coefficient x3 is the
second digit of the product and so on.
(Contd.)
1

2. Urdhva-Tirayk Sutra
Thumb Rule: A close observation of the equation of the previous slide would
reveal the following thumb rules for multiplication of two numbers with three
digits each;
- The first digit of the product is got by the vertical multiplication of the first digits from
the left side.
- The second digit is got by the cross-wise multiplication of the first two
digits and by the addition of the two products
- The third digit is obtained by summing the results of multiplication of the first digit of
the multiplicand by the last digit of the multiplier, of the middle one by the middle one
and the last one by the first one.
- The fourth digit is obtained by summing the results of multiplication of the second
digit of the multiplicand by the third digit of the multiplier and the third digit of the
multiplicand by the second digit of the mutiplier
-The last digit of the product is obtained by multiplying the last digit of the
multiplicand by the first digit of the multiplier.
This rule can be extended to numbers containing any number of
digits.
2

3. Urdhva-Tirayk Sutra
Pictorial Explanation of the
rule:
142 * 327
First digit: Second digit Third digit
1 4 2 1 4 2 1 4 2
3 2 7 3 2 7 3 2 7
3 14 21
Fourth digit Fifth digit
1 4 2 1 4 2
= 3 | 14 | 21 | 32 | 14
3 2 7 3 2 7
32 14
= 3+1 | 4+2 | 1+3 | 2+1 | 4
= 46434
3

4. Urdhva-Tirayk Sutra
Examples:
Note:In Example 3, the sum of the
1. 12*13 = 1*1|1*3+2*1|2*3 products gives 30 and the digit in
= 1 | 5 | 6 = 156 ten’s place i.e. 3, is Carried forward
to the left of the ‘ | ’ and added to 9.
Similarly, 7*3 gives 21 and the 2,
2. 23*21 = 2*2 | 2*1+3*2 | 3*1 being a carry, is added to the 0 on
= 4 | 8 | 3 = 483 the left of the ‘ | “.In these
3 37*33 = 3*3 | 3*3+7*3 | 7*3 exercises,
the ‘Carry’s’ are shown as
= 9 |30 | 21 subscripts and ‘negative digits’ with
= 9+3 | 0 +2| 1 a bar on the top of the digit, for
= 1221 better understanding.
4. 111*111 = 1*1 | 1*1+1*1 | In Example 4, the digits of the
1*1+1*1+1*1 | multiplicand and multiplier are
1*1+1*1 | 1*1 having the same color code, but the
Digits in the multiplier are in Bold’.
=1 | 2 | 3 | 2 | 1 This would enable a better
=12321 understanding of the pattern of
cross multiplications of digits from
multiplicand and multiplier in a
systematic manner.
4

5. Urdhva-Tirayk Sutra
5. 108*108
108
108
1*1 | 1*0+0*1 | 1*8+0*0+8*1 | 0*8+8*0 | 8*8
= 1 | 0 | 16 | 0 | 64
= 1 | 0+1 | 6 | 0+6 | 4
= 1| 1 | 6 | 6 | 4
= 11664
Note the subscripts, which are carried and
Added to the previous digits.
6. 321*52
= 321*052
321
052
3*0 | 3*5+2*0 | 3*2+2*5+1*0 | 2*2+1*5 | 1*2
= 0 | 15 | 16 | 9 | 2
= 0+1 | 5+1 | 6 | 9 | 2
= 16692
5

6. Urdhva-Tirayk Sutra
Owing to their relevance to this context, a few algebraic examples of the
Urdhva-Tiryak type are given below:
1. (a+b)*(a+9b)
a+b
a+9b
a2+10ab+b2
2. (a+3b)*(5a+7b)
a+3b
5a+7b
5a2+22ab+21b2
3. (3x2+5x+7)*(4x2+7x+6)
3x2+5x+7
4x2+7x+6
12x4+41x3+81x2+79x+42
6

7. Urdhva-Tirayk Sutra
The Use of the Vinculum
The multiplications by digits higher than 5 may some times be facilitated by the
use of the vinculum.
Note: A negative digit in a number (not negative number) is represented by a bar on the top of the
digit. This is called vinivulum
Ex: 1. In the number 576, the digits 7 and 6 can be written with viniculum as follows;
500+76
=500+(100-24)
=600-20-4
=624
Ex 2. The number 73 can be written as follows;
73 = 70+3
= (100-30)+3
= 1+(-3)+3
=1 3 3
But the vinculum process is one which one must very carefully practice,
before one resorts to it and relies on it.
7

8. Urdhva-Tirayk Sutra
Miscellaneous Examples:
1. 73*37
By urdhva-tirayak method: By vinculum method
73 133
37 043
2181 04519
52 12
2701 2701
Obviously, this method is
better.
2. 94*81
urdhva-tirayak : Vinculum: Nikhilam:
94 114 94 -6
81 121 81 – 19
72414 13794 (175-100) |
114
= 7614 = 7614 = 7614
8

9. Urdhva-Tirayk Sutra
Practical Applications in Compound Multiplication:
A. Area of a rectangle
Examples:
1. 5’11” * 7’8”
Suppose 12” = x, then the above multiplication can be written as
( 5x+11) * (7x+8) 5x+11
= 35x2+117x+88 7x+8
= 35x2+(108x+9x)+88
= 35x2+(9x*x+9x)+(7x+4)
= 35x2+9x2+16x+4
= 44x2+ (12+4)x+4
= 44x2 + (x+4)x + 4
= 45x2 + 4*12 + 4
= 45x2 + 52
= 45sq.ft+ 52 sq.in
2. Similarly, work out 3’7” * 5’10”
The answer should be 44 sq.ft + 124 sq.in
9

10. Urdhva-Tirayk Sutra
Volume of Parallelpipeds
Examples:
1. 3’7” * 5’10” * 7’2”
Suppose 12” = x, then the above multiplication can be written as
(3x+7) * (5x+10) * (7x+2)
= (15x2+65x+70) * (7x+2)
=((15x2+(5x+5)*x+ (5x+10)) * (7x+2)
= (20x2+10x+10) * (7x+2)
=140x3+110x2+90x+20
= 140x3+(9x+2)x2+(7x+6)x+(x+8)
= 149x3+9x2+7x+8
= 149 cuft +9*144 cuin+7*12 cuin+8 cuin
= 149cuft and 1388 cuin
Questions relating to paving, carpeting, ornamenting etc can be
Readily answered by this method.
Example: At the rate of 7 annas, 9 pices per foot, how much 8 yards,
1 foot, 3 inches costs?
( Note: 1 Re=16 Annas and 1 Anna = 12pices
8 yards and 1 foot = 25 ft)
Answer; 25 ft – 3 in
7 annas – 9 pies
= 175 annas| 246 pies | 9*3/12 pies
= 175 annas + ( 20 annas + 6 pies) + (2 1¼ pies)
or 195 annas, 8 1/4 pies or Rs 12/3/8¼
End of chapter 1B – Urdhva-tiravk sutra 10