Cube Roots in Vedic Mathematics

Contributed by:

Harshdeep Singh Fri, Apr 08, 2022 12:54 PM UTC

This PDF contains :
Abstract,
Key Words,
1. INTRODUCTION,
2. HISTORY OF CUBE ROOTS,
3. VEDIC METHODS TO FIND CUBE ROOTS
OF PERFECT CUBES,
4. DIFFERENT CASES FOR PERFECT CUBES,
4.1. Rules for all three cases,
4.1.1. Illustration for Case I,
4.1.2. Illustration for Case II,
4.1.3. Illustration for Case III,
5. CONCLUSION,
REFERENCES

1. ISSN NO. 2456-3129
International Journal of Engineering, Pure and Applied Sciences,
Vol. 3, No. 1, March-2018
Cube Roots in Vedic Mathematics
Krishna Kanta Parajuli
Associate Professor (Mathematics), Nepal Sanskrit University, Nepal.
E-mail: kknmparajuli@gmail.com
Abstract– Finding the cube roots of a number by conventional method is time consuming and tedious job with
more complexity. However, using Vedic methods it becomes interesting and fast too. There are special and
general methods to find cube roots in Vedic Mathematics. This article only focuses on special cases and
explains how Vedic method can be applied to extract the cube roots of any perfect cube number as well as
bigger and bigger numbers to make it easier for students to understand. This article only concentrates to
illustrate real part of cube roots not for imaginary roots. The method for determining the cube root was firstly
provided by the great Indian Mathematician or astronomer Aryabhatta in 556 B.S. and father of Vedic
Mathematics Bharati Krishna Tirthaji Maharaja's method is likely to Symmetry with Aryabhatta's method. Here
this article focuses on only the Vedic methods not on Aryabhatta’s.
Key Words:– Cube Roots; Perfect cube; ljnf]sgd; ljhfÍ; gjz]if; Vedic Mathematics; Aryabhatta.
Vedic mathematics ljnf]sgd and ljhfÍ are mostly
1. INTRODUCTION
used for it.
The third power of the number is called its cube, and
the inverse operation of finding a number whose cube
2. HISTORY OF CUBE ROOTS[1]
is n is called extracting the cube root of n. In
mathematics, a cube root of a number x is a number y The method for determining the cube root was firstly
such that y3 = x. In Mathematics, each real number provided by the great Mathematician or Astronomer
(except 0) has exactly one real cube root (which is Aryabhatta in 556 B.S. . Brahmagupta (in 685 B.S.),
known as principal cube root) and two imaginary cube Shreedhar (in nearly 9th Century of B.S.), Shreepati
roots, and all non-zero complex number have three (in 1096 B.S.), Bhaskar (in 1207 B.S.), Narayan (in
distinct complex (imaginary) cube roots. For example, 1413 B.S.) and other scholars followed the method of
the cube root of 8 are: 2, –1 + i 3 , –1 - i 3 whereas Aryabhatta's to find cube root of numbers. Aryabhatta
cube roots of 8i are: –2i, 3 + i, – 3 + i. Thus, the has also mentioned in his book, ul0ftkfb, a method to
cube root of a number x are the numbers of which extract the cube root of any number, but the method is
satisfy the equation y3 = x. too complex to understand the fifth sloka of
The present existing curriculum taught in our school Aryabhatta's book æul0ftkfbÆ mentioned as:
level uses only the method of factorization of a
number to extract cube roots. Finding the cube roots c3gfb eh]b låtLoft lqu'0f]g 3g:o d"nju]{0f .
of a number by conventional method is time ju{ l:qk"j{ u'l0ft M zf]Wo M k|ydfb 3g:o 3gft\ .. [2]
consuming and tedious job with more complexity. -cfo{e6Losf] ul0ftkfb_
However, using Vedic methods it becomes interesting
and fast too. We can find the cube root of any number With a slight modification of method of Aryabhatta,
within very short period by just looking at the number Samanta Chandra Sekhar came up with his own way
with perfect practice. There are special and general to determine the cube root in 1926 B.S. Even the
methods for calculating cube root of the number. modern mathematical practices follows the same way
There is a general formula for n terms, where n being to find the cube root as Aryabhatta used, with few
any positive numbers. Special parts also specific for tweaks and turns. Bapudev Shastri was a latter scholar
special numbers of cube with certain digits of to Chandra Sekhar who came up with his method,
numbers. This article only concentrates to illustrate accompanied by about half of the method provided by
the specific methods of Vedic Mathematics for real Aryabhatta in 1940 B.S.. However, 1952 B.S. became
part of cube roots not for imaginary roots. It explains the most wonderful year for "cube root" as Gopal
how Vedic method can be applied to extract the cube
[1]
roots of any perfect cube number as well as bigger Pant, Nayaraj [2037 B.S.]: ækl08t uf]kfn kf08] / pgsf]
and bigger numbers to make it easier for students to 3gd"n Nofpg] /LltÆ, g]kfn /fhsLo k|1f k|lti7fg, pp. 86–87.
[2]
understand. Among the 16 Sutras and 13 sub-sutras of Pant, Nayaraj [2037 B.S.]: ækl08t uf]kfn kf08] / pgsf]
3gd"n Nofpg] /LltÆ, g]kfn /fhsLo k|1f k|lti7fg, pp. 64
49

2. ISSN NO. 2456-3129
International Journal of Engineering, Pure and Applied Sciences,
Vol. 3, No. 1, March-2018
Pandey came up with the must excellent and
Table 2
practicable method for determining cube roots.
Brahmangupta express his method for finding cube ljhfÍ of Cube
roots as: Number Cube
(Using gjz]if)
1 1 1
5]bf]83gfb\ åLltofb\ 3gd"ns[ltl:q;ª u'0ffKts[ltM . 2 8 8
zf]Wof lqk"j{u'l0ftf k|ydfb\ 3gtf] 3gf] d"nd\ .. & .. [3] 3 27 0
-a|fXd:k'm6l;4fGt ul0ftfWofo_ 4 64 1
5 125 8
All the mathematician after Aryabhatta followed his 6 216 0
method with some modification. Specially, father of 7 343 1
Vedic Mathematics Bharati Krishna Tirthaji 8 512 8
Maharaja's method is likely to Symmetry with 9 729 0
Aryabhatta's method.
From the table, it can be concluded that, if the digit
3. VEDIC METHODS TO FIND CUBE ROOTS sum of the number is found to be 0, 1 or 8 then the
OF PERFECT CUBES given number is a perfect cube. It should also be
noted that this is necessary but not sufficient condition
To calculate cube root of any perfect cube quickly, we to test the perfect cube number. i.e. if the ljhfÍ of the
need to remember the cubes from1 to 10. cube does not match with the ljhfÍ (digit sum) of
13 = 1; 23 = 8; 33 = 27; 43 = 64; 53 = 125 cubes of cube roots then the answer is definitely
6 = 216; 7 = 343; 8 = 512; 9 = 729; 103 = 1000
3 3 3 3
wrong. But if they match each other than the answer
From the above cubes of 1 to 10, we need to is most likely correct and not definitely correct.
remember the following facts:
i) If the last digit of perfect cubes are 1, 4, 5, 6, 9 or
10 then the last digit of cube roots are same. Example: 2744 may be a perfect cube because
ii) If the last digit of perfect cubes are 2, 3, 7 or 8 ljhfÍ of this number is 2 + 7 + 4 + 4 = 8
then the last digit of cube roots are 8, 7, 3 or 2 (by using gjz]if method)
respectively.
It's very easy to remember the relations given above The left digit of a cube root having more than 7 digits,
as follows: or ten's digit of cube root having less than 7 digits can
be extracted with the help of the following table:
Table 1
Table 3
1 ⇒1 Same numbers
Left-most pair of the Nearest Cube Roots
8⇒2 Complement of 8 is 2
Cube Roots
7⇒3 Complement of 7 is 3 1–7 1
4⇒4 Same numbers 8 – 26 2
5⇒5 Same numbers 27 – 63 3
6⇒6 Same number 64 – 124 4
3⇒7 Complement of 3 is 7 125 – 215 5
2⇒8 Complement of 2 is 8 216 – 342 6
9⇒9 Same number 343 – 511 7
0⇒0 Same number 512 – 728 8
729 – 999 9
i.e. 8 ⇔ 2 and 7 ⇔ 3, remaining are same. Moreover,
it can be concluded that the cubes of the first nine
4. DIFFERENT CASES FOR PERFECT CUBES
natural numbers have their own distinct endings; and
there is no possibility of overlapping or doubt as in There are three cases:
the case of square. (i) Cube root of a number having less than 7 digits.
There is another question arises that the given number (ii) Cube root of a number having more than 7 digits
is perfect cube or not? Is there any rule to notify it? but less than 10 digits.
Yes, the Vedic method gjz]if determined the number (iii) Cube root of a number greater than 7 digits but
whose cube root is to be extracted is a perfect cube or ending with even numbers.
not. Observe the table:
Pant, Nayaraj [2037 B.S.]: ækl08t uf]kfn kf08] / pgsf]
3gd"n Nofpg] /LltÆ, g]kfn /fhsLo k|1f k|lti7fg, pp. 64
50

3. ISSN NO. 2456-3129
International Journal of Engineering, Pure and Applied Sciences,
Vol. 3, No. 1, March-2018
4.1. Rules for all three cases (d) Subtracting R3 from 12977875 i.e. 12977875 –
125 and eliminate last zero is 1297775.
I. Make group of 3 digits, starting from the right.
(e) Middle digit of cube root is obtained by 3R2M =
(a) The numbers having 4, 5 to 6 digits will have
3 × 52 × M = 75 M.
a 2 digits cube root.
(f) We should be looking for a suitable value of M
(b) The numbers having 7, 8 or 9 digits will have
so that the unit digit of 75M becomes equal to
a 3 digits cube root.
the unit digit of 1297775 (which is obtained in
(c) The number having 10, 11 or 12 digits will
(d)).
have a 4 digits cube root.
II. Table [1] will help us to determine the unit digit of If we obtain more than one value of M, we use ljhfÍ
cube root and table [3] will give the left digit of method and test which value of M is best suited in this
the cube root. case. In this problem, here is 5 options for M, i.e. 1,
3, 5, 7 or 9.
Where, ljhfÍ of given number 12977875 is 1.
4.1.1. Illustration for Case I Again, ljhfÍ of (215)3 = 8; ljhfÍ of (235)3 = 1;
Vilokanam -ljnf]sgd_ method is used to extract the ljhfÍ of (255) = 0;
3
cube root of a number having less than 7 digits. ljhfÍ of (275)3 = 8 and ljhfÍ of (295)3 = 1
Here, ljhfÍ of (235)3 & (295)3 are equal to the ljhfÍ
of given number, between them (235)3 is best
Example: Find the cube root of 17576
3
suited. ∴ 12977875 = 235
Stepwise: (a) Placing bar as: 17 576
(b) It will have a two digit cube root.
(c) The first bar falls on unit digit 6, so 4.1.3. Illustration for Case III
from above table [1] & [2], we can The cube root of a number greater than 7 digits but
say that the unit digit of the root is 6. ending with even number can be obtained by dividing
(d) From table [3], left digit of cube root the number whose cube root has to be extracted by 8
is 2. until odd cubs to be obtained, and can be used ljhfÍ
3 method to ascertain the cube roots as in case II.
Hence, 17576 = 26.
4.1.2. Illustration for Case II 3
Example: Find 5414689216 .
The cube root of more than 7 digits number will Since, this is 10 digits even numbers we need to
contain 3 digits. Let it be denoted by unit digit (R), divide by 8 until odd cube to be obtained.
left digit (L) and middle digit (M). L and R can be 8 5414689216
determined by ljnf]sgd method, whereas M can be 8 676836152
determined by ljhfÍ . 84604519
Steps: (a) Subtract R3 from the number and Here, the cube root of 84604519 can be find as in
eliminate the last zero.
(b) The middle digit of cube root is obtained 3
case II. i.e. 84604519 = 439.
by 3R2M. Substituting different values of
3 3
M. So that we may reach to the unit digit Hence, 5414689216 = 8 × 8 × (439)3
of the number obtained in the previous = 2 × 2 × 439 = 1756.
step.
If we obtained more values of M, we use ljhfÍ
method which should equal to the given number and 5. CONCLUSION
choosing best suited. It is very easy to find the cube roots of a number
having less than 7 digits. But, it needs a little practice
and patience initially to extract the cube roots of a
3 number having more than 7 digits but less than 10
Example: Find 12977875 .
digits by Vedic methods. Also for the number greater
Here, ljnf]sgd and ljhfÍ methods are used: than 7 digits but ending with even number, while
(a) Placing bar as: 12 977 875 extracting the cube root of a number having its unit
(b) Unit digit of cube root is 5 i.e. R = 5. digit, even we may get two values of M and it
(c) Here, left group is 12, so from the table [3], becomes difficult to ascertain the exact value of M,
L = 2. Vedic method provided the ljhfÍ method to
determine it. It was not found the concrete proper and
practical method to determine the cube roots of the
51

4. ISSN NO. 2456-3129
International Journal of Engineering, Pure and Applied Sciences,
Vol. 3, No. 1, March-2018
number before Aryabhtta's. All the Mathematician • Kapoor, S. K., Kapoor, R. P. (2010). Practice
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method of Aryabhatta and Bharatikrishna Tirtharaj Delhi-02.
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Upkar Prakasan, Agra-2.
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