Squares of numbers closer to the base by Bhārati Kṛṣṇa Tīrtha
A number, when multiplied by the number itself the product obtained, is called "Square of that number ". We will come to know the squares for numbers which are closer to the bases like 10,100,1000.
Square of Number closer to the Bases
For squares of numbers close to the base we use a sub sūtra of the "Nikhilam" sūtra.[1]
यावदूनं तावदूनीकृत्य वर्गं च योजयेत्
" Yāvadūnaṃ Tāvadūnīkṛtya Vargaṃ Ca Yojayet "
" Whatever the extent of its deficiency, lessen by that amount, and set of the square of that deficiency"
This will be explained through the below examples.
Base : The whole number system is made up of only 9 numbers (1-9) and a zero. All these numbers repeat themselves in a specific order after numbers like 10, 100, 1000, and so on. These numbers which have one as the first digit, followed by zeros are called bases.
Complement : Complement of a number is the difference between that number and its nearest base.
Ex: 1) Complement of 103 is 103 - 100 = 3 as 100 is the nearest base. 2) Complement of 95 is 100 - 95 = 5 as 100 is the nearest base.
Square of a number
Square of Number more than the Base
Square of number more than the base consists of two parts as mentioned below.
Surplus is a complement of the number.
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| (Number + Surplus) | Square of Surplus |
Example : 1032
Base for 103 is 100 and 3 more than the base (100).
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| 103 is 3 more than the base (100).
Add the excess number (3) to the number (103) 103 + 3 = 106 |
Take the square of the complement (103 -100 = 3) as the nearest base is 100 for 103
32 = 9 . Since 100 is the base we write it as 09 |
| 106 | 09 |
Answer : 1032 = 10609
Example : 1252
Here is the base is 100. Number = 125 ; Surplus = 25
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| (Number + Surplus) | Square of Surplus |
| 125 + 25 | 252 |
| 150 | 625 as the base is 100 , keep last two digits and carry over 6 |
| 150 + Carry over (6) | 25 |
| 156 | 25 |
Answer : 1252 = 15625
Example : 10162
Here is the base is 1000. Number = 1000 ; Surplus = 16
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| (Number + Surplus) | Square of Surplus |
| 1016+16 | 162 |
| 1032 | 256 , as the base is 1000, keep last 3 digits |
| 1032 | 256 |
Answer : 10162 = 1032256
Square of Number less than the Base
Square of number less than the base consists of two parts as mentioned below.
deficiency is a complement of the number.
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| (Number - deficiency) | Square of deficiency |
Example : 972
Base for 97 is 100 and 3 less than the base (100).
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| 97 is 3 less than the base (100).
subtract the deficiency from the given number (97) 97 - 3 = 94 |
Take the square of the complement (100 - 97 = 3) as the nearest base is 100 for 97
32 = 9 . Since 100 is the base we write it as 09 |
| 94 | 09 |
Answer : 972 = 9409
Example : 822
Here is the base is 100. Number = 82 ; Deficiency = 18
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| (Number - deficiency) | Square of deficiency |
| 82-18 | 182 |
| 64 | 324 , as the base is 100, keep last 2 digits and carry over 3 |
| 64 + Carry over (3) | 24 |
| 67 | 24 |
Answer : 822 = 6724
Example : 99892
Here is the base is 10000. Number = 9989 ; Deficiency = 11
| Left Hand Side (LHS) | Right Hand Side (RHS) |
|---|---|
| (Number - deficiency) | Square of deficiency |
| 9989 - 11 | 112 |
| 9978 | 121, as the base is 10000, keep last 4 digits - 0121 |
| 9978 | 0121 |
Answer : 99782 = 99780121
See Also
आधार के निकट संख्याओं का वर्ग - भारती कृष्ण तीर्थ
References
- ↑ Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. pp. 210–215. ISBN 978-81-208-3230-5.
