Tips and Tricks: Squaring | Improve Your Calculations: Vedic Maths (English) - Class 6 PDF Download

Square of numbers ending in 5

Sutra: ‘By one more than previous one” 

Example: 75 × 75 or 75² 
As explained earlier in the chapter of multiplication we simply multiply 7 by the next number i.e. 8 to get 56 which forms first part of answer and the last part is simply 25 = (5)². So, 75 × 75 = 5625
This method is applicable to numbers of any size.

Example 2: 605² 
60 × 61 = 3660 and 5² = 25
∴ 605² = 366025
Square of numbers with decimals ending in 5 

Example 3: (7.5)² 
7 × 8 = 56, (0.5²) = 0.25
(7.5)² = 56.25 (Similar to above example but with decimal) 

Squaring numbers above 50

Example: 52² 
Step 1: First part is calculated as 5² + 2 = 25 + 2 = 27
Step 2: Last part is calculated as (2)² = 04 (two digits)
∴ 52² = 2704

Squaring numbers below 50 

Example: 48² 
Step 1: First part of answer calculated as: 5² – 2 = 25 – 2 = 23
Step 2: second part is calculated as: 2² = 04
∴ 48² = 2304

Squaring numbers near base

Example: 1004² 
Step 1: For first part add 1004and 04 to get 1008
Step 2: For second part 4² = 16 = 016 (as, base is 1000 a three digit no.)
∴ (1004)² = 1008016

Squaring numbers near sub-base

Example: (302)² 
Step 1: For first part = 3 (302 + 02) = 3 × 304 = 912 [Here sub – base is 300 so multiply by 3]
Step 2: For second part = 2² = 04
∴ (302)² = 91204

General method of squaring

The Duplex 

Sutra: “Single digit square, pair multiply and double” we will use the term duplex,` D’ as follows: 
For 1 figure(or digit) Duplex is its square e.g. D(4) = 4² = 16
For 2 digits Duplex is twice of the product e.g. D(34) = 2 (3 x 4) = 24
For 3 digit number: e.g. (341)² 
D(3) = 3² = 9
D (34) = 2 (3 × 4) = 24
D (341) = 2 (3 × 1 ) + 4² = 6 + 16 = 22

D (41 ) = 2 (4 × 1 ) = 8
D (1) = 1² = 1
∴ (341)² = 116281

Algebraic Squaring

Above method is applicable for squaring algebraic expressions: 

Example 1: (x + 5)² 
D (x) = x2 D(x + 5) = 2 (x × 5) = 10x
D (5) = 5² = 25
∴ (x + 5)2 = x² + 10x + 25 

Example: (x – 3y)² 
D (x)= x² 
D(x – 3y) = 2 (x) × – 3y) = – 6xy
D(–3y ) = (–3y)² = 9y²
∴ (x – 3y)² = x² – 6xy + 9y²