Mentally square of any number quickly in 2 sec | Vedic math tricks | Full proof

 Sohel Sahoo: Hello guys,

      We quite often come across to find out square of any number in various competitive exam &daily basis. so, to solve this I have brought an easy & effective noteworthy method in the wake of making very long, tedious & cumbersome problems in hands down.

  The efficacy is based on the mental one line method of arithmetic.In order to make you understand I have put forward my efforts to express it algebraically.

            Let's get started.

    Here we can find square of any no.close to multiples of 10 in more formally powers of 10 i.e.10^n where n£N=1,2,3...(decimal representation)     The modus operandi i.e.mode of operation is based upon taking base number.

* Note:- 'Deficit' means number less than the base taken and 'surplus' means no. more than the base taken here.

(1) Numbers near and less than the powers of 10.:-

*Let X=100(say) be the base.Any number near and less than it can be treated as (X-Y) ; where,Yis the deficit:-

Now,,(X-Y)²=X²-2XY+Y²

                           =X(X-2Y)+Y²

                            =X{(X-Y) --Y}+Y²

                         =Base(No.--deficit)+(deficit)²

                  =100(number--deficit)+(deficit)²

                          =Number--deficit//(deficit)²

N.B:-Any no.can be expressed by its place value decimal representation.e.g.987=9×100+8×10+7.So,the last two digit are obtained by leaving 100 th place or better you can get it by the remainder so obtained by dividing through 100.Here divisor=100, remainder=deficit^2 which is last two digits.

  # So,in more general the square of deficit should be that no of digits equal to no.of zeroes in base number taken.Other digit(s) should be taken as carried over to left hand side.

Eg 1:  9² Here base is 10.

       The answer is separated into two parts by a ‘/’(slash)

        Note that deficit is 10 - 9 = 1

        Multiply the deficit by itself or square it 1² = 1.

        As the deficiency is 1, subtract it from the number i.e., 9–1 = 8.

        Now put 8 on the left and 1 on the right side of the vertical line or slash i.e., 8/1.

        Hence 81 is answer. 

Eg. 2:  96² Here base is 100.

        Since deficit is 100-96=4 and square of it is 16 and the deficiency subtracted from the number 96 gives 96-4 = 92, we get the answer 92 / 16  Thus 96² = 9216.

Eg. 3:  994² Base is 1000

        Deficit is 1000 - 994 = 6. Square of it is 36.

        Deficiency subtracted from 994 gives 994 - 6 = 988

        Answer is 988 / 036     [since base is 1000]

Eg. 4:   9988²   Base is 10,000.

Deficit = 10000 - 9988 = 12.

        Square of deficit = 122 = 144.

        Deficiency subtracted from number = 9988 - 12 = 9976.

        Answer is 9976 / 0144     [since base is 10,000]. 

Eg. 5:   88²  Base is 100.

        Deficit = 100 - 88 = 12.

        Square of deficit = 122 = 144.

        Deficiency subtracted from number = 88 - 12 = 76.        Now answer is 76 / 144 = 7744    [since base is 100]

   e.g. 97²=97--3//3²      =94//9       =94//09=9409

  86²=86--14//14²    =72//196          =73//96=7396

    994²=994--6/6²     =988//36        =988//036=988036   [Here base is 1000]

 Do rigorous practice to become expert in applying this formula.

(2)Numbers near and greater than the powers of 10.:-

*Any number near &greater than X=100(say) can be written as (X+Y),0<Y<100:-

  Hence, (X+Y)²=X²+2XY+Y²

                                 =X(X+2Y)+Y²

                                 =X{(X+Y)+Y}+Y²

                            =Base (No.+surplus)+(surplus)²

                     =100(No.+more value)+surplus ²

                                  =No.+surplus//surplus²

Eg.(1):  13²

        Instead of subtracting the deficiency from the number we add and proceed as in Method-1.

        For 13² , base is 10, surplus is 3.

        Surplus added to the number = 13 + 3 = 16.

        Square of surplus = 3² = 9.Answer is 16 / 9 = 169.

Eg.(2):   112²

           Base = 100, Surplus = 12,

        Square of surplus = 12² = 144

        add surplus to number = 112 + 12 = 124.        Answer is 124 / 144 = 12544

 e.g. 102²=102+2//2²=104/4 =104//04=10404 etc as above said likewise.

         114²=114+14//14² =   128/196 =129/96 = 12996

     1007²=1007+7//7²= 1014/049  =1014049 etc. [Here base is 1000]

(3)*Squaring of numbers ending in 5:-

Any number ending in 5 is more conveniently written as a5 where,a is a natural number.

 Now , (a5)²=(10a+5)²   (Decimal representation of a5=10a+5)

       =(10a)²+2×10a×5+5²

        =100a²+100a+25

        =100a(a+1)+25

       =a(a+1)//25

# Henceforth,owing to above formula we conclude that in order to get square of numbers ending in 5 we have to multiply the previous digit(s) of 5 by one more than itself&then just merely placing 25.

  e.g.  (35)² =3×4/25=1225

   (105)² =10×11/25=11025

    (165)² =16×17/25=27225

(485)² = 48 x 49//25 = 2352//25 = 235225(Ans)

(4)*Square of any number near to submultiples of 10:-

Square of any no. near to submultiples of 10 can also be calculated by above two bold formulae.

(X+Y)² =Base (No.+surplus)+(surplus)²

 (X-Y)² = Base(No.--deficit)+(deficit)²

  This can be obtained only by multiplying the digit multiplied in base of 10^n as explained below.

Example 1:   388²

 Nearest base = 400.

        We treat 400 as 4 x 100. As the number is less than the base we proceed as follows

        Number 388, deficit = 400 - 388 = 12

        Since it is less than base, deduct the deficit

            i.e. 388 - 12 = 376.

        multiply this result by 4 since base is 4 X 100 = 400.

                    376 x 4 = 1504

        Square of deficit = 12² = 144.

 Hence answer is 1504 / 144 = 150544    [since we have taken base as multiples of 100]

Example 2:   67² Nearest base = 70.                                    

Example 3:   416² Nearest ( lower ) base.

        Here surplus = 16 and 400 = 4 x 100           

Example 4:   5012² Nearest lower base is 5000 = 5 x 1000& Surplus = 12.                          

e.g. 397² =4(397--3)//3²  [Here base can be taken as 400=4*100]

                =4 x 394//9
                =1576//09=157609

503²=5(503+3)//3²
         =5 x 506//9
        =2530//09=253009
Likeise you can solve lots of problem.If further any doubt remains contact me.

# Special case of squaring a number between 30 to 80 :-

Here the middle number is 50 & this can be taken as the submultiple of 10&by the help of this we can calculate easily square of any no. between 100 as rest are covered by above two rules.

   Any number between 30 to 80 can be considered as 50+a or 50-a.Let's discuss in detail.

Squaring numbers between 30 to 50:-

Any number between 30 to 50 can be written as (50-a) 

         (50-a)²

          = 50²-2x50a+a²

          =2500-100a+a²

          =(25-a)100+a²

          = (25-a)//a²

   ^Note:- a^{2 should be in two digit if not make it so by adding a zero before it.}

e.g.  48² = 25-2//2² = 23//04 = 2304

       37² = 25-13//13² = 12//196 = 13//96 = 1396

       42² = 25-8//8² = 17//64 = 1764

        33² = 25-17//17² = 8//289 = 10//89 = 1089 etc.

Squaring numbers between 50 to 80:-

Any number between 50 to 80 can be written as (50+a) 

         (50+a)²

          = 50²+2x50a+a²

          =2500+100a+a²

          =(25+a)100+a²

          = (25+a)//a²

   ^Note:- a^{2 should be in two digit if not make it so by adding a zero before it.}

e.g.  52² = 25+2//2² = 27//04 = 2704

       63² = 25+13//13² = 38//196 = 39//96 = 3996

       59² = 25+9//9² = 34//81 = 3481

        67² = 25+17//17² = 42//289 = 44//89 = 4489 

       71² = 25+21//21² = 46//441 = 50//41 = 5041 etc.

For better understanding&fast calculation it is recommended to learn(cram) square of numbers from 1 to 20 or more appropriately from 1 to 30.

Numbers	Square of numbers	Method of base number(10 or 20 or 30)  20 = 2 x 10 & 30 = 3 x 10
1	1	1 [Base:10]
2	4	4 [Base:10]
3	9	9 [Base:10]
4	16	16 [Base:10]
5	25	25 [Base:10]
6	36	36 [Base:10]
7	49	49 [Base:10]
8	64	64 [Base:10]
9	81	81 [Base:10]
10	100	100
11	121	112=11+1/12=121 [Base:10]
12	144	122=12+2/22=14/4=144 [Base:10]
13	169	132=13+3/32=16/9=169 [Base:10]
14	196	142=14+4/42=18/16=19/6=196 [Base:10]
15	225	1 x 2/25=225
16	256	162=2(16-4)/42=2x12/16=24/16=256 [Base:20]
17	289	172=2(17-3)/32=2x14/9=28/9=289 [Base:20]
18	324	182=2(18-2)/22=2x16/4=32/4=324 [Base:20]
19	361	192=2(19-1)/12=2x18/1=36/1=361 [Base:20]
20	400	400 [Base:20]
21	441	212=2(21+1)/12=2x22/1=44/1=441 [Base:20]
22	484	222=2(22+2)/22=2x24/4=48/4=484 [Base:20]
23	529	232=2(23+3)/32=2x26/9=52/9=529 [Base:20]
24	576	242=2(24+4)/42=2x28/16=56/16=57/6=576 [Base:20]
25	625	2 x 3/25 = 625
26	676	262=3(26-4)/42=3x22/16=66/16=67/6=676 [Base:30]
27	729	272=3(27-3)/32=3x24/9=72/9=729 [Base:30]
28	784	282=3(28-2)/22=3x26/4=78/4=784 [Base:30]
29	841	292=3(29-1)/12=3x28/1=84/1=841 [Base:30]
30	900	900

 NOTE:-  

1.     A number with 2,3,7or8 at unit's place will never be a perfect square as you see above from 0 to 9 each digit's square ends only in 0/1/4/5/6/9 .

2.       No. ending in an odd no. of zeros is not a perfect square.

3.       Square of even numbers  are even; since, (2n)²=4n²

4.       Square of odd numbers are odd; since, (2n+1)²=4n²+4n+1=2n(2n+2)+1,which is odd.

          Hope this will serve my Moto to unearth the concepts & logic behind each statement of mathematics and eventually make you a mathophilic as well as your purpose of study in a wonderful perspective....

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