Square: Some more examples

(101)^2 = (10^2) (2×10×1) (1^2) or (1^2) (2×1×01) {(01) ^2}

               = (100) (20) (1) or (1) (02) (01)

               = (100+2) (0) (1) or 10201

               = (102) (0) (1) or 10201

                = 10201 or 10201.

(102)^2 = (10^2) (2×10×2) (2^2) or (1^2) (2×1×02) {(02) ^2}

               = (100) (40) (4) or (1) (04) (04)

               = (100+4) (0) (4) or 10404

               = (104) (0) (4) or 10404

               = 10404 or 10404.

(105)^2 = (10^2) (2×10×5) (5^2) or (1^2) (2×1×05) {(05) ^2}

               = (100) (100) (25) or (1) (10) (25)

               = (100) (100+2) (5) or 11025

               = (100) (102) (5) or 11025

               = (100+10) (2) (5) or 11025

               = (110) (2) (5) or 11025

               = 11025 or 11025.

(124)^2 = (12^2) (2×12×4) (4^2) or (1^2) (2×1×24) (24^2)

               = (144) (96) (16) or (1) (48) (576)

               = (144) (96+1) (6) or (1) (48+5) (76)

               = (144) (97) (6) or (1) (53) (76)

               = (144+9) (7) (6) or 15376

               = (153) (7) (6) or 15376

                = 15376 or 15376.

(1835)^2 = (18^2) (2×18×35) (35^2)

                 = (324) (1260) (1225)

                 = (324) (1260+12) (25)

                 = (324) (1272) (25)

                 = (324+12) (72) (25)

                 = (336) (72) (25)

                 = 3367225.

(1836)^2 = (18^2) (2×18×36) (36^2)

                 = (324) (1296) (1296)

                 = (324) (1296+12) (96)

                 = (324) (1308) (96)

                 = (324+13) (08) (96)

                 = (337) (08) (96)

                 = 3370896.

Square: Try it

Square: Try it We know that
(a+b)^2 = a^2 + 2ab + b^2.
With the help of above we can find out the square of a number in the following way…
(11)^2 = (1^2) (2×1×1) (1^2)
= (1) (2) (1)
=121.
(12)^2 = (1^2) (2×1×2) (2^2)
= (1) (4) (4)
= 144.
(13)^2 = (1^2) (2×1×3) (3^2)
= (1) (6) (9)
= 169.
(14)^2 = (1^2) (2 ×1×4) (4^2)
= (1) (8) (16)
= (1) (8+1) (6)
= (1) (9) (6)
= 196.
(15)^2 = (1^2) (2 ×1×5) (5^2)
= (1) (10) (25)
= (1+1) (0+2) (5)
= (2) (2) (5)
=225.
(16)^2 = (1^2) (2×1×6) (6^2)
= (1) (12) (36)
= (1+1) (2+3) (6)
= (2) (5) (6)
= 256.
(17)^2 = (1^2) (2×1×7) (7^2)
= (1) (14) (49)
= (1+1) (4+4) (9)
= (2) (8) (9)
= 289.
(18)^2 = (1^2) (2×1×8) (8^2)
= (1) (16) (64)
= (1+1) (6+6) (4)
= (2) (12) (4)
= (2+1) (2) (4)
= (3) (2) (4)
= 324
(19)^2 = (1^2) (2×1×9) (9^2)
= (1) (18) (81)
= (1+1) (8+8) (1)
= (2) (16) (1)
= (2+1) (6) (1)
= 361.
(20)^2 = (2^2) (2×2×0) (0^2)
= (4) (0) (0)
= 400.
(21)^2 = (2^2) (2×2×1) (1^2)
= (4) (4) (1)
= 441.
(25)^2 = (2^2) (2×2×5) (5^2)
= (4) (20) (25)
= (4+2) (0+2) (5)
= (6) (2) (5)
= 625.
(98)^2 = (9^2) (2×9×8) (8^2)
= (81) (144) (64)
= (81) (144+6) (4)
= (81) (150) (4)
= (81+15) (0) (4)
= (96) (0) (4)
= 9604.

square of a natural number is the sum of two definite natural numbers.

Square of a natural number can be found with the help of the following equation

(xy)^2 = x(x+1)y^2 + 20x(y−5).

Where y = 0 to 9; x = 0 to ∞.

Explanation:- value of first number is obtained by putting the value of x(x+1) next to the value of y^2.

Example:- square of 37 can be obtained by putting the value of 3(3+1) next to the value of 7^2 and add into them 20 times the value of 3(7−5).

Now 3(3+1) = 3×4 = 12 and 7^2 =49, put them next to each other. We get 1249.
In 1249 add 20 times the value of 3×(7−5).

Now 20×3(7−5)= 20×3×2 = 120

So 1249+120=1369. This is the required square of 37.

Another example
596^2 = 59(59+1)6^2 +20×59(6−5).
=(3540)36 + 1180
=354036+1180
=355216. This is the required square of 596. Here value of y is 6 and value of x is 59.

Another example
594^2 = 59(59+1)4^2 + 20×59(4−5)
= (59×60)4^2 + 20×59×(−1)
= (3540)16 − 1180
=354016 – 1180
=352836. This is the required square of 594. Here value of y is 4 and value of x is 59.

Note:- Numbers has its mirror image.