Square of Two-Digit Numbers

Square of Two-Digit Numbers

Dr Himanshu Shekhar

Kids! This post has another aspect of mathematical calculation simplified and the method can be executed as mental mathematics with a little efforts.

Square of Two-digit numbers is explored with of the help of squares of numbers having 0 or 5 at unit place. Square of any number with five at unit place is explored in previous posts (right part is product of tens-digit and one more than tens-digit and left part is 25). A recap of known squares of two digit numbers from previous posts and from mathematical simplification are given below and known to kids.

15x15 = 1x2|5x5 = 2|25 = 225	20x20 = 2x2|00 = 4|00 = 400
25x25 = 2x3|5x5 = 6|25 = 625	30x30 = 3x3|00 = 9|00 = 900
35x35 = 3x4|5x5 = 12|25 = 1225	40x40 = 4x4|00 = 16|00 = 1600
45x45 = 4x5|5x5 = 20|25 = 2025	50x50 = 5x5|00 = 25|00 = 2500
55x55 = 5x6|5x5 = 30|25 = 3025	60x60 = 6x6|00 = 36|00 = 3600
65x65 = 6x7|5x5 = 42|25 = 4225	70x70 = 7x7|00 = 49|00 = 4900
75x75 = 7x8|5x5 = 56|25 = 5625	80x80 = 8x8|00 = 64|00 = 6400
85x85 = 8x9|5x5 = 72|25 = 7225	90x90 = 9x9|00 = 81|00 = 8100
95x95 = 9x10|5x5 = 90|25 = 9025

The two-digit numbers, whose squares are given in previous table are called reference for this post. All two-digit numbers are one or two more or less than the numbers given in above table.

For any number ‘n’, one more than reference, the square is given by square of reference plus odd number derived from reference (odd number explained in first post on Introduction to Numbers). First odd number is 3, given by 2x1+1, second odd number is (2x2+1=) 5, 3^rd odd number is (2x3+1+) 7, 4^th odd number is (2x4+1) 9 and so on h’th odd number is given by (2xh+1). The odd number derived from reference is given by one more than twice the reference. This is explained by the following examples.

16x16 = 15x15 + (2x15+1) = 225 + 31 = 256

21x21 = 20x20 + (2x20+1) = 400 + 41 = 441

26x26 = 25x25 + (2x25+1) = 625 + 51 = 676

31x31 = 30x30 + (2x30+1) = 900 + 61 = 961

36x36 = 35x35 + (2x35+1) = 1225 + 71 = 1296

41x41 = 40x40 + (2x40+1) = 1600 + 81 = 1681

46x46 = 45x45 + (2x45+1) = 2025 + 91 = 2116

81x81 = 80x80 + (2x80+1) = 6400 + 161 = 6561

86x86 = 85x85 + (2x85+1) = 7225 + 171 = 7396

91x91 = 90x90 + (2x90+1) = 8100 + 181 = 8281

96x96 = 95x95 + (2x95+1) = 9025 + 191 = 9291

For any number ‘n’, one less than reference, the square is given by square of reference minus n’th odd number. The n’th odd number is given by 2n+1.

14x14 = 15x15 – (2x14+1) = 225 – 29 = 196

19x19 = 20x20 – (2x19+1) = 400 – 39 = 361

Ir is clear that for such numbers reference number is n+1 and the second term of 2n+1 can be written as [2x(n+1)–1] or one less than twice the reference. Rather than considering (2n+1) for finding difference from square of reference, it is better to write n’th odd number as one less than twice the reference number. This is explored below.

74x74 = 75x75 – (2x75-1) = 5625 – 149 = 5476

79x79 = 80x80 – (2x80-1) = 6400 – 159 = 6241

84x84 = 85x85 – (2x85-1) = 7225 – 169 = 7056

89x89 = 90x90 – (2x90-1) = 8100 – 179 = 7921

94x94 = 95x95 – (2x95-1) = 9025 – 189 = 8836

99x99 = 100x100 – (2x100-1) = 10000 – 199 = 9801

For finding squares of numbers, which are 2 more than reference number, it is square of reference plus 4 times the (i) average of reference and number or (ii) one more than reference or (iii) one less than the number or (iv) reference plus 4. For 17 reference number is 15 and square of 17 is equal to square of 15 plus 4 times 16. All are explored below.

17x17 = 15x15 + (4x16) = 225 + 64 = 289

22x22 = 20x20 + (4x21) = 400 + 84 = 484

27x27 = 25x25 + (4x26) = 625 + 104 = 729

77x77 = 75x75 + (4x75+4) = 5625 + 304 = 5929

82x82 = 80x80 + (4x80+4) = 6400 + 3204 = 9604

97x97 = 95x95 + (4x95+4) = 9025 + 384 = 9409

For finding squares of numbers, which are 2 less than reference number, it is square of reference minus 4 times the number, in between reference and number. This is explored with examples below.

28x28 = 30x30 – (4x29) = 900 – 116 = 784, or alternately

28x28 = 30x30 – (4x30-4) = 900 – 116 = 784

43x43 = 45x45 – (4x45-4) = 2025 – 176 = 1849

63x63 = 65x65 – (4x65-4) = 4225 – 256 = 3969

88x88 = 90x90 – (4x90-4) = 8100 – 356 = 7744

This is just practice and mental exercise, which can lead to mastering the mental mathematics with square numbers, explained in this post. Although, this method is explained for two-digit numbers, but it is equally applicable for larger numbers.

998x998 = 1000x1000 – (4x1000-4) = 1000000 – 3996 = 996004

9999x9999 = 10000x10000 – (2x10000-1) = 100000000 – 19999 = 9998001

Practice and enjoy. Do give feedback for improvement.

Dr Himanshu Shekhar