Consecutive Number Multiplied Again

Consecutive Number Multiplied Again

Dr Himanshu Shekhar

Kids! As previous post entitled “Multiplication of Consecutive Numbers” got very good response (more than 100 in one day) from readers, I am encouraged to extend the topic and explore another quick, easy, universal and better method of multiplying consecutive numbers. By practice this can be perfected for mental mathematics.

As per new method illustrated in this post, the multiplication of consecutive numbers is carried out using partition, as discussed in many previous posts. The product of consecutive numbers will have three partitions:

• Left partition is multiplication of left digits
• Right partition has 1-digit and it is given by product of right digits.
• Central partition is product of left digit of bigger number and the sum of right side digits.
• Central partition is also allocated 1-digit only.
• If number of digits is more than 1 for right and central partition, they are carried over to left side partition.

This is explained with examples, as done in other posts, earlier.

26x27 = 2x2|2x(6+7)|6x7 = 4|26|42 = 4|26+4|2 = 4|30|2 = 4+3|0|2 = 702

93x94 = 9x9|9x(3+4)|3x4 = 81|63|12 = 81|63+1|2 = 81|64|2 = 81+6|4|2 = 8742

72x73 = 7x7|7x(2+3)|2x3 = 49|35|6 = 49+3|5|6 = 5256

68x69 = 6x6|6x(8+9)|8x9 = 36|102|72 = 36|102+7|2 = 36|109|2 = 36+10|9|2 = 4692

57x58 = 5x5|5x(7+8)|7x8 = 25|75|56 = 25|75+5|6 = 25|80|6 = 25+8|0|6 = 3306

49x50 = 4x5|5x(9+0)|9x0 = 20|45|0 = 20+4|5|0 = 2450

80x81 = 8x8|8x(0+1)|0x1 = 64|8|0 = 6480

Although the process is illustrated for explanation in written form, but the multiplication can be explored mentally, without writing. Some practice is needed for that. This method is independent on type of end digits available and there is no need to remember the unit place digit of the numbers, to be multiplied (as needed in previous post for similar operation)

Corollary 1: The process can be extended to product of any two 2-digit numbers, but some modification is needed in central partition.

Central partition is sum of cross-multiplication of digits of the number. The left digit of one number is multiplied to right digit of the other and vice-versa and then added.

29x43 = 2x4|2x3+9x4|9x3 = 8|6+36|27 = 8|42|27 = 8|42+2|7 = 8|44|7 = 8+4|4|7 = 1247

48x87 = 4x8|4x7+8x8|8x7 = 32|28+64|56 = 32|92|56 = 32|92+5|6 = 32|97|6 = 32+9|7|6 = 4176

73x94 = 7x9|7x4+3x9|3x4 = 63|28+27|12 = 63|55|12 = 63|55+1|2 = 63|56|2 = 63+5|6|2 = 6862

87x37 = 8x3|8x7+7x3|7x7 = 24|56+21|49 = 24|77|49 = 24|77+4|9 = 24|81|9 = 24+8|1|9 = 3219

Corollary 2: This process of multiplication can be extended to three digit numbers also. It must be kept in mind that for multiplication of three digit numbers, each multiplicand is partitioned into 1-digit right partition and 2-digit left partition before multiplication. The reverse of this should not be done. First some simple examples are illustrated.

123x124 = 12x12|12x(3+4)|3x4 = 144|84|12 = 144|84+1|2 = 144|85|2 = 144+8|5|2 = 15252.

117x118 = 11x11|11x(7+8)|7x8 = 121|165|56 = 121|165+5|6 = 121|170|6 = 121+17|0|6 = 13806

113x114 = 11x11|11x(3+4)|3x4 = 121|77|12 = 121|77+1|2 = 121|78|2 = 121+7|8|2 = 12882

Corollary 3: The process of multiplication for larger 3-digit numbers may need multiple application of partition in getting the product. Examples are given below to illustrate the point.

432x234 = 43x23|43x4+2x23|2x4 = 4x2|4x3+3x2|3x3| 172+46|8 = 8|18|9| 218|8 = 8+1|8|9| 218|8 = 989+21|8|8 = 101088

456x824 = 45x82|45x4+6x82|6x4 = 4x8|4x2+5x8|5x2| 180+492|24 = 32|48|10| 672+2|4 = 32+4|8+1|0| 674|4 = 3690+67 |4|4 = 375744

Kids! Enjoy the new methods in mathematics. It is all mental mathematics with minimum writing, but practice is the key to such expertise. Guardians and teachers may contribute in augmenting the posts for better acceptability by kids.

Dr Himanshu Shekhar

Multiplication of Consecutive Numbers

Multiplication of Consecutive Numbers

Dr Himanshu Shekhar

Kids! This time, I am exploring another aspect of Multiplication through examples, where method to find out product of consecutive integers will be illustrated.

In fact, in one of the post entitled “Multiplication of Numbers with Complimentary Digits”, such requirements are spelt out, which is reproduced below:

122x128 = 12x13 | 2x8 = 156 | 16 = 15616.

145x145 = 14x15 | 5x5 = 21025

The left partition invariably require, multiplication of two consecutive numbers. Rather than partitioning, as discussed in previous posts, the current post explores addition and some other properties based on last digits of the products. For ease of calculation, explanation using product of 2-digit consecutive numbers is discussed in the post, which can be extended further by practice.

The product of consecutive numbers, with unit place digit of one of the numbers as 0 or 5 can be obtained by direct multiplication, easily, as given below, with slight modification (extraction of 2&5 to give 2x5 = 10 from numbers).

89x90 = 8010; 
90x91 = 8190; 
94x95 = 47x19x10 = 8930; 
95x96 = 10x19x48 = 9120;

79x80 = 6320; 
80x81 = 6480; 
84x85 = 42x17x10 = 7140; 
85x86 = 10x17x43 = 7310;

The product of consecutive integers, where unit place digits of smaller number are 1 and 2, the product is obtained by addition of two parts:

• One part is obtained by product of equal separation of both consecutive numbers
• Second part is product of unit place digits.

Equal separation from consecutive numbers means either (i) 1 less than smaller and 1 more than larger number or (ii) 2 less than smaller and 2 more than larger number. This is illustrated by examples:

First examples with 1 less than smaller and 1 more than larger number is taken:

91x92 = 90x93+2 = 8370+2 = 8372;

81x82 = 80x83+2 = 6640+2 = 6642;

71x72 = 70x73+2 = 5110+2 = 5112;

61x62 = 60x63+2 = 3780+2 = 3782;

Examples with 2 less than smaller and 2 more than larger number is taken:

92x93 = 90x95+6 = 8550+6 = 8556;

82x83 = 80x85+6 = 6800+6 = 6806;

72x73 = 70x75+6 = 5250+6 = 5256;

62x63 = 60x65+6 = 3900+6 = 3906;

The product of consecutive integers, where unit place digits of the smaller number are 6 and 7, the product is obtained by addition of two parts:

• One part is obtained by product of equal separation of both consecutive numbers
• Second part is product of 5 less than each unit place digits.

Equal separation from consecutive numbers means either (i) 1 less than smaller and 1 more than larger number or (ii) 2 less than smaller and 2 more than larger number. This is illustrated by examples:

First examples with 1 less than smaller and 1 more than larger number is taken:

96x97 = 95x98+1x2 = 10x19x49+2 = 9310+2 = 9312;

86x87 = 85x88+1x2 = 10x17x44+2 = 7480+2 = 7482;

76x77 = 75x78+1x2 = 10x15x39+2 = 5850+2 = 5852;

66x67 = 65x68+1x2 = 10x13x34+2 = 4420+2 = 4422;

Examples with 2 less than smaller and 2 more than larger number is taken:

97x98 = 95x100+6 = 9500+6 = 9506;

87x88 = 85x90+6 = 7650+6 = 7656;

77x78 = 75x80+6 = 6000+6 = 6006;

67x68 = 65x70+6 = 4550+6 = 4556;

Now two type of consecutive number multiplication is left, where unit place digit of larger number is 4 or 9.

Main part of addition will be product of one less than smaller and one more than larger number and 2 is added to it. Examples are given below.

93x94 = 92x95+2 = 10x46x19+2 = 8740+2 = 8742;

98x99 = 97x100+2 = 9700+2 = 9702;

83x84 = 82x85+2 = 10x41x17+2 = 6970+2 = 6972;

88x89 = 87x90+2 = 7830+2 = 7832;

73x74 = 72x75+2 = 10x36x15+2 = 5400+2 = 5402;

78x79 = 77x80+2 = 6160+2 = 6162;

Now any set of multiplication of consecutive numbers can be carried out easily. It is just practice, which makes this method perfect. It is mental mathematics, which needs quick action by mind to get product of consecutive numbers.

Kids! Explore this method and compare with known conventional method. Enjoy.

Dr Himanshu Shekhar

Multiplication with Reference: Part 4

Multiplication with Reference: Part 4

Dr Himanshu Shekhar

Kids! Multiplication is such a beautiful mathematical operation that it can be explored in several different ways.

The previous few posts are exploring multiplication of numbers using a reference, which are invariably some power of 10. However, nearness to a power of 10 is one pre-requisite of such calculation. In the current post, reference number is taken as some multiple of power of 10, thus expanding the domain of this mental mathematical technique.

In this post reference can be 20 (2x10), 50 (5x10 or ½ x 100), 60 (6x10), 300 (3x100), 700 (7x100) and so on. The product calculation is again carried out under two heads – left partition and right partition.

Ø  Left partition is decided by multiplication of (i) multiplier of power of 10 and (ii) sum of multipliers minus reference.

Ø  Right partition has number of digits as number of zeros in the base of reference and is given by product of differences of multipliers from reference.

In nutshell, only difference will come in the left partition, where it will be multiplied by a number given by multiple of power of 10. Let us solve examples.

54x56 = 5x6|4x6 = 30|24 = 3024. (Mental mathematics explained in earlier post “Multiplication of Numbers with Compliment Digits”)

54x56 = 5x(54+56-50)|4x6 = 5x60|24 = 300|24 = 300+2|4 = 302|4 = 3024. (Reference is 5x10:, Multiplier 5: Right partition has 1-digit only).

54x56 = ½ x (54+56-50)|4x6 = ½ x60|24 = 30|24 = 3024. (Reference is ½ x100:, Multiplier ½ : Right partition has 2-digit only).

48x63 = 4x(48+63-40)|8x23 = 4x71|184 = 284|184 = 284+18|4 = 302|4 = 3024. (Reference is 4x10:, Multiplier 4: Right partition has 1-digit only).

48x63 = 5x(48+63-50)|-2x13 = 5x61|-26 = 305|-26 = 305|4-30 = 305|34 = 305-3|4 = 302|4 = 3024. (Reference is 5x10:, Multiplier 5: Right partition has 1-digit only: -26 = 34).

48x63 = ½ x(48+63-50)|-2x13 = ½ x61|-26 = 30.5|-26 = 30|50-26 = 30|24 = 3024.  (Reference is ½ x100:, Multiplier ½: Right partition has 2-digit only: 0.5 on left partition = 0.5x100 = 50 on right partition).

48x63 = 6x(48+63-60)|-12x3 = 6x51|-36 = 306|-36 = 306|4-40 = 306|44 = 306-4|4 = 302|4 = 3024. (Reference is 6x10:, Multiplier 6: Right partition has 1-digit only).

312x318 = 31x32|2x8 = 992|16 = 99216. (Mental mathematics explained in earlier post “Multiplication of Numbers with Compliment Digits”)

312x318 = 3x(312+318-300)|12x18 = 3x330|216 = 990+2|16 = 992|16 = 99216. (Reference is 3 x 100:, Multiplier 3 : Right partition has 2-digit only).

798x805 = 8x(798+805-800)|-2x5 = 8x803|-10 = 6424|90-100 = 6424|190 = 6424-1|90 = 6423|90 = 642390. (Reference is 8 x 100: Multiplier 8 : Right partition has 2-digit only: -10 = 190).

798x805 = 4/5 x(798+805-800)|-2x5 = 4/5 x803|-10 = 3212/5 |-10 = 642.4|-10 = 642|400-10 = 642|390 = 642390. (Reference is 4/5 x 1000:, Multiplier 4/5 : Right partition has 3-digit only: 0.4 on left partition = 0.4x1000 = 400 on right partition).

237x245 = ¼ x (237+245-250)|-13x-5 = ¼ x232|065 = 58|065 = 58065. (Reference is ¼ x 1000:, Multiplier ¼ : Right partition has 3-digit only).

758x753 = ¾ x (758+753-750)|8x3 = ¾ x 761 | 024 = 2283/4 | 024 = 570.75 | 024 = 570| 750+024 = 570|774 = 570774. (Reference is ¾  x 1000:, Multiplier ¾ : Right partition has 3-digit only: 0.75 on left partition is 750 on right partition).

Kids can explore themselves with the help of their guardians and teachers to understand the method. Practice makes everything perfect. So, practice and master the technique of multiplication involving references.

Each example can be solved by a number of method and reference. The end result remains same but in between mathematical calculation varies little bit. This mental mathematical techniques, can be helpful in cracking all competitive examinations and also for cracking online examinations, involving numerical aptitude.

Dr Himanshu Shekhar

Multiplication with Reference: Part 3

Multiplication with Reference: Part 3

Dr Himanshu Shekhar

Kids! Multiplication with reference is explained in previous 2-posts (Multiplication with a Reference & Multiplication using Reference: Part 2) using 100 as reference. In fact, any power of 10 can be used as reference for quick mental multiplication, provided, both numbers are in the vicinity of the reference (however not mandatory, as explained with some examples below). 

In the current post, mental multiplication, explained in previous few posts, have been extended further to numbers, which are near to other powers of 10. As usual, the product is partitioned into two parts – left part and right part.

• The left part of the product is given by sum of both the numbers minus reference. It is written in previous 2-posts as subtracting cross-difference from any of the multiplier. 
• The right part of the product given by product of difference of multipliers from reference, taking proper sign (+ or -).
• The power of 10, used as reference indicates number of digits used for right part of the product. Number of digits for right part is equal to power of 10, used as reference.
• If right part is more than number of allocated digits, the provision of carry over to left part is there.
• If right part is less than number of allocated digits, leading zeros can be added.
• The numbers can be written as partial negative digits (shown by using underlined digits), as explained in previous post (Multiplication with Reference: Part 2).

12x17 = 12+17-10|2x7 = 19|14 = 19+1|4 = 204. (Reference 10; 1 digit for right partition)

12x17 = 12+7|2x7 = 19|14 = 19+1|4 = 20|4 + 204. (Technique used in previous 2-posts for left part: 1-digit for right partition)

12x17 = 17+2|2x7 = 19|14 = 19+1|4 = 20|4 + 204. (Technique used in previous 2-posts for left part: 1-digit for right partition)

14x16 = 14+16-10|4x6 = 20|24 = 20+2|4 = 22|4 = 224.

14x16 = 1x2|4x6 = 2|24 = 224 (Mental mathematics explained in post “Multiplication of Numbers with Compliment Digits)

8x6 = 8+6-10|-2x-4 = 4|8 = 48.

13x7 = 13+7-10|3x-3 = 10|-9 = 10|11 = 10-1|1 = 91. (Reference 10; 1 digit for right partition: -9 = -10+1 = 11)

1012x1004 = 1012+1004-1000|12x4 = 1016|048 = 1016048. (Reference 1000: 3-digits right partition)

998x987 = 998+987-1000|-2x-13 = 985|026 = 985026.

1016x992 = 1016+992-1000|16x-8 = 1008|-128 = 1007|1000-128 = 1007|872 = 1007872. (Reference 1000: 3-digits right partition: Negative carryover for right partition)

998x92 = 998+92-1000|-2x-908 = 90|1816 = 90+1|816 = 91|816 = 91816. (Reference 1000: 3-digits right partition: Carry over to left partition: One number too small from reference)

1013x111 = 1013+111-1000|13x-889 = 124|-11557 = 124-11|-557 = 113|-557 = 113-1| 1000-557 = 112|443 = 112443. (Reference 1000: 3 digits for right, Carry over jugglery to be understood).

1013x111 = 1013+111-1000|13x-889 = 124|-11557 = 124|12000-443 = 124-12|443 = 112|443 = 112443. (Reference 1000: 3 digits for right, Carry over jugglery to be understood).

991x87 = 991+87-1000| -9x-913 = 78|8217 = 78+8|217 = 86|217 = 86217.

9999x9983 = 9999+9983-10000|-1x-17 = 9982|0017 = 99820017. (Reference 10000: 4-digits right partition)

999999998x1000000018 = 999999998+1000000018-1000000000|-2x18 = 1000000016|-36 = 1000000016-1|1000000000-36 = 1000000015|999999964 = 1000000015999999964. (Reference 1000000000: 9-digits right partition: Negative carry over to right partition)

As, I believe in illustrating through examples, less English is written. However, the key points are indicated and training mind to solve multiplications is an art, to be mastered by individuals. Hope that this extension of multiplication with reference is liked by kids. I will post, multiplication with reference as multiples of powers of 10, as next post. Enjoy. Do solve and explore.

Dr Himanshu Shekhar

Multiplication using Reference: Part 2

Multiplication Using reference: Part 2

Kids! This section is added after getting motivation from the Daughter of Shri Dayanand Kumar, a senior scientist is DRDO. This part is an extension to previous post and many similar additional posts will come, as the topic has many diversions.

This post discusses mental multiplication using reference 100. In previous post, either both the multipliers were less than 100 or both were more than 100. In this section, those multiplications will be explained, where, one number is less than reference and other is more than reference.

For 108x98, differences from reference (100) are +8 and -2, respectively. The product has two parts. The left side digit will be 108-2 = 98+8 = 106. The right partition has two digits, given by product of these differences, +8x-2 = -16. This can be written as -16 = 84 -100 = -100+84 = 184. (underline denotes negative term). Now it has three terms, but only 2-digits are specified for right side. So, -1 is carried over to left side and left side digit becomes 106-1 = 105. So, 108x98 = 10584. Some more examples are solved.

108x98 = 108-2|+8x-2 = 106|-16 = 106|84-100 = 106|184 = 106-1|84 = 105|84 = 10584

108x98 = 98+8|+8x-2 = 106|-16 = 106|84-100 = 106|184 = 106-1|84 = 105|84 = 10584

112x97 = 112-3|+12x-3 = 109|-36 = 109|164 = 109-1|64 = 108|64 = 10864

112x97 = 97+12|+12x-3 = 109|-36 = 109|164 = 109-1|64 = 108|64 = 10864

107x88 = 107-12|+7x-12 = 95|-84 = 95|116 = 95-1|16 = 94|16 = 9416

107x88 = 88+7|+7x-12 = 95|-84 = 95|116 = 95-1|16 = 94|16 = 9416

112x88 = 112-12|+12x-12 = 100|-144 = 100|256 + 100-2|56 = 9856

112x88 = 88+12|+12x-12 = 100|-144 = 100|256 = 100-2|56 = 9856

119x83 = 119-17|+19x-17 = 102|-323 = 102|477 = 102-4|77 = 9877

119x83 = 83+19|+19x-17 = 102|-323 = 102|477 = 102-4|77 = 9877

This post is mainly to introduce the concept of negative part in a number in mental mathematics. The concept, if understood can be used with ease for solving the above multiplications in slightly different way also.

108x98 = 108x102 = 108-2|8x-2 = 106|-16 = 106|184 = 106-1|84 = 10584

112x97 = 112 x 103 = 112-3|12x-3 = 109|-36 = 109|164 = 109-1|64 = 108|64 = 10864

107x88 = 107x112 = 107-12|7x-12 = 95|-84 = 95|116 = 95-1|16 = 94|16 = 9416

112x88 = 112x112 = 112-12|+12x-12 = 100|-144 = 100|256 = 100-2|56 = 9856

119x83 = 119x117 = 119-17|+19x-17 = 102|-323 = 102|477 = 102-4|77 = 9877

67x 114 = 133x114 = 114-33|14x-33 = 81|-462 = 81|538 = 81-5|38 = 7638

More of similar examples can be solved and power of mental mathematics can be felt. Hope that introduction of negative part in a number is understood by kids. It needs practice as concept is entirely different from conventional approach. Enjoy it.

Dr Himanshu Shekhar

Multiplication with a Reference

Multiplication with a Reference

Kids! This post is to explain another mental mathematics technique, which can be mastered with practice.

As described in previous post of finding square of any 2-digit number, using a reference, this section is also utilizing a reference number to find product. Being decimal system of numbers, again powers of 10 (like 10, 100, 1000, etc) are taken as reference. For two digit numbers, 100 can be taken as reference. Since, reference added multiplication is explained, nearness to reference is one of the pre-requisites. Again, rather than writing in English, example is solved for explanation and illustration.

To find, 112x108, both the numbers are near 100, which is taken as reference. The product is partitioned into two parts. Right part has two allocated space and is given by product of difference of numbers from reference. First number is more by (112-100=) 12 and second is less by (108-100=) 8. So, last two digits (left two digits) of the product is 12x8=96. The right side of the product is given by cross-addition of difference. Right side is 112+8 = 108+12 = 120. So, product is 12096.

112x108 = 112+8|12x8 = 120|96 = 12096

Some more examples are shown, below.

116x104 = 116+4|16x4 = 120|64 = 12064

116x104 = 104+16|16x4 = 120|64 = 12064

108x108 = 108+8|8x8 = 116|64 = 11664

107x111 = 107+11|7x11 = 118|77 = 11877

107x111 = 111+7|7x11 = 118|77 = 11877

If both numbers are less than reference then also, the method is applicable. To find, 93x98, both numbers are near 100, which is taken as reference. The product is again partitioned into two parts. Right part has two allocated space and is given by product of difference of numbers from reference. First number is less by (93-100=) -7 and second is less by (98-100=) -2. So last two digits will be -2x-7= 14. The right side of the product is given by cross-addition of difference. Right side is 93-1 = 98-7 = 91. So, product is 9114.

93x98 = 93-2|-7x-2 = 91|14 = 9114.

93x98 = 98-7|-7x-2 = 91|14 = 9114.

99x89 = 99-11|-1x-11 = 88|11 = 8811

99x89 = 89-1|-1x-11 = 88|11 = 8811

95x94 = 95-6|-5x-6 = 89|30 = 8930

95x94 = 94-5|-5x-6 = 89|30 = 8930

88x97 = 88-3|-12x-3 = 85|36 = 8536

88x97 = 97-12|-12x-3 = 85|36 = 8536

77x96 = 77-4|-23x-4 = 73|92 = 7392

77x96 = 96-23|-23x-4 = 73|92 = 7392

The process is extended little further, where left side digit is more than 100 and two spaces are not sufficient to accommodate them. In that case, carry over provisions exits and hundreds place digits can be carried over to other side of partition. Some examples are illustrated below.

127x112 = 127+12|27x12 = 139|324 = 139+3|24 = 142|24 = 14224

133x116 = 133+16|33x16 = 149|528 = 149+5|28 = 154|28 = 15428

118x114 = 118+14|18x14 = 132|252 = 132+2|52 = 134|52 = 13452

119x111 = 119+11|19x11 = 130|209 = 130+2|09 = 132|09 = 13209

75x88 = 75-12|-25x-12 = 63|300 = 63+3|00 = 66|00 = 6600

84x88 = 84-12|-16x-12 = 72|192 = 72+1|92 = 73|92 = 7392

82x88 = 82-12|-18x-12 = 70|216 = 70+2|16 = 72|16 = 7216

The last multiplication can be solved by method explained in earlier post (Multiplication of Numbers with Compliment Digits) because unit place digits of both the numbers add to 10. For recapitulation of the method, explained in earlier post.

82x88 = 8x9|2x8 = 72|16 = 7216

The post explains a metal mathematics method for those multiplications, where numbers are near 100. It can be practiced and more extension of the method will be posted in subsequent posts, to extend the method to different references and different combinations of multiplicands.

These all are parts of metal mathematics, which can be used to crack competitive examinations. Please do post, your comments to improve the content.

Dr Himanshu Shekhar

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