Multiplication of Numbers with Compliment Digits

Multiplication of Numbers with Compliment Digits

Kids! This post is for finding multiplication of numbers, fulfilling certain criteria.

In mathematics, Ancient India is credited with invention of Zero and that led to development of decimal system. This is because there are only 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to constitute entire mathematics. The topic in this post discusses simple method for multiplication of two-digit numbers, whose unit place digits are 10-compliment of each other and other digit is the same. This means that the last (Unit-place) digits of the multipliers add to 10. The steps are as follows:

• Partition the product into two parts.
• Left part is given by product of tens-place digit and one more than the 10s-place digits
• Right side is given by product of unit place digit and two places are allocated for this.
• If right side partition exceeds 100, carry over to left side is permitted.

Since it is mathematics and not literature, without much writing, some examples are solved directly for those multiplications, where unit place digits of multipliers are 10s compliment and other digits are the same.

Suppose 92x98 is to be obtained. Both numbers have 9 at tens place and sum of unit place digit is (2+8 =) 10. Left side of the product is 9x10 = 90 and right side of the product is 2x8 = 16. So, 92x98 = 9016.

92x98 = 9x10 | 2x8 = 90 | 16 = 9016

83x87 = 8x9 | 3x7 = 72 | 21 = 7221

64x66 = 6x7 | 4x6 = 42 | 24 = 4224

71x79 = 7x8 | 1x9 = 56 | 09 = 5609

52x58 = 5x6 | 2x8 = 30 | 16 = 3016

More examples, like 91x99 (= 9009), 93x97 (= 9021), 94x96 (= 9024), 81x89 (= 7209), 82x88 (= 7216), 84x86 (7224), 72x78 (= 5616), 73x77 (= 5621), 74x76 (= 5624), 61x69 (= 4209), 62x68 (= 4216), 63x67 (= 4221), 51x59 (= 3009), …. can be tried by kids and the power of quick mathematics can be explored.

This type of mathematical ease has many types of corollaries.

TYPE I: The process can be extended to three digit numbers also. 113x117 = 11x12 | 3x7 = 132 | 21 = 13221. 122x128 = 12x13 | 2x8 = 156 | 16 = 15616.

TYPE II: If the last two digits of three digit number add to 100, the same process is repeated with last four digit earmarked for right hand side. 191x109 = 1x2 | 91x9 = 2 | 0819 = 20819, 198x102 = 1x2 | 98x02 = 2 | 0196 = 20196. 189x111 = 1x2 | 89x11 = 2 | 0979 = 20979. It can be extended to 4-digit numbers too. 1191x1109 = 11x12 | 91x9 = 132 | 0819 = 1320819, 1289x1211 = 12x13 | 89x11 = 156 | 0979 = 1560979.

Type III: The same technique can be used for finding square of a number with 5 at unit place. 25x25 = 2x3 | 5x5 = 6 | 25 = 625, 115x115 = 11x12 | 5x5 = 132 | 25 = 13225, 145x145 = 14x15 | 5x5 = 21025.

TYPE IV: The process can be extended to any number of digits, but the criteria of compliment of power of 10 is to be satisfied. This means, last few digits must add to 10, 100, 1000, 10000, 100000, and so on. 12399 x 12301 = 123x124 | 99x01 = 15252 | 0099 = 152520099, 129999x120001 = 12x13 | 9999x0001 = 132 | 00009999 = 13200009999.

Kids! Explore this and intimate concern, if any.

Dr Himanshu Shekhar