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

Complimentary Multiplication : Extension 1

Complimentary Multiplication : Extension-1

It is observed that in the previous post, the calculation for two-digit numbers are understood by many kids. However, with higher number of digits, some more clarity is needed. This post is added to have more clarity for specially Type II Corollary of previous post.

If we have more than two digits, then decimal complimentation can be observed in several ways.

Unit place digits are complimentary, like 1239 and 1231, 437 and 433, 254 and 256, ..

Last two digits are complimentary, like 1239 and 1261, 437 and 463, 254 and 246, …

Last three digits are complimentary, like 1239 and 1761, 4998 and 4002, …

This trend can continue and pair of numbers having similar characteristics can be explored, obtained and considered. It must be noted that in the pair of numbers, leaving complimentary digits, other digits are the same.

The method resembles that explained in the previous post. Product will contain two parts. Left part will be same part of the two numbers (to be multiplied) and one more than the same part. Right part is product of complimentary digits and number of digits in right part is twice the number of digits in the complimentation. The multiplication by the method explained in previous post is carried out below.

1239x1231 = 123x124 | 9x1 = 15252 | 09 = 1525209

437x433 = 43x44 | 7x3 = 1892 | 21 = 189221

254x256 = 25x26 | 4x6 = 650 | 24 = 65024

1239x1261 = 12x13 | 39x61 = 156 | 2379 = 1562379

437x463 = 4x5 | 37x63 = 20 | 2331 = 202331

254x246 = 2x3 | 54x46 = 6 | 2564 = 62564

1239x1761 = 1x2 | 239x761 = 2 | 181879 = 2181879

4998x4002 = 4x5 | 998x002 = 20 | 001996 = 20001996

This post is initiated and created at the behest of Diptasha Das, a great all round achiever in academics and sports.

Any kid can make comments for clarity and share it with their friends for getting more feedback.

Dr Himanshu Shekhar

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

Introduction to Numbers

Introduction to Numbers

“Mathematics is the queen of sciences and the theory of numbers is the queen of mathematics”. – Gauss.

Anybody and everybody, who has undergone formal education at any level has faced the frightening effects of draconian subject called “Mathematics”. Although Mathematics has only 10 numerals ranging from zero to nine (0,1,2,3,4,5,6,7,8,9), the types of problems that can trouble any individual are magnanimous. Mathematics deals with properties and interaction of numbers and ten characters enlisted above gave the most popular number system a name “Decimal System”.

Numbers are symbolic representations for counting objects. It brings uniformity and makes various dimensions repeatable, independent of time, place, person or domain. Whether it is cost of objects or number of items or weight of an object or volume or area of a domain, numerals are frequently used to denote quantitative parameter for any dimension. But contrary to the popular believe that numbers in general follow a random sequence, a well recognized pattern evolves, when one deciphers the beauty of numerals. In this first part, introduction to numbers is presented in a systematic way.

Each of the ten symbols or characteristic, representing numbers is called digit. But list of numbers does not end with single digit numbers. Biggest single digit number is 9, beyond which two-digit numbers start. Smallest two digit number is 10 and largest one is 99, which becomes a threes digit number from 100 and so on. The smallest digit of any number of digits contain ‘1’ at its extreme left followed by as many zeros as required to complete number of digits. Largest number of given number of digit contains all ‘9’s. For example largest five digit number is 99999 and smallest five digit number is 10000. A very large number is called infinity and very small number is replaced with zero generally. In addition to 10 symbolic representations of numbers, they can be also expressed on a number line, which varies from minus infinity on extreme left to plus infinity on extreme right with centre named as origin situated at number zero. A number lying on left side is smaller.

Numbers are classified under different heads, with certain basic properties. The smallest such set is natural number (denoted by N). All countable numbers from 1 to plus infinity is included in this set (1,2,3,4,5,6,7, …….). If zero is also added to the list of natural number, it is called whole number (denoted by W). In the domain of whole numbers, if negative integers are also added, Integers (denoted by Z; German Zehlan) are formed. A further superset of number is rational numbers (denoted by Q: Quotient). All the numbers, which can be represented in the fractional form p/q, where q is not equal to zero, are included in this series. Naturally this includes all integers. For a fractional form p/q, ‘p’ is called numerator, while ‘q’ is called denominator. Another way of representing rational numbers is the decimal form of either terminating (0.32, 0.25 …) or recurring (0.33333…, 0.166666….) decimal. The numbers, which are not satisfying the fractional or decimal form requirement of rational numbers, are called irrational numbers (denoted by I). It includes square roots (Ö2, Ö3, Ö15 etc), pi (p), base of Naperian log (e) etc. Combination of both rational and irrational numbers forms a set of real numbers (denoted by R). The biggest set of numbers is complex number (denoted by C), which include imaginary part (i = Ö-1) in the set of real numbers.

Based on properties of various numbers, all numbers divisible by ‘2’, are called even numbers (2,4,6,8…….) otherwise they are called odd numbers (1,3,5,7,9….). A number, having only two factors namely ‘1’ and the number itself, are called prime numbers (2,3,5,7,11,13….), while numbers having more than two factors are called composite numbers (4,6,8,9,10,12…). If sum of factors of a number is equal to number itself, it is called perfect number (6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14 …). A number, which satisfies polynomial algebraic equation with integer coefficients are called algebraic number (Integers, square roots, cube roots, trigonometric functions of an angle etc) else they are named transcendental number (log 2, p, 2^Ö2 etc). Reversing digits of a number is writing digits of the number in opposite turned around way. e.g. 14537 is reverse of 73541. If a number and its reverse are same, they are called Palindromes, like 13631, 161 etc.

Do you know?

1.      ‘1’ is neither a prime number nor a composite number.

2.      ‘2’ is only even prime number.

3.      Zero is a rational number.

4.      ‘p’ and ‘e’ are irrational numbers.

5.      ‘p’ is approximated to 22/7 or 355/113 or 3.1415926…

6.      ‘6’ is the smallest perfect number.

7.      There are infinite rational numbers between two rational numbers.

8.      All real numbers can be represented on a number line.

9.      A rational fraction a/b in lowest terms has a terminating decimal expansion if and only if the integer b has no prime factors other than 2 and 5. For integers, ‘1’ remains in the denominator.

10.  Complex numbers cannot be represented on a number line. They are represented in a plane by “Argand Diagram”.

11.  All transcendental numbers are irrational but all irrational numbers (Ö2) are not transcendental numbers.

12.  Trigonometric functions (Sin20, Cos35, …) are algebraic irrational numbers and not transcendental numbers.

13.  Hilbert number (2^Ö2 ) is transcendental irrational number.