Most students’ introduction to mathematics has been
bereft of appreciation of its beauty. This article is written to glass case
just a few beautiful examples of mathematical marvels with a view to enable
students develop deeper interest in Numbers and Mathematics for their gain:

**Example (1)**: When we use a calculator to determine the following sum, we find it to be zero.

123,789

^{2}+ 561,945

^{2}+ 642,864

^{2}- 242,868

^{2}-761,943

^{2}- 323,787

^{2}= 0

This may not be quite appealing at the first glance; since we have the squares of large numbers and they seem to show no particular pattern. Yet when we begin to manipulate these numbers in an orderly manner, the sum amazingly remains equal to zero in all the cases!

In the first case, let us delete the hundred-thousand place (the left-most digit) from each number:

23,789

^{2}+ 61,945

^{2}+ 42,864

^{2}- 42,868

^{2}-61,943

^{2}- 23,787

^{2}= 0

3,789

^{2}+ 1,945

^{2}+ 2,864

^{2}- 2,868

^{2}-1,943

^{2}- 3,787

^{2}= 0

789

^{2}+ 945

^{2}+ 864

^{2}- 868

^{2}-943

^{2}- 787

^{2}= 0

89

^{2}+ 45

^{2}+ 64

^{2}- 68

^{2}-43

^{2}- 87

^{2}= 0

9

^{2}+ 5

^{2}+ 4

^{2}- 8

^{2}-3

^{2}- 7

^{2}= 0

Let us now follow a similar process. This time we shall delete the unit’s digit from each of the numbers. Yet again, amazingly, we see each time that the resulting sum is zero:

123,789

^{2}+ 561,945

^{2}+ 642,864

^{2}- 242,868

^{2}-761,943

^{2}- 323,787

^{2}= 0

123,78

^{2}+ 561,94

^{2}+ 642,86

^{2}- 242,86

^{2}-761,94

^{2}- 323,78

^{2}= 0

123,7

^{2}+ 561,9

^{2}+ 642,8

^{2}- 242,8

^{2}-761,9

^{2}- 323,7

^{2}= 0

123

^{2}+ 561

^{2}+ 642

^{2}- 242

^{2}-761

^{2}- 323

^{2}= 0

12

^{2}+ 56

^{2}+ 64

^{2}- 24

^{2}-76

^{2}- 32

^{2}= 0

Let us now combine the two types of deletions into one by removing the right and left

digits from each number and, yet again, we retain zero sums!

123,789

^{2}+ 561,945

^{2}+ 642,864

^{2}- 242,868

^{2}-761,943

^{2}- 323,787

^{2}= 0

2378

^{2}+ 6194

^{2}+ 4286

^{2}- 4286

^{2}-6194

^{2}- 2378

^{2}= 0

37

^{2}+ 19

^{2}+ 28

^{2}- 28

^{2}-19

^{2}- 37

^{2}= 0

**Example (2)**

The beauty of mathematics lies in the surprising nature of its numbers. Not many words are needed to demonstrate this appeal. Just look, enjoy, and spread these amazing properties to all.

1 X 1 = 1

11 X 11 = 121

1111 X 1111 = 1234321

11111 X 11111 = 123454321

111111 X 111111 = 12345654321

1111111 X 1111111 = 1234567654321

11111111 X 11111111 = 123456787654321

111111111 X 111111111 = 12345678987654321

**Example (3)**

1 X 8 + 1 = 9

12 X 8 + 2 = 98

123 X 8 + 3 = 987

1234 X 8 + 4 = 9876

12345 X 8 + 5 = 98765

123456 X 8 + 6 = 987654

1234567 X 8 + 7 = 9876543

12345678 X8 + 8 = 98765432

123456789 X 8 + 9 = 987654321

**Example (4)**

Another interesting number is 142,857, which is a cyclic number and is obtained as 0.142857142857142857… when we divide 1 by 7. When it is multiplied by the numbers 2 through 8, the results are amazing.

142857 X 2 = 285714

142857 X 3 = 428571

142857 X 4 = 571428

142857 X 5 = 714285

142857 X 6 = 857142

We can see symmetries in the products and also notice that the same digits are used in the product as in the first factor. Further, consider the order of the digits. With the exception of the starting point, they are in the same sequence. Now look at the product, 142857 X 7 = 999999!

It gets even stranger with the product, 142857 X 8 = 1142856. If we remove the millions digit and add it to the units digit, the original number is formed!

**Example (5)**

Here are some number charmers!

12345679 X 9 = 111111111

12345679 X 18 = 222222222

12345679 X 27 = 333333333

12345679 X 36 = 444444444

12345679 X 45 = 555555555

12345679 X 54 = 666666666

12345679 X 63 = 777777777

12345679 X 72 = 888888888

12345679 X 81 = 999999999

In the following pattern, notice that the first and last digits of the

products are the digits of the multiples of 9.

987654321 X 9 = 08 888 888 889

987654321 X 18 = 17 777 777 778

987654321 X 27 = 26 666 666 667

987654321 X 36 = 35 555 555 556

987654321 X 45 = 44 444 444 445

987654321X 54 = 53 333 333 334

987654321 X 63 = 62 222 222 223

987654321 X 72 = 71 111 111 112

987654321 X 81 = 80 000 000 001

**Example (6)**

0 X 9 + 1 = 1

1 X 9 + 2 = 11

12 X 9 + 3 = 111

123 X 9 + 4 = 1111

1234 X 9 + 5 = 11111

12345 X 9 + 6 = 111111

123456 X 9 + 7 = 1111111

1234567 X 9 + 8 = 11111111

12_345_678 _ 9 + 9 = 111111111

**Example (7)**

0 X 9 + 8 = 8

9 X 9 + 7 = 88

98 X 9 + 6 = 888

987 X 9 + 5 = 8888

9876 X 9 + 4 = 88888

98765 X 9 + 3 = 888888

987654 X 9 + 2 = 8888888

9876543 X 9 + 1 = 88888888

98765432 X 9 + 0 = 888888888

**Example (8)**

1 X 8 = 8

11 X 88 = 968

111 X 888 = 98568

1111 X 8888 = 9874568

11111 X 88888 = 987634568

111111 X 888888 = 98765234568

1111111 X 8888888 = 9876541234568

11111111 X 88888888 = 987654301234568

111111111 X 888888888 = 98765431901234568

1111111111 X 8888888888 = 987654321791234568

**Example (9)**

Numbers form beautiful relationships! There is much more to numbers than meets the eye.

135 = 1

^{1}+ 3

^{2}+ 5

^{3}

175 = 1

^{1}+ 7

^{2}+ 5

^{3}

518 = 5

^{1}+ 1

^{2}+ 8

^{3}

598 = 5

^{1}+ 9

^{2}+ 8

^{3}

Now, taken one place further, we may get:

1306 = 1

^{1}+ 3

^{2}+ 0

^{3}+ 6

^{4}

1676 = 1

^{1}+ 6

^{2}+ 7

^{3}+ 6

^{4}

2427 = 2

^{1}+ 4

^{2}+ 2

^{3}+ 7

^{4}

The next ones are equally amazing.

3435 = 33 + 44 + 33 + 55

438579088 = 4

^{4}+ 3

^{3}+ 8

^{8}+ 5

^{5}+ 7

^{7}+ 9

^{9}+ 0

^{0}+ 8

^{8}+ 8

^{8 }

(For convenience and for the sake of amusement, 0

^{0 }has been taken as 0, though in fact, it is indeterminate.)

**Example (10)**

1 = 1!

2 = 2!

145 = 1! + 4! + 5!

40_585 = 4! + 0! + 5! + 8! + 5!

**Example (11)**

There are times when the numbers speak more effectively than any explanation.

Here is one such case

1

^{1}+ 6

^{1}+ 8

^{1}= 15 = 2

^{1}+ 4

^{1}+ 9

^{1}

1

^{2}+ 6

^{2}+ 8

^{2}= 101 = 2

^{2}+ 4

^{2}+ 9

^{2}

1

^{1}+ 5

^{1}+ 8

^{1}+ 12

^{1}= 26 = 2

^{1}+ 3

^{1}+ 10

^{1}+ 11

^{1}

1

^{2}+ 5

^{2}+ 8

^{2}+ 12

^{2}= 234 = 2

^{2}+ 3

^{2}+ 10

^{2}+ 11

^{2}

1

^{3}+ 5

^{3}+ 8

^{3}+ 12

^{3}= 2366 = 2

^{3}+ 3

^{3}+ 10

^{3}+ 11

^{3}