The Beauty of Numbers

 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.
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123,789 ² + 561,945²  + 642,864²  - 242,868² -761,943²  - 323,787² = 0 
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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! 
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 In the first case, let us delete the hundred-thousand place (the left-most digit) from each number:
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23,789 ² + 61,945²  + 42,864²  - 42,868² -61,943²  - 23,787² = 0

Let us repeat this process by deleting the left-most digit of each number and look at the results: 
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3,789 ² + 1,945²  + 2,864²  - 2,868² -1,943²  - 3,787² = 0
­789 ² + 945² + 864² - 868² -943² - 787² = 0
­89 ² + 45² + 64² - 68² -43² - 87² = 0
­9 ² + 5² + 4² - 8² -3² - 7² = 0
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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 + 561,945²  + 642,864²  - 242,868² -761,943²  - 323,787² = 0
123,78 ² + 561,94²  + 642,86²  - 242,86² -761,94²  - 323,78² = 0
123,7 ² + 561,9²  + 642,8²  - 242,8² -761,9²  - 323,7² = 0
123 ² + 561²  + 642²  - 242² -761²  - 323² = 0
12 ² + 56²  + 64²  - 24² -76²  - 32² = 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 + 561,945²  + 642,864²  - 242,868² -761,943²  - 323,787² = 0
2378 ² + 6194² + 4286² - 4286² -6194² - 2378² = 0
37 ² + 19² + 28² - 28² -19² - 37² = 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 = 1¹ + 3² + 5³
175 = 1¹ + 7² + 5³
518 = 5¹ + 1² + 8³
598 = 5¹ + 9² + 8³

Now, taken one place further, we may get:

1306 = 1¹ + 3² + 0³ + 6⁴
1676 = 1¹ + 6² + 7³ + 6⁴
2427 = 2¹ + 4² + 2³ + 7⁴

The next ones are equally amazing.

3435 = 33 + 44 + 33 + 55
438579088 = 4⁴ + 3³ + 8⁸ + 5⁵ + 7⁷ + 9⁹ + 0⁰ + 8⁸ + 8⁸

 (For convenience and for the sake of amusement, 0⁰   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

1¹ + 6¹ + 8¹ = 15 = 2¹ + 4¹ + 9¹
1² + 6² + 8² = 101 = 2² + 4² + 9²

1¹ + 5¹ + 8¹ + 12¹ = 26 = 2¹ + 3¹ + 10¹ + 11¹
1² + 5² + 8² + 12² = 234 = 2² + 3² + 10² + 11²
1³ + 5³ + 8³ + 12³ = 2366 = 2³ + 3³ + 10³ + 11³