Surprising Number Patterns Part I – Pola Bilangan Menakjubkan Bag. I

There are times when the charm of mathematics lies in the surprising
nature of its number system. There are not many words needed to demonstrate
this charm. It is obvious from the patterns attained. Look, enjoy, and
spread these amazing properties to your students. Let them appreciate the
patterns and, if possible, try to look for an “explanation” for this. Most
important is that the students can get an appreciation for the beauty in
these number patterns.
1 . 1 = 1
11 . 11 = 121
111 . 111 = 12321
1111 . 1111 = 1234321
11111 . 11111 = 123454321
111111 . 111111 = 12345654321
1111111 . 1111111 = 1234567654321
11111111 . 11111111 = 123456787654321
111111111 . 111111111 = 12345678987654321

1 . 8 + 1 = 9
12 . 8 + 2 = 98
123 . 8 + 3 = 987
1234 . 8 + 4 = 9876
12345 . 8 + 5 = 98765
123456 . 8 + 6 = 987654
1234567 . 8 + 7 = 9876543
12345678 . 8 + 8 = 98765432
123456789 . 8 + 9 = 987654321

Notice (below) how various products of 76,923 yield numbers in the same
order but with a different starting point. Here the first digit of the product
goes to the end of the number to form the next product. Otherwise, the
order of the digits is intact.
76923 . 1   = 076923
76923 . 10 = 769230
76923 . 9   = 692307
76923 . 12 = 923076
76923 . 3   = 230769
76923 . 4   = 307692

Notice (below) how various products of 76,923 yield different numbers
from those above, yet again, in the same order but with a different starting
point. Again, the first digit of the product goes to the end of the number
to form the next product. Otherwise, the order of the digits is intact.

76923 . 2  = 153846
76923 . 7  = 538461
76923 . 5  = 384615
76923 . 11 = 846153
76923 . 6  = 461538
76923 . 8  = 615384

Another peculiar number is 142,857. When it is multiplied by the numbers
2 through 8, the results are astonishing. Consider the following products
and describe the peculiarity.

142857 . 2 = 285714
142857 . 3 = 428571
142857 . 4 = 571428
142857 . 5 = 714285
142857 . 6 = 857142

You can see symmetries in the products but notice also that the same
digits are used in the product as in the first factor. Furthermore, 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 . 7 = 999999. Surprised?
It gets even stranger with the product, 142857 . 8 = 1142856. If we
remove the millions digit and add it to the units digit, the original number is formed.
It would be wise to allow the students to discover the patterns themselves.
You can present a starting point or a hint at how they ought to start and
then let them make the discoveries themselves. This will give them a sense
of “ownership” in the discoveries. These are just a few numbers that yield
strange products.

Source: Math' Wonders to Inspire Teachers and Students (by Alfred S. Posamentier)

Posted on April 19, 2013, in Mathematics and tagged , , , . Bookmark the permalink. Leave a comment.

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