Surprising Number Patterns III

Here are some more charmers of mathematics that depend on the surprising
nature of its number system. Again, not many words are needed to
demonstrate the charm, for it is obvious at first sight. Just 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. You
might ask them why multiplying by 9 might give such unusual results.
Once they see that 9 is one less than the base 10, they might get other
ideas to develop multiplication patterns. A clue might be to have them
consider multiplying by 11 (one greater than the base) to search for a
pattern.
0 x 9 + 1 = 1
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1 111
1 234 x 9 + 5 = 11 111
12 345 x 9 + 6 = 111 111
123 456 x 9 + 7 = 1 111 111
1 234 567 x 9 + 8 = 11 111 111
12 345 678 x 9 + 9 = 111 111 111
A similar process yields another interesting pattern. Might this give your
students more impetus to search further?

0 x 9 + 8 = 8
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8 888
9 876 x 9 + 4 = 88 888
98 765 x 9 + 3 = 888 888
987 654 x 9 + 2 = 8 888 888
9 876 543 x 9 + 1 = 88 888 888
98 765 432 x 9 + 0 = 888 888 888