The beauty of nine time table

Pardon me if my writing is confusing since I am not a Mathematician.

Every child would have to memorize multiplication time tables in Mathematics lessons. Digits 6 and above were a challenge to commit to memory.

Now, I share with you the beauty of nine time table. The left column is nine time table and the right column lists the sum of digits for each multiplication.

When you represent the multiplication of nine time table as two- or three-digit forms, the sum of the digits will always add up to nine except those of multiplications by 11, 21, 22, 31, 32, 33, 41, 42, 43, 44 etc. which give 18.

Interestingly, if you observe a pattern, what is the summation of the digits when you multiply 9 by 55? The answer is 18 i.e. a ten-fold increased will give one multiplication where the digits add up to 18, a twenty-fold increased will give two multiplications with digits adding to 18 etc.

Another pattern for multiplications from one to ten is that the digits are completely opposite when you use an imaginary line to divide at multiplications of five (45) and six (54), four (36) and seven (63), etc.

My last observation for multiplications of nine from one to ten is the first digit is one less than the number you multiply. For example, for multiplication of nine by two, the first digit of the multiplication is 2 - 1 = 1. Since it always add up to nine, the second digit is 9 - 1 = 8.

I know it is a long way to do multiplication by nine, but this is just another way of looking at it instead of memorizing.