Wednesday, December 13

What Your Teacher Never Taught You About Nine Times Tables.

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All of us may remember learning our multiplication tables when we very small. I myself remember saying by rote “1 x 2 is 2, 2 x 2 is 4, 3 x 2 is 6” etc. The easiest one was multiplying by 10 because all you had to do was add a 0 to get the correct answer. Other than that the higher the figure you multiplied by the harder the multiplication became. Perhaps there are those who find that nine times tables are hard to remember. There is a fascinating way to check if you are correct. Let us look at what your teacher never taught you about nine times tables.

If you were to look at your nine times tables carefully, you will notice something very unusual.  Starting from the very first digit of your first answer and going downwards you will see that it goes like this 0 (from 09), 1, 2, 3, 4, 5, 6, 7, 8, 9:

  1 x 9 = 09  

  2 x 9 = 18  

  3 x 9 = 27  

  4 x 9 = 36  

  5 x 9 = 45  

  6 x 9 = 54  

  7 x 9 = 63  

  8 x 9 = 72  

  9 x 9 = 81

10 x 9 = 90

The one abnormality is the second 9 when we go to 11. Similarly if we look at the second digit from the first answer in 1 x 9 is 09 and you go straight down you will see the reverse order of what you have just noticed.  

  1 x 9 = 09

  2 x 9 = 18

  3 x 9 = 27

  4 x 9 = 36

  5 x 9 = 45

  6 x 9 = 54

  7 x 9 = 63

  8 x 9 = 72

  9 x 9 = 81

 10 x 9 = 90

You will get 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, then the sequence starts again 9, 8, etc. This is not the only unusual thing about the nine times table. If you were to add all the digits on the answer side of the table they usually add up to 9, the exception usually being wherever more than one 1 shows up like 11 x 9. So let us look at the multiplication table again: 

  1 x 9 = 09 (0 + 9 = 9)  

  2 x 9 = 18 (1 + 8 = 9)  

  3 x 9 = 27 (2 + 7 = 9)  

  4 x 9 = 36 (3 + 6 = 9)  

  5 x 9 = 45 (4 + 5 = 9)  

  6 x 9 = 54 (5 + 4 = 9)  

  7 x 9 = 63 (6 + 3 = 9)  

  8 x 9 = 72 (7 + 2 = 9)  

  9 x 9 = 81 (8 + 1 = 9)

10 x 9 = 90 (9 + 0 = 9)

Wherever there is more than one 1 in a number and you multiply it by 9 the sum of the digits will not be nine but a multiple of 9 like 18. For instance 11 x 9 = 99 (9 + 9 = 18). But other than that the great tendency is to get 9 as the sum of the digits. Note what happens with bigger figures: 

123 x 9         = 1107         (1 + 1 + 0 + 7 = 9)

1234 x 9       = 11106       (1 + 1 + 1 + 0 + 6 = 9)

12345 x 9     = 111105     (1 + 1 +1 + 1 + 0 + 5 = 9)

123456 x 9   = 1111104   (1 + 1 +1 + 1 + 1 + 0 + 4 = 9)

1234567 x 9 = 11111103 (1 + 1 + 1 +1 + 1 + 1 + 0 + 3 = 9) 

You could even predict what 12345678 x 9 is. Add an extra 1 to the front and lessen the 3 to a 2. It is incredible that no matter how big the number is you still get the digits adding up to 9 when there is only one 1 in the figure multiplied by 9.

So if you cannot remember if 5 x 9 is 45, 46 or 47, ask yourself which one has digits that add up to 9. The answer will be 45. Of course if you cannot remember what 6 x 9 is and you know that 5 x 9 is 45, then just add 9 to 45 and you will get 54. Addition can help with multiplication. By the way, that last principle works with all multiplication tables. I believe after this people will see 9s everywhere as their eyes been opened. After reading this people should have confidence and work out their tables with NImbleNEss. You have now been geNuINEly eNlIghteNEd as to the eNgINEering of the nine times table. This is a mnemonic coNtINgEncy plan if you forget or are not sure of your answers.

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