8

Yesterday while I was watching my 8 year old son doing his math-exercises which was multiplications with 9, I noticed this symmetry in the results of multiplying 9 with the range 1 to 10, which I think is very beautiful! Where does this symmetry in 9 comes from, and are there studies on this whether it is a coincident, does such symmetry apply only to 9, etc.? I would be very grateful if someone could point me to more reads about this. Thank you in advance!


  9      9      9      9      9      9      9      9      9      9
  x1     x2     x3     x4     x5     x6     x7     x8     x9     x10

(0)[9] (1)[8] (2)[7] (3)[6] (4)[5] (5)[4] (6)[3] (7)[2] (8)[1] (9)[0]
|   |   |  |   |  |   |  |   |  |   |  |   |  |   |  |   |  |   |  |
|   |   |  |   |  |   |  |   |  -----  |   |  |   |  |   |  |   |  |
|   |   |  |   |  |   |  |   -----------   |  |   |  |   |  |   |  |
|   |   |  |   |  |   |  -------------------  |   |  |   |  |   |  |
|   |   |  |   |  |   -------------------------   |  |   |  |   |  |
|   |   |  |   |  ---------------------------------  |   |  |   |  |
|   |   |  |   ---------------------------------------   |  |   |  |
|   |   |  -----------------------------------------------  |   |  |
|   |   -----------------------------------------------------   |  |
|   -------------------------------------------------------------  |




(0)    (1)    (2)    (3)    (4)    (5)    (6)    (7)    (8)    (9)
   [9]    [8]    [7]    [6]    [5]    [4]    [3]    [2]    [1]    [0]
amte
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  • See: https://math.stackexchange.com/a/65409/1203597 – Soham Saha Jun 10 '24 at 06:55
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    It works in base $b$ with $b-1$. – JMP Jun 10 '24 at 06:56
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    Adding 9 is the same as adding 10 and subtracting 1. Therefore the tens digit goes up while the units digit goes down. They change at the same speed leading to this pattern. – Jaap Scherphuis Jun 10 '24 at 08:39
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    I just want to say that while the answer is pretty simple, I remember being very struck by this as a young person, and it was some years before I was able to see the reasoning behind it. I'm happy to see someone else going on this journey. – Richard Rast Jun 10 '24 at 16:31

1 Answers1

11

This symmetry will always happen with multiplications of the largest possible digit, i.e., if you are working in base $b$, then multiples of $b-1$ will be

$$0n_{b-1}\\ 1n_{b-2}\\ 2n_{b-3}\\ \vdots\\ n_{b-3}2\\ n_{b-2}1\\ n_{b-1}0$$

where $n_k$ defines the $k$-th digit in your system.

In other words, in base $8$, multiples of $7$ are $$07_8=07_{10}\\ 16_8=14_{10}\\ 25_8=21_{10}\\ 34_8=28_{10}\\ 43_8=35_{10}\\ 52_8=42_{10}\\ 61_8=49_{10}\\ 70_8=56_{10}$$

5xum
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