Dozenal Multiplication Table – Post 42(36)

This post uses multiple number bases

1 2 3 4 5 6 7 8 9 X E 10 – Base R

1 2 3 4 5 6 7 8 9 10 – Base X

Base X – Whenever you’re multiplying by a number, and you wind up with something ending with a 0, it means as you continue going on, you’re going to repeat the same pattern you got since yoru last 0. For instance, 5, 10, 15 20, 25 30. Each number is a 5, then the whole 0 after. Or for 4, you get 4, 8, 12, 16, 20. Then that pattern repeats, 24, 28 32 36 40. In decimal, this repeating happens either every other step(only for things divisible by 5), every five steps(even numbers), or every ten steps(Everything else)

Base R – Most of dozenal multiplication starts repeating very quickly. In fact, everything repeats either

every 2(divisible by 6),

3(divisible by 4),

4(divisible by 3),

6(divisible by 2), or

10(everything else).

And so, 2/3 of all numbers are very easy to work with, regardless of anything else.

2, 4, 6, 8, X, 10

3, 6, 9, 10

4, 8, 10

6, 10

8, 14, 20

9, 16, 23, 30

For X and E, you get very similar patterns as 8 and 9 in base X

X, 18, 26, 34, 42, 50

E, 1X, 29, 38, 47, 56, 65, 74, 83, 92, X1, E0.

10, of course(10, 20, 30, 40, 50….)

11 has literally exactly the same patterns as 11 in base X

11, 22, 33, 44, 55, 66, 77, 88, 99, XX, EE, 110, 121….

And so does 12.

12, 24, 36, 48, 5X, 60

In fact, any situation where a given digit doesn’t have to cross the 10 boundary into another digit, anything you do is going to be the same.  33 * 3 still equals 99, for example.

And now the proper table in a format you’re probably used to.

1 2 3 4 5 6 7 8 9 X E 10 11 12
2 4 6 8 X 10 12 14 16 18 1X 20 22 24
3 6 9 10 13 16 19 20 23 26 29 30 33 36
4 8 10 14 18 20 24 28 30 34 38 40 44 48
5 X 13 18 21 26 2E 34 39 42 47 50 55 5A
6 10 16 20 26 30 36 40 46 50 56 60 66 70
7 12 19 24 2E 36 41 48 53 5X 65 70 77 82
8 14 20 28 34 40 48 54 60 68 74 80 88 94
9 16 23 30 39 46 53 60 69 76 83 90 99 X6
X 18 26 34 42 50 5X 68 76 84 92 X0 XX E8
E 1X 29 38 47 56 65 74 83 92 X1 E0 EE 10X
10 20 30 40 50 60 70 80 90 X0 E0 100 110 120
11 22 33 44 55 66 77 88 99 XX EE 110 121 132
12 24 36 48 5A 70 82 94 X6 E8 10X 120 132 144

 

 

And in case you also want the decimal multiplication table…

1 2 3 4 5 6 7 8 9 10 11 12
2 4 6 8 10 12 14 16 18 20 22 24
3 6 9 12 15 18 21 24 27 30 33 36
4 8 12 16 20 24 28 32 36 40 44 48
5 10 15 20 25 30 35 40 45 50 55 60
6 12 18 24 30 36 42 48 54 60 66 72
7 14 21 28 35 42 49 56 63 70 77 84
8 16 24 32 40 48 56 64 72 80 88 96
9 18 27 36 45 54 63 72 81 90 99 108
10 20 30 40 50 60 70 80 90 100 110 120
11 22 33 44 55 66 77 88 99 110 121 132
12 24 36 48 60 72 84 96 108 120 132 144

 

 

February 26th, 2014 | Categories: Math | Comments: No Comments |