Let’s begin by looking at multiplications. When you multiply two different numbers you always have to place as many numbers of zeros in front of the multiplicant as there a digits in the multiplier. So, if you should multiply 123 with 34 you have to look at 123 as 00123. 1234 * 4 then it’s 01234. Ok?
Each number has a “neighbour” to the right, except the last digit. 1234, where 2 is the neighbour to 1, 3 is the neighbour to 2, 4 is the neigbour to 3 and no one is the neighbour to 4.
Ok, we begin with multiplication by 12. Double each digit, starting from the right, and add the neighbour.
379 * 12 = ?
00379 * 12
9 has no neighbour. 9*2 = 18
(7*2) + 9 = 14 + 9 = 23 (and then the 1 from 18) = 24
(3*2) + 7 = 6 + 7 = 13 (and then the 2 from 24) = 15
(0*2) + 3 = 3 (and then the 1 from 15) = 4
(0*0) + 0 = 0
379 * 12 = 4 548
Multiplication by 11.
First recopy the last digit, then add the number with the one before on the rest.
4 862 * 11 = ?
004862 * 11
2 = 2
6 + 2 = 8
8 + 6 = 14
8 + 4 = 12 (and add the 1 from 14) = 13
4 + 0 = 4 (and then add the 1 from 13) = 5
0 + 0 = 0
4 862 * 11 = 53 482
Multiplication by 3.
Rules:
1. Take ten minus the first number and double this. Add on 5 if the number is odd.
2. Take each successive number from 9 and double this. Add half the neighbour. Add on 5 if the number is odd. (rule: “half”)
3. Write down a zero next to the last number. Subtract two from half its neighbour. (rule: “half”)
4 932 * 3 = ?
04932 * 3
(10 - 2) * 2 = 16
((9 - 3) * 2) + (2/2) + 5 = 12 + 1 + 5 = 18 (add the 1 from 16) = 19
((9 - 9) * 2) + (3/2) + 5 = 0 + 1 + 5 = 6 (add the 1 from 19) = 7
((9 - 4) * 2) + (9/2) = 10 + 4 = 14
(4/2) - 2 = 0 (add the 1 from 14) = 1
4 932 * 3 = 14 796
Multiplication by 4.
Rules:
1. Take ten minus the first number. Add on 5 if the number is odd.
2. Take each successive number from 9 and add half the neighbour. Add on 5 if the number is odd. (rule: “half”)
3. Write down a zero next to the last number. Subtract one from half its neighbour.
4 932 * 4 = ?
04932 * 4
10 - 2 = 8
(9 - 3) + (2/2) + 5 = 6 + 1 + 5 = 12
(9 - 9) + (3/2) + 5 = 0 + 1 + 5 = 6 (add the 1 from 12) = 7
(9 - 4) + (9/2) = 5 + 4 = 9
4 - 3 = 1
4 932 * 4 = 19 728
Multiplication by 5.
Rules:
1. Take half of each numbers neighbour. (rule: “half”)
2. Add on another 5 if the number is odd.
3. Write down a zero to the last (left) number.
4 932 * 5 = ?
04932 * 5
0/2 = 0 (has no neighbour)
(2/2) + 5 = 6
(3/2) + 5 = 6
9/2 = 4
4/2 = 2
0/2 = 0
4 932 * 5 = 24 660
Multiplication by 6.
In this case you have to think about a rule called “half”. Half is the number divided by another minus the rest. For example 7/2 = 3 and not 3,5.
This is what you do:
1. Add half the neighbour to each number.
2. Add on another 5 if the number is odd.
3. Write down a zero next to the last (left) number and repeat the above.
4 932 * 6 = ?
04932 * 6
2 (has no neighbour)
3 + 1 + 5 = 9
9 + 1 + 5 = 15
4 + 4 = 8 (and add the 1 from 15) = 9
0 + 2 = 2
4 932 * 6 = 29 592
Multiplication by 7.
3 steps:
1. Double each number and add half its neighbour. (rule: “half”)
2. Add on another 5 if the number is odd.
3. Write down a zero next to the last (left) number.
4 932 * 7 = ?
04932 * 7
2 * 2 = 4 (has no neighbour)
(3 * 2) + 1 + 5 = 12
(9 * 2) + 1 + 5 = 24 (add the first 1 from 12) = 25
(4 * 2) + 4 = 12 (add the first 2 from 25) = 14
(0 * 2) + 2 = 2 (add the first 1 from 14) = 3
4 932 * 7 = 34 524
Multiplication by 8.
Rules:
1. Take ten minus the first number and double this.
2. Take each successive number from 9 and double this. Then add its neighbour.
3. Write down a zero next to the last number and subtract two from its neighbour.
4 932 * 8 = ?
04932 * 8
(10 - 2) * 2 = 16
((9 - 3) * 2) + 2 = 12 + 2 = 14 (add the 1 from 16) = 15
((9 - 9) * 2) + 3 = 0 + 3 = 3 (add the 1 from 15) = 4
((9 - 4) * 2) + 9 = 10 + 9 = 19
4 - 2 = 2 (add the 1 from 19) = 3
4 932 * 8 = 39 456
Multiplication by 9.
Rules:
1. Write down ten minus the first number.
2. Take each successive number from 9 and add its neighbour.
3. Write down a zero next to the last number and subtract one from its neighbour.
4 932 * 9 = ?
04932 * 9
10 - 2 = 8
(9 - 3) + 2 = 8
(9 - 9) + 3 = 3
(9 - 4) + 9 = 14
4 - 1 = 3 (and add 1 from 14) = 4
4 932 * 9 = 44 388
Multiplication by 10, 2, 1 and 0.
Well, I’m not sure if I have to go thru this one but here it is. 
To multiply by two, double each number.
To multiply by ten, write down a zero at the end of the number.
Any number multiplied by 1 is itself and multiplied by zero is zero.
So, that was it for now. I hope you have learned something and start to use your head more when calculating. 
Related links:
Practice Trachtenberg multiplications
Trachtenberg system on Wikipedia
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