I haven’t read the other responses but the Trachtenberg method of speed mathematics is excellent. I bought a used copy but I believe you can access the PDF version free online.
I remember having this same thought before, how an 8x8 could be reduced to a 4x4. The prospect of memorizing all 2-digit multiplication tables is daunting though to say the least. Do you know by chance if Willem memorized the tables using mnemonics? Or was it just normal spaced repetition? I guess if you memorized 1,000 digits/day you could learn the tables in about a month. I think it’s the retrieval process that would be tricky.
I just wanted to quickly interject that I think the duplex method of roots provides a better algorithm. I read through the Trachtenberg book a while ago.
Keep in mind that although you can change a 8X8 into a 4X4 by using 2 digit numbers, the additions also become much longer.
So it is not certain that you gain time that way.
I know Willem personally, have been to his house many times.
He explains that he remembered the 2x2’s just by doing them (as a kid).
Now he just knows them by heart.
I know; I envy that too.
Absolutely. I’d be interested to know what the time benefit is. I’m sure there’s a way to get a good approximation. You would just have to add the times from each step.
That’s really cool. I wish I was more productive as a kid. I could have been learning my 2x2’s instead of playing all those video games.
What are you now focusing on?
Multiplication? 2x2? 3x3?
Actually now I’m focusing in multiplication in general and particualry I went a little deeper with squaring problems, its way more interesting knowing that arthur benjamin is actually using the “major system” in his calculations processes! And he also know all the squares of all the numbers between 11-99 by heart! Making squaring a 4 digit number very easier adding to that if he is using the major system as a memory devise the helps him holding all these numbers in his mind vivid while multipling/summing them a very interesting way to do it actually I knew it , the key for mental math is how to make these numbers vivid for you while calcualting them.
This is from How to be Clever:
Take the multiplication one digit at a time, starting from the end. For the sum 396 x 184, we know the final digit of the answer will be the final digit of 6 x 4 – which, if you know your times tables you’ll know to be 24. So our last digit is 4 – write it down or remember it, depending on how hard you want to make things for yourself, and carry the 2 into the next calculation.* Now for the second-last digit. It’s less obvious, but this one is made up of the tens from the previous calculation (that 2 we just carried), plus 6 x 8, plus 9 x 4. That is, we’re multiplying the last digit of each number with the second-to-last digit of the other number. 2 + 48 + 36 = 86.
Once again, we take the last digit of the answer, write it down in front of the 4 we already got, and carry the 8. Or, to look at it another way, we’re taking the first number (396) and gradually multiplying each digit by each digit of the second number (184), working backwards. Carrying on in the same way, our third-fromlast digit will be 8 + (6 x 1) + (9 x 8) + (3 x 4), which all adds up to 98.
Now, we’ve already multiplied the 6 in 396 by each digit of the 184, so we can forget about it now. We’ve carried the 9 from the third calculation, and we add to it 9 x 1 and 3 x 8, giving us 42. We should now have an answer that ends with 2864, just waiting for one last calculation. Carry the 4, and add 3 x 1, and we’re done. 396 times 184 is 72,864.
The most difficult seeming calculations usually have an easy way of getting to the answer if you think about it longer.
Because I would do the previous calculation as follows:
In my mind I see:
400 - 4
180 + 4
Now do criss cross multiplication. From left to right.
400 X 180 = 72,000 (in my mind I do 4x18, then double the first and halve the second number = 8x9 = 72. Add zeros.)
4 X 400 = 1,600
-4 X 180 = -720 (same calculation as the first step}
-4 X 4 = -16
Putting it all together.
2: 1,600 - 720 = 1,600 - 800 + 80 = 880.
(Alternatively, start with 720. Add 80 to get to 800 and add another 800 To get to 1,600.)
3: 880 -16 = 864. (880 -20 +4)
Also observe that the calculation is done without carrying.
Even if you don’t know these, you can easily move from one (known) square to an unknown one.
Let’s say you need to know 24^2 and you know 25^2 = 625.
We use: (x-1)^2 = x^2 -2x +1
in this case (25-1)^2 = 25^2 -2*25 +1 = 625 -50 +1 = 576.
Going upwards, use (x+1)^2 = x^2 +2x +1
Starting with 20^2=400:
21^2 = 20^2 + 40 +1 =441
22^2 = 441 + 42 + 1 = 484
23^2 = 484 + 44 + 1 = 529
See how quick you can do this?
Use this as an exercise while waiting for a bus or so.
Squares of two digit numbers ending in 5 can be easily calculated in this fashion:
1) they all end in 5 x 5 = 25 (right hand side)
2) multiply the tenth digits by itself-plus-one (left hand side)
15 → 1 x 2 = 2 … 225
25 → 2 x 3 = 6 … 625
35 → 3 x 4 = 12 … 1225
in fact in general, you can use above method if the tenth digits are the same and the unit digits add to 10…
28 x 22
left hand side: 2 x 3 = 6
right hand side: 8 x 2 = 16
73 x 77
lhs: 7 x 8 = 56
rhs: 3 x 7 = 21