Let’s revisit the basic way of this system. If you understand this, you can do most - if not all - of the calculations Art does.
Let’s do 23 X 62.
You can change the ‘23’ into ‘20’, so go down in amount, then the 62 must come up, and becomes 65. So do 20 X 65 = 1300. Easy start, right?
Now the difference.
We subtracted 3 from 23 to get 20. The difference between 20 and 62 is 42.
Now multiply the differences: 3 X 42 = 126.
Add the two results:
1300 + 126 = 1426. This is your answer.
You can also change the ‘23’ into ‘25’, so go up in amount, then the 62 must come down, and becomes 60. So do 25 X 60 = 1500. Easy start, right?
Now the difference.
We added 2 to 23 to get 25. The difference between 25 and 62 is 37.
Now multiply the differences: 2 X 37 = 74.
Subtract the two result numbers:
1500 - 74 = 1426. This is your answer.
Compare the two. Either you go up - with the lowest number - or you go down.
Try this out on a number of multiplications and see if this clicks.
If it does, try to learn both ways (of going either up or down).
Usually either one of these two gets you the easiest numbers to work with.
This is the most generic multiplication system. It work always and will generate digits. If all shortcuts fail, this will generate the digits.
For me it is a last resort way. First I like to find shortcuts. If that does not turn up anything, I use this.
The drawback is that sometimes it leads to a lot of carrying.
Take 99 X 98 for example.
An example of the criss-cross method:
12
34 X
Start with 10 X 30 = 300
Then do 10 X 4 + 2 X 30 = 40 + 60 = 100. This is the cross.
Add the last two results: 300 + 100 = 400.
Last we do 2 x 4 = 8.
Add 400 and 8 to get 408.
This is true.
Use 2X2 to get reacquainted with the systems.
Then venture to 2X3 and 3x3.
Arthur Benjamins book - if I recall correctly - does not do division.
However; I have written a lot lately about about different ways of speeding this up.
For division, read this (in opposite order):
You said you need to be able to do mental calculation in your work. Can you give examples?