I am requesting advice on if I should chose one of these systems or use a combination of two I am looking for a system that is fast easy and requires no paper apart from some minor on paper maths but still helps to make it easier (in other words mostly mental maths but also helps to simplify on paper mathematics)
No system can be called the best actually.
If you are a beginner, perhaps try the “cross method”.
I have three in mind and am trying to choose between trachtenberg, vedic maths and arthur benjamins style when they are online I hope user @kinma can be of some help
“Best” would be if you understood the algebra behind these techniques, so that you understand why they make the arithmetic faster. Take the squares of 2-digit numbers for example… have a look here:
…which leans in the Trachtenberg direction and compare that to a more Vedic approach here:
Of course if you want to be able to do 2-digit squares in the fastest way possible to use them to calculate 3-digit and 4-digit squares, you should simply memorize the squares up to 99. Arthur Benjamin has quite a few of the calculations he describes actually memorized.
Either way, these “systems” are not mutually exclusive and they all simply use different algebraic approaches to speed up arithmetic computation steps. The general idea is to be able to identify the shortcut given the problem; whereas, school math teaches you the algorithm that always works.
Say the problem is 27x23 and you know that if the unit digits add to 10 and the tens digit are the same (which is the case here), you can simply get the left hand side by going 2x(2+1)=6 and the right hand side by going 7x3=21; so you know 27x23=621 with very little effort.
The reason that they teach 27x20 and add 27x3 in school is because that approach always works. You can do 27x20 + 25x20 just the same when the problem is 27x25 and you’d get the correct answer. You can’t use the aforementioned approach here because the unit digits don’t add to 10.
To be faster here, you’d find the midpoint of 27 and 25 which is 26 and square it for 676 and then subtract the distance to the midpoint squared for 676-1=675. I doubt that you learned in school that 5*7=6^2-1^2 or 4*8=6^2-2^2 etc.
In summary, it is far more important to know why something works than it is to know who said that something would work a certain way.