# Which system is best? \[mental calculation\] **URL:** \ **Category:** Mental Calculation, Mathematics, and Puzzles\ **Tags:** trachtenberg-method\ **Created:** [April 22, 2021, 9:10am UTC](https://forum.artofmemory.com/t/which-system-is-best-mental-calculation/62113 "2021-04-22T09:10:46Z")\ **Posts on this page:** 1\ **Showing post:** 4
**Author:** ![bjoern.gumboldt](https://forum.artofmemory.com/user_avatar/forum.artofmemory.com/bjoern.gumboldt/32/16747_2.png) [@bjoern.gumboldt](https://forum.artofmemory.com/u/bjoern.gumboldt)\ **Post date:** [April 22, 2021, 11:09am UTC](https://forum.artofmemory.com/t/which-system-is-best-mental-calculation/62113/4 "2021-04-22T11:09:14Z")
> [@\_Shadow01](#): > > trying to choose between trachtenberg, vedic maths and arthur benjamins style “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: > [@How to calculate squares quickly](https://forum.artofmemory.com/t/how-to-calculate-squares-quickly/29839): > > Let’s focus on calculating squares. They can be used to speed up regular multiplication, like we discuss in the [difference of squares](https://artofmemory.com/forums/difference-of-squares-even-faster-multiplication-5152.html) thread. The tens are easy, 10 squared = 100, 20 squared = 400, 30 squared = 900, etc. The fives are easy: 35 squared is 3 times 4 times 100 and add 25 = 1225. 45 squared is 4 times 5 times 100 and add 25 = 2025. 55 squared is 5 times 6 times 100 and add 25 = 3025. In other words take the first digit, multiply this by one more and concatenate ‘25’ to the res… …which leans in the **Trachtenberg** direction and compare that to a more **Vedic** approach here: > [@How can I start mental calculating](https://forum.artofmemory.com/t/how-can-i-start-mental-calculating/34895/33): > > I’d argue that most people use some kind of number-to-image system when doing mental math with a few more significant digits that require keeping track of. That could be the major system, number shapes, or what have you… most people doing blindfolded Rubik’s cube use a double letter system with speffz. So whilst interesting maybe, that he picked the major system, it’s really not that surprising. Okay, so there’s 89 squares that he knows (11-99) but did you ever try to structure them in any … 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. --- _[View the full topic](https://forum.artofmemory.com/t/which-system-is-best-mental-calculation/62113)._