How much faster are various mathematical operations using a Trachtenberg method rather than a conventional method?
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It depends on what area of arithmetic you are talking about
It's really useful in some places whereas at times it just involves a lot of learning by rote where normal conventional methods could have their way much more easily.
According to the book 'The Trachtenberg Speed System of Basic Mathematics', "A thin, studious-looking boy wearing silver-rimmed spectacles was told to multiply 5132437201 times 452736502783. He blitzed through the problem computing the answer—2323641669144374101785—in seventy seconds. The class was one where the Trachtenberg system of mathematics is taught." The book also states that "Educators have found that the Trachtenberg system shortens time for mathematical computations by twenty percent.", although this statement does not seem consistent with the previous example.
From playing with the system a little myself, 20% sounds close for day to day use and not putting in much dedicated practice. As with most mental math (and most math in general) speed will increase based on practice. I enjoy that there are only a small number of techniques used for a large number of applications, rather than a large number of application specific techniques to memorize as in some systems.
The more extreme example mentioned sounds more like a stage demonstration than an average improvement.
Studying and practicing the Trachtenberg System has helped me DRASTICALLY improve my day to day mental arithmetic, especially noticeable when out shopping and working out the cost savings in buying bulk and cost comparisons per kilo.
Your question is hard to answer but I will say that where as before I'd simply give up on trying to do large multiplication in my head (as it would be impossible), now I actually am able to do the large multiplications and check them with good accuracy in my head. 3-4 digit multiplications (ie: 2873 x 231) in my head now generally take me around 15-30 seconds to work out (depending on the numbers) with almost 100% accuracy, about a month ago I would be able to do them in about 30-40 seconds ~80% correct. Give me a pen and paper and I can now do the same sums often in under 10 seconds.
The claims of a boy doing very large multiplications doesn't sound like it's impossible but would take a good few months of practice. Good memory recall is obviously an important skill to have and will need improving if you're going to stand a chance doing sums of that length.
I study once a week for a couple of hours and have been for about 2 months but also practice when out shopping like I said.
A good free site I found helpful to use alongside the book is http://www.trachtenbergsystem.org
...Same info as in the book but easier to understand (for me) and talks you through every step.