I’m looking for something to memorize number-wise. But there is a problem: I’m also interested in learning some speed maths in the future.
What is really necessary to memorize? What will be really helpful, even after I know good speed calculation tricks?
For example, I thought I should learn all of the squares up to 100 and primes. But it seems that there are systems which will let you calculate any 2 digit number times any other 2 digit number in less than 10 seconds without much practise. And there seems to be similar systems for primes?
Someone who knows a lot of tricks: What is really worth memorizing, numbers-wise? I don’t fear any challenge in this department. In fact, I’m looking forward to memorizing something numbers-wise, because I want to practise my number memorizations in a useful way.
But I don’t want to waste my time learning how to hit nails with my fist when there already exists a hammer.
So the question is: What is something that is both useful and can’t be done with some relatively easy-to-learn speed calculating?
I realise that I should probably just go learn speed calculating instead. But I’m more interested in memorizing right now. If I can’t find a conclusive answer to this question I guess I’ll go read a book or two about speed calculating.
The first thing that comes to mind is logarithms.
With logarithms you can do amazing things.
For example how would you mentally calculate 2.39 ^ 0.12?
Or if you think that an investment pays 8.4% per year, how long does it take to double or triple your investment?
Memorizing squares is handy too.
They can also be used in other calculations.
An example.
Suppose you memorized that 56^2 = 3136.
If you have to calculate 54 X 58, you can immediately call out 3132, because you know it is 4 less than the number you memorized.
sure, but for the squares: 54*58 = 50 * 62 + 4 * 8 = 2500 + 600 + 32 = 3132, so it’s not that much harder. It’s the same idea btw, but a different usage. The conjugation rule rocks the speed maths!
The logaritms I could look into, I saw some people talking about using it as a tool in here, but I didn’t quite understand it. Seemed like it was mostly used for big approximations, which is kinda cool, but wasn’t my initial plan.
What about primes? Seems like it’s useful pretty often… No? I saw a couple of techniques for finding primes though, the trick is ofcourse if there is a good system for finding out if something is divisible by 7, which I have not found yet. (Atleast that works up to 100. But memorizing primes up to 1000 should definately be achievable.)
I feel like the only reason to use something that is already fairly easily done, like squares, would be if that technique could then be used to drastically improve a step of a bigger system. However I’m not so sure that will happen, as I will first have to recall the memorization que images for squares and move back and forth and the method becomes increasingly difficult when the numbers are wide appart. If I’m unlucky I might even have to use two memorized squares, and then subtract from eachother. Also only works on half of the multiplications.
True, the conjugation rule rocks.
I feel though you are missing the point.
The more you memorize the bigger the calculations that you can do.
Willem Bouman knows the tables up to 100 X 100 by heart.
Wim Klein knew the logarithms of all integers up to 150 with 5 digit precision.
They both know the squares up to 1000.
Think about it, why would this be?
With that knowledge memorized, one can do amazing things.
For example one can do cross multiplication using 2 digits at the same time.
You will then be twice as fast.
What do you mean by big approximations? If you memorize the logarithms with 5 digits precision, you can calculate with those 5 digits.
Of course, if you want to be able to achieve 100% correct answers, then they cannot always be used.
Some things can be calculated using logarithms a lot faster than with methods that calculate with 100% precision.
7^17 comes to mind for example. How long would it take you to calculate this?
To be honest, I don’t use primes a lot. But a lot of other do, so I guess it has it’s use.
Divisibility by 7 btw is almost always done by doing the actual division.
Ok that sounds pretty cool, kinma. Would it be very easy to calculate squares to, then? If you knew this logarithm trick?
Do you have a good source for learning this technique? I actually want to memorize a huge amount of numbers to practise my number memorizations, but I don’t want to make sure that whatever I learn is, if not useful, atleast fun.
If the numbers are not too big, try to do the calculations yourself.
This is extremely good training.
See where your limits are, then expand them.
But to answer your question, yes, if you know logarithms, then calculating the square of a number becomes multiplying by 2, and this is indeed easy.
It takes a lot of dedication though.You need to be able to calculate the log of a number, then double it, then convert it back to the answer. But, once you can do this, 7^17 can be calculated as well. Same principle.
To be honest, there are not a lot of sources about this.
Just a couple of books and a couple of people doing this.
Again, start by doing the squares yourself and see how far you can go calculating them in your head.
You need a good amount of practice to solve 2*2s under 15 seconds. Sometimes an easier way can be used, but some of them are just that tiny bit too tricky. For big problems, logs seem good- can`t do them myself but I see the power of the system :-).
Have you checked out books about the subject already?
You have got me interested in logarithms now. I guess that it’s really difficult to tell exactly what will become useful, especially since there aren’t really any real world problems that this solves.
I just want to make sure that I’m not learning something that then turns out to never be used. Everything must atleast have some kind of show-off factor. Even better if I can actually use it in real life or when I study physics.
How long time does it take you to calculate 84**2 with this system? And what do you use for 34*81?
RetoCH:
No, I haven’t looked at any books regarding the topic of using logarithms as speed mathematics trick.
I have read maybe 70 pages from Bill Handley - Speed mathematics, and that’s it.
But it does solve real world problems. Here is an example. Let’s say that something grows with 5% per day.
How long does it take to become 3 times as big? This is difficult to calculate without logarithms.
Answer: 22 days.
Or: if you double your money in a year, how much is that per month? Answer: 6% (assuming compound interest).
How is this for show-off factor?
You will see a lot of logarithms when you study physics. Just Google ‘Logarithms physics’.
This is quicker the old-fashioned way. When you get to higher powers or different problems, that is when you really need logs.
I meant it doesn’t solve problems, because if I’m faced with such difficult problems, I probably have my cell phone that I can do these calculations on anyway.
Either way, the big problem for me with memorizing mathematics is that given enough time we can calculate this even in our head, and there are always a billion different ways to calculate stuff. Difficult to choose paths for me atleast.
The log thing is pretty cool because it is actually something that takes way too long time to do in your head , or even on paper, without tricks, and I guess primes have the same appeal, although maybe not the same usage possibilities.
I’m really interested how you deal with problems like 34*81, actually. A big part of memorizing squares for me was that they can be used as mental math tricks even for other problems. But when the numbers are far appart it becomes a huge subtraction, and might be easier to just calculate old-fashioned?
The trick in the speed calculating book I’m read, I have no idea why he put there, as straight multiplication is much faster, and harder to fail… I guess there’s some appeal to showing that the system can be extended beyond where it’s efficient…
Usually, when you think about it, there are quicker algorithms. In the case of 34 X 81, I would use 34 minus 10%, minus 10%. This is the same as (34 X 9) X 9.
If you can do this calculation mentally, you go (disregarding decimal point): 34 minus 10% = 34 - 3.4 = 306.
306 - 30.6 - 2754. For this calculation I do ‘306 - 31’ first = 275. Then I correct the difference from 31 and 30,6.
So I add .4 to 275 = 2754. Then I decide where to put the decimal point.
The point is, learn a lot of different ways and decide per calculation what algorithm you will be taking.
Squares have a special place in my heart because of a thing when I was younger, but they’re actually pretty easy to calculate anyway. Either with anchor, or they’re ±1 from an even 10 or 5, or just straight (a+b)**2 = etc rule is pretty fast. I feel like I’m spending so much time deciding wether to start that I could’ve been halfway through memorizing them now if I didn’t though ^^ Either way, I’m feeling very compelled to learn the logarithms. Squares are very cool, but their use are just way too narrow. Let’s just say like this, I will continue reading my book, I’m like halfway through, I’ve found some useful things, but I also skip a lot. When I’m done I will go for the logarithms, unless I have any reason to change my mind.
regarding the “what trick to use” discussion - I definately think that there’s truth in the quote “I don’t fear the man who has practised 10 000 kicks. I fear the man who has practised one kick 10 000 times”. I do think that there’s a lot of value in learning a system that can do EVERYTHING, and do it really fast. I don’t want to learn and then relearn and relearn. If I’m learning a system I will probably use it for all its possible uses until the day that I die ^^.
But… It seems like it’s difficult to find such wonderformulas.
Anyway, where do you suggest I start with the logarithm-thing? I have seen that you have done more threads about it.
In this case, I would use cross multiplication.
This can do everything. And also very fast.
You already know the Anchor Method. This can also be used as a general method.
I love logarithms. One passion of mine is that I want to be able to mentally calculate them. This of course also helps in memorizing them. Where to start? How about memorizing the logarithms 1- 10? Base 10.
yeah… But to be honest, it seems faster to just straight up multiplicate in that case. So much to remember back and worth when the numbers are far appart. If there is any speed to be gained from that, I’ll have to work a long time on it.
Is that really enough decimals? only 3? I’m almost dissapointed I’ll start with an even amount of digits anyway, so 4 it is I guess.
I think it seems a bit lazy to only learn the primes, when it’s so easy to memorize numbers. I don’t want to have to start by calculating my key.
But start with 4 if you want.
Memorize all numbers, but never forget how to derive numbers from other numbers.
Here is a nice exercise to help you start doing this.
From these numbers one can easily derive the numbers in between.
How would you derive the numbers 1.5, 2.5, 3.5, etc.?
Your asked ‘why the base 10?’. Base 10 has a couple of properties that make converting numbers easy.
If log(2) = 0.3010, then log(20) = 1.3010 and log (200) = 2.3010
See why this is easy?
We can use this thread and cater to your specific needs.
I would suggest you start working on your mental agility.
Try to find the numbers in between: 1.5, 2.5, 3.5, etc. See if you can find a way to interpolate.
See if you can come up with a method to create these numbers from the knowledge you have memorized.
In other words; put the knowledge to the test.
This breeds 2 things. One, by doing these mental calculations you are reinforcing the numbers you memorized.
You can check your answers by fitting the results into your knowledge.
(you can also just use a calculator to check your results, but that is less romantic ).
Two, by working with the numbers you start to understand the system better.
You will also learn which calculations are easier for you to do than others.
In time you might want to memorize and/or calculate all logs from 1 to 100.
If you find a good way to interpolate, you will never be far off.
Otherwise you are just memorizing a bunch of numbers without being able to actually use it.
I don’t get how to calculate the fractions of primes that I haven’t memorized. All calculations were done in the head in the spirit of mental calculation, so might be some mistake, especially for 7.5.
I think addition and subtraction with 4 digit numbers in the head is really difficult to keep track of, actually… Maybe I should start by working on that xD
