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Tips to Improve Your Mental Math Ability

153 points by artsandsci 10y ago

discombobulate 10y ago
> For instance, you can multiply by 8 by doubling three times. So instead of trying to figure out 12x8, just double 12 three times: 24, 48, 96 This can be more generally thought of as using exponentiation. 12 * 8 = 12 * 2^3 Using the more general rule, again. 12 * 9 = 12 * 3^2. Or 12 * 3 * 3.
- cgriswald 10y ago
  I find both of these easier and faster using other methods. Maybe just because I've been doing them since childhood. 12 * 8 = 10 * 8 + 2 * 8 = 96 12 * 9 = 12 * 10 - 12 = 108 That said, most of the things in the article are things I've picked up more or less automatically. If you could train yourself to do them, I think it could help, but I think most people who do them just figured them out on their own. I disagree entirely with "Memorizing building blocks", mostly because I see the huge dichotomy among students between those who try to memorize and those who attempt to learn concepts. Those who memorize do worse on tests, because if they forget, they can't just apply their knowledge. They also seem to forget the material faster after the class. Those who learn the concepts are usually okay, even if they forget a detail they can often work it out. I also think that someone who needs to memorize that 3/5 = 0.6 is unclear on some deeper concept somewhere. Edit: formating
  - discombobulate 10y ago
    > I find both of these easier and faster using other methods. Maybe just because I've been doing them since childhood. Sure. It's just another tool. > I disagree entirely with "Memorizing building blocks", mostly because I see the huge dichotomy among students between those who try to memorize and those who attempt to learn concepts. Do you find the best students are fluent with the techniques and have an understanding? It would seem to me, spending time pondering meaning also has the benefits of remembering 'building blocks'; because they're the primitives involved in the struggle for understanding.
    - cgriswald 10y ago
      There are three components: facts, techniques, concepts. Being fluent with techniques is not strictly necessary, but is generally important and can aid in helping to understand concepts by demonstrating relationships. But a student who prioritizes concepts is probably better off than one who only understands techniques.[0] My objection to the article is that it is advocating memorizing facts. Figuring out that 3/5 == 0.6 is trivial if you already understand the concepts and knowing that 3/5 == 0.6 is pointless if you don't know what it means. [0] - Anecdotally, I breezed through calc 1 in high school. First done with every test, always aced those tests, never studied. Had no idea what I was doing, because I was just memorizing and applying rules. Knowing just the techniques was equivalent to knowing nothing.
      - discombobulate 10y ago
        > Had no idea what I was doing, because I was just memorizing and applying rules. Knowing just the techniques was equivalent to knowing nothing. I suppose so. If you wrote some code to carry out the rules, you wouldn't say the computer knows anything.
      - dwringer 10y ago
        If you were always first done and aced your tests, but think you had no idea what you were doing, then think how everyone else must have felt.
  - deleted 10y ago
    [deleted]
  - mathiium 10y ago
    128=(11+1)8=88+8=96 1111128 = 1111118+8=888896
jflowers45 10y ago
reminds me of one of my favorite books from when I was a kid - https://www.amazon.com/Math-Magic-Everyday-Problems-Revised/dp/0060726350 https://www.amazon.com/Math-Magic-Everyday-Problems-Revised/...
- dwe3000 10y ago
  One of my favorites from eighth grade was Isaac Asimov's Realm of Algebra - I read that before any of his other books.
pklausler 10y ago
An amusing way to keep your mental math skills working is to compute the day of the week for calendar dates. (There's a few ways to do this that are published on the Web, e.g. the "Doomsday" method, but one can derive much faster methods with a little analysis.) You can quickly gain a kind of "Rain Man" reputation if somebody mentions a date and you instantly mutter "Saturday". But it's just a matter of adding three numbers and taking the remainder modulo 7.
- oh_sigh 10y ago
  Can you provide a link to a method that is faster than the doomsday algorithm? If I know the doomsday for a given year, I can get the day of week for a date in about 3 seconds. I'd love to get that even faster(or have a faster method when I don't know the doomsday)
  - pklausler 10y ago
    Memorize this table of offsets: 0 Oct, Jan 1 May 2 Aug 3 Feb, Mar, Nov 4 Jun 5 Sep, Dec 6 Apr, Jul Precalculate the offset for the year (2017 is 117 + trunc(117/4) == 146 == 6 mod 7). Then the day of the week for, say, today, is 6 (for the year) + 3 (for Feb) + 23 (day of the month) == 32 == 4 mod 7, which is Thursday . Other examples: D-Day, 6 June 1944, was (44+11 == 55 == 6 mod 7 for the year) + 4 (Jun) + 6 == 16 == 2 mod 7 == Tuesday. The only twist is needing to subtract 1 from the result for dates in Jan and Feb on leap years, since those dates precede the leap day.
    - pmoriarty 10y ago
      I don't understand how you got the "6 mod 7" in the two examples above. Everything else makes sense, though. Thanks.
- dahart 10y ago
  Taking a number of any size mod 7 can also be done in your head, from left to right! First time I saw this was in intro to theoretical CS, where the exercise is create a state machine for mod 7. It was a big surprise for me that it's super simple and you just feed it digits til you run out. Almost trivial to do mentally on very large numbers as fast as someone can speak the number, with just a little practice.
Existenceblinks 10y ago
Those who grown up in asian schools would see these as super easy tricks. Not intend to make superiority sound here but just the well-known fact. There was a trend in my high school when math teacher ask to solve (expect elaborate steps clearly) a problem on blackboard. Ones who can make a shortest steps possible is cool. I always love the moment that I can skip 5 steps with a magic line of equation and go straight to the answer. And then the teacher will remember you, not to appreciate but to revenge you with semester exam! Sorry for being irrelevant.
- dualogy 10y ago
  Interesting indeed! In the west maths is widely not taught/practiced in such "fun" ways sadly, and by and large mostly formulaically and by rote. Unsurprisingly, it instils mostly a yawn reflex in a majority of people for the rest of their lives. Not a good outcome. Asia is a big continent, what kinda countries or areas are you talking of?
  - Existenceblinks 10y ago
    I am in Thailand where their education system is crappy. IMO, west math is better when it comes to systematic thinking, less hacky, more stable. Asian is good at mess thing up in hacky ways (or in the black side, cheating!), system designed is not stable in general. Chinese and korean are more serious about teaching in my perspective. It's probably not that bad that Thailand's education system is crappy, it's more relax.
  - gpawl 10y ago
    There's a widespread effort in USA to teach calculation using multiple algorithms (thanks to Common Core), and the unsurprising result is that parents are howling mad about the "nonsense" their kids are being taught.
Pete_D 10y ago
There's a great segment in one of Richard Feynman's books where he talks about picking up tricks like this, and ends up beating an abacus salesman in a cube-root-finding race. I think it involved memorizing log tables.
- pmoriarty 10y ago
  The story of "Feynman vs The Abacus" can be read here: [1] While it's a great story and is very entertainingly written, as is typical of Feynman, I don't think he was being very fair to the merits of the two approaches to mental math -- his own vs that of the soroban. Feynman clearly hadn't done any research on the merits of the soroban method, he just had this single run in with a random soroban salesman. From this one experience, and without very much thought on the subject, he just kind of dismisses the method almost out of hand. Such offhand dismissals are unfortunately also pretty typical of Feynman regarding subjects he is mostly ignorant of and has little experience with (such as his dismissals of philosophy in other talks). In the story, the random soroban salesman, who clearly hasn't even very much proficiency with the soroban (as evidenced by his need for a physical soroban, vs soroban-trained practitioners who no longer need the physical device but can apply the soroban methods in their head) is matched against one of the greatest physicists in history, a Physics Nobel Prize winner who has advanced math training. That's really not a fair match. It would have been great to see how Feynman would have fared against someone like this guy: [2] or against one of the national soroban competition winners, or against a top mathematician or physicist who was soroban-trained. That would be a fairer match and the results more indicative of the merits of the two approaches. [1] - http://www.ee.ryerson.ca/~elf/abacus/feynman.html http://www.ee.ryerson.ca/~elf/abacus/feynman.html [2] - https://www.youtube.com/watch?v=rPTKZ4PLkMc https://www.youtube.com/watch?v=rPTKZ4PLkMc
  - gpawl 10y ago
    You miss the point of the story. The story is about "calculating" vs "thinking", and how refusing to think holds one back.
dahart 10y ago
> Add and Subtract From Left to Right This one's my favorite, and I like to combine it with doing all borrows/carries in a separate pass, which means intermediate "digits" are sometimes no longer 0..9. For example: 78 + 89 = 7+8|8+9 = 15|17 = 1|6|7 = 167
- mathiium 10y ago
  78+89 = 80+90-3=167
- Existenceblinks 10y ago
  90 + 78 - 1 is faster (to me). Actually, I believe it just depends on ways which either fits your brain or you gets used to it.
  - dahart 10y ago
    For this example, totally. What I'm talking about is more useful with larger numbers, when working on paper, and simply as an alternative way to think. I'm not making any claims about what's "faster", that's up to individual experience and preference. That said, rounding tricks get hard to use on 4+ digit numbers.
george_ciobanu 10y ago
I'm a bit surprised - I do almost all these things in my head anyway, never thought them as tricks just that I'm weird and cannot model school methods like others do. (Not trying to brag)
- LordKano 10y ago
  I thought pretty much the same thing as I was reading this. I don't think of these as "tricks", just "how I do it in my head".
- irrational 10y ago
  But, how did you learn them? I didn't do many of these things naturally, but I've read about them over the years and have learned to do them from reading so now I do almost all these things in my head too.
  - george_ciobanu 10y ago
    Laziness? I didn't like remembering to carry when multiplying 45x16 so I did 40x16 + 5x16 - easy to add two numbers in your head, hard to remember two shifted partial results (the way it's done on paper). Different media require different methods. 45x 16 on paper: 270+ 45 It's easy to see here but in your head you have to remember that 45 is shifted left when added to 270. Visually it makes sense but otherwise hard. Breaking things into multiplications by 10, five or other easy substeps is recursive and easy to trace back up. Truth is I don't know how I came up with it, I think I was just lazy.
dahart 10y ago
While these are simple tricks almost everyone here knows, there are great books on advanced techniques. I heard about this one from a math blog and got it. Goes into gory detail about to manually calculate logs and roots, reciprocals and transcendentals. It was a fascinating read, I expected it to help me sleep at night and it did the opposite. "Dead Reckoning: Calculating Without Instruments" https://www.amazon.com/gp/product/0884150879/ https://www.amazon.com/gp/product/0884150879/
- pmoriarty 10y ago
  How would you respond to the negative Amazon reviews that say the book is very difficult, not well explained, and has no exercises?
  - dahart 10y ago
    All those things are true. ;) And the author admits them in the introduction, which you should be able to read from the amazon page. It is still an amazing book. A lot of the reviews clearly expected this to be a way to learn quick tricks to impress their friends. There are lots of books like that, but this isn't one of them, this one goes deep into the theory and long lost methods for doing difficult problems without a computer. For this audience, the difficulty level will be mixed and not necessarily all hard. Anyone who's done some programming will find plenty of material that is both accessible and super interesting.
dancek 10y ago
There was a great MITx course called Street-Fighting Math. Most of the techniques and ideas taught apply well to mental math. The course materials are freely readable after logging into edX and enrolling. https://www.edx.org/course/street-fighting-math-mitx-6-sfmx https://www.edx.org/course/street-fighting-math-mitx-6-sfmx EDIT: Apparently the similarly-titled MIT Press book is freely available as PDF. Now that the course isn't currently running, I'd probably prefer the book. https://mitpress.mit.edu/books/street-fighting-mathematics https://mitpress.mit.edu/books/street-fighting-mathematics
- imranq 10y ago
  this is great - thanks!
pmoriarty 10y ago
Lately I've been picking up the soroban[1], the Japanese abacus, and it's been tons of fun. It feels a little like solving a Rubik's cube with arithmetic, or maybe like working with a finite state machine. There are different algorithms to apply depending on the state of the soroban, and applying the right sequence of these algorithms will get you to the right result. I find it to be a little addictive, and sometimes find it a bit hard to stop. I always feel like wanting to improve my skills a little more, become a little faster at it, and increase the number of digits and terms I can handle without making a mistake. The soroban is a great tool for developing concentration, a memory for numbers, a facility for performing a relatively complex series of steps in a certain sequence, and eventually for lightning fast mental arithmetic. In Japan, soroban use is taught to young kids[2], who after a while develop enough proficiency not to need the physical device any longer and can perform the calculations on an imaginary soroban, and eventually can achieve some really amazing feats of mental arithmetic, such as this example from their national competitions: [3] Forward to the 1'45" mark in the following video for another really impressive display of soroban-trained mental math prowess: [4] Finally, there's a great subforum for mental math in the Art of Memory Forum: [5] [1] - https://en.wikipedia.org/wiki/Soroban https://en.wikipedia.org/wiki/Soroban [2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y https://www.youtube.com/watch?v=Px_hvzYS3_Y [3] - https://www.youtube.com/watch?v=7ktpme4xcoQ https://www.youtube.com/watch?v=7ktpme4xcoQ [4] - https://www.youtube.com/watch?v=rPTKZ4PLkMc https://www.youtube.com/watch?v=rPTKZ4PLkMc [5] - http://mt.artofmemory.com/forums/mathematics-and-mental-calculation http://mt.artofmemory.com/forums/mathematics-and-mental-calc...
- rabboRubble 10y ago
  Agree. I taught myself in order to teach a young relative, and it really improved my ability to handle mental calculations at the simple end of the mathematical operational spectrum. What was really weird was doing the abacus work, moving beads, all in chunks of 5 or 10, and then looking at the final results without any preconceived notion of what the results were. Totally fresh eyes as I counted the final set of beads. The manual movement was one thing, but the motion didn't cause me to visualize "numbers" as I understand them. Completely separate pathway to the same answer.
Tycho 10y ago
I don't care too much about mental arithmetic, but I care a lot about my "mathematical literacy." I find that reading anything "mathy" can quickly become hard to follow. Sometimes it seems the authors are being very obtuse by introducing variables and not specifying what they represent (also doesn't help that they are all greek letters which I forget the names of). But regardless, I'd love to be able to read pages of mathematics as easily as I can read pages of source code. Anyone got tips for that? Is there some sort of bootcamp for learning the language of maths? I feel the language part can be disentangled from the problem solving part. The same way one doesn't need to be able to come up with brand new algorithms to read and understand the algorithms used in computer science. I don't want math superpowers, just math fluency...
- cshenton 10y ago
  I've found a good basis in abstract math makes a really wide range of mathy stuff accessible. The first two chunks I would tackle from a high school math + college calculus foundation would be set theory and group theory, and though it's an oldie, I'd honestly recommend Russel and Whitehead's "Principles of Mathematics" for the former as a start.
- westoncb 10y ago
  The 'Princeton Companion to Mathematics' has the best introduction I've seen to understanding what mathematics is, and what the most important, orientation-giving concepts are. I think your problem is missing implicit context; once you begin understanding what mathematicians value and how various branches interrelate, and how parts of it developed historically—it becomes much easier to follow all kinds of mathematical writing. The intro is about 150 pages, and somewhat dense, but with work definitely accessible to someone with little knowledge of mathematics. I wish they would sell it separately from the rest of the book because it's that good, but unfortunately it comes attached to another thousand pages of material you probably won't have a use for unless you get into mathematics research.
- nandemo 10y ago
  Try "What is Mathematics?" by Courant et al. However, you might be mistaken about language and problem solving being independent.
penglish1 10y ago
Worth a read: "Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks" by Arthur Benjamin, one of my wife's college professors.
vitohuang 10y ago
'Mathemagics' https://www.amazon.co.uk/Mathemagics-Genius-without-Really-Trying/dp/0737300086 https://www.amazon.co.uk/Mathemagics-Genius-without-Really-T... This is awesome book, it teach you all sort of tricks from 3 digits addition to 3 digits multiplication. I know its written for school teens, but still pretty helpful life skills.

153 points by artsandsci 10y ago

> For instance, you can multiply by 8 by doubling three times. So instead of trying to figure out 12x8, just double 12 three times: 24, 48, 96 This can be more generally thought of as using exponentiation. 12 * 8 = 12 * 2^3 Using the more general rule, again. 12 * 9 = 12 * 3^2. Or 12 * 3 * 3.

cgriswald 10y ago
I find both of these easier and faster using other methods. Maybe just because I've been doing them since childhood. 12 * 8 = 10 * 8 + 2 * 8 = 96 12 * 9 = 12 * 10 - 12 = 108 That said, most of the things in the article are things I've picked up more or less automatically. If you could train yourself to do them, I think it could help, but I think most people who do them just figured them out on their own. I disagree entirely with "Memorizing building blocks", mostly because I see the huge dichotomy among students between those who try to memorize and those who attempt to learn concepts. Those who memorize do worse on tests, because if they forget, they can't just apply their knowledge. They also seem to forget the material faster after the class. Those who learn the concepts are usually okay, even if they forget a detail they can often work it out. I also think that someone who needs to memorize that 3/5 = 0.6 is unclear on some deeper concept somewhere. Edit: formating
- discombobulate 10y ago
  > I find both of these easier and faster using other methods. Maybe just because I've been doing them since childhood. Sure. It's just another tool. > I disagree entirely with "Memorizing building blocks", mostly because I see the huge dichotomy among students between those who try to memorize and those who attempt to learn concepts. Do you find the best students are fluent with the techniques and have an understanding? It would seem to me, spending time pondering meaning also has the benefits of remembering 'building blocks'; because they're the primitives involved in the struggle for understanding.
  - cgriswald 10y ago
    There are three components: facts, techniques, concepts. Being fluent with techniques is not strictly necessary, but is generally important and can aid in helping to understand concepts by demonstrating relationships. But a student who prioritizes concepts is probably better off than one who only understands techniques.[0] My objection to the article is that it is advocating memorizing facts. Figuring out that 3/5 == 0.6 is trivial if you already understand the concepts and knowing that 3/5 == 0.6 is pointless if you don't know what it means. [0] - Anecdotally, I breezed through calc 1 in high school. First done with every test, always aced those tests, never studied. Had no idea what I was doing, because I was just memorizing and applying rules. Knowing just the techniques was equivalent to knowing nothing.
    - discombobulate 10y ago
      > Had no idea what I was doing, because I was just memorizing and applying rules. Knowing just the techniques was equivalent to knowing nothing. I suppose so. If you wrote some code to carry out the rules, you wouldn't say the computer knows anything.
    - dwringer 10y ago
      If you were always first done and aced your tests, but think you had no idea what you were doing, then think how everyone else must have felt.
- deleted 10y ago
  [deleted]
- mathiium 10y ago
  128=(11+1)8=88+8=96 1111128 = 1111118+8=888896

jflowers45 10y ago

reminds me of one of my favorite books from when I was a kid - https://www.amazon.com/Math-Magic-Everyday-Problems-Revised/dp/0060726350 https://www.amazon.com/Math-Magic-Everyday-Problems-Revised/...

dwe3000 10y ago
One of my favorites from eighth grade was Isaac Asimov's Realm of Algebra - I read that before any of his other books.

pklausler 10y ago

An amusing way to keep your mental math skills working is to compute the day of the week for calendar dates. (There's a few ways to do this that are published on the Web, e.g. the "Doomsday" method, but one can derive much faster methods with a little analysis.) You can quickly gain a kind of "Rain Man" reputation if somebody mentions a date and you instantly mutter "Saturday". But it's just a matter of adding three numbers and taking the remainder modulo 7.

oh_sigh 10y ago
Can you provide a link to a method that is faster than the doomsday algorithm? If I know the doomsday for a given year, I can get the day of week for a date in about 3 seconds. I'd love to get that even faster(or have a faster method when I don't know the doomsday)
- pklausler 10y ago
  Memorize this table of offsets: 0 Oct, Jan 1 May 2 Aug 3 Feb, Mar, Nov 4 Jun 5 Sep, Dec 6 Apr, Jul Precalculate the offset for the year (2017 is 117 + trunc(117/4) == 146 == 6 mod 7). Then the day of the week for, say, today, is 6 (for the year) + 3 (for Feb) + 23 (day of the month) == 32 == 4 mod 7, which is Thursday . Other examples: D-Day, 6 June 1944, was (44+11 == 55 == 6 mod 7 for the year) + 4 (Jun) + 6 == 16 == 2 mod 7 == Tuesday. The only twist is needing to subtract 1 from the result for dates in Jan and Feb on leap years, since those dates precede the leap day.
  - pmoriarty 10y ago
    I don't understand how you got the "6 mod 7" in the two examples above. Everything else makes sense, though. Thanks.
dahart 10y ago
Taking a number of any size mod 7 can also be done in your head, from left to right! First time I saw this was in intro to theoretical CS, where the exercise is create a state machine for mod 7. It was a big surprise for me that it's super simple and you just feed it digits til you run out. Almost trivial to do mentally on very large numbers as fast as someone can speak the number, with just a little practice.

Existenceblinks 10y ago

Those who grown up in asian schools would see these as super easy tricks. Not intend to make superiority sound here but just the well-known fact. There was a trend in my high school when math teacher ask to solve (expect elaborate steps clearly) a problem on blackboard. Ones who can make a shortest steps possible is cool. I always love the moment that I can skip 5 steps with a magic line of equation and go straight to the answer. And then the teacher will remember you, not to appreciate but to revenge you with semester exam! Sorry for being irrelevant.

dualogy 10y ago
Interesting indeed! In the west maths is widely not taught/practiced in such "fun" ways sadly, and by and large mostly formulaically and by rote. Unsurprisingly, it instils mostly a yawn reflex in a majority of people for the rest of their lives. Not a good outcome. Asia is a big continent, what kinda countries or areas are you talking of?
- Existenceblinks 10y ago
  I am in Thailand where their education system is crappy. IMO, west math is better when it comes to systematic thinking, less hacky, more stable. Asian is good at mess thing up in hacky ways (or in the black side, cheating!), system designed is not stable in general. Chinese and korean are more serious about teaching in my perspective. It's probably not that bad that Thailand's education system is crappy, it's more relax.
- gpawl 10y ago
  There's a widespread effort in USA to teach calculation using multiple algorithms (thanks to Common Core), and the unsurprising result is that parents are howling mad about the "nonsense" their kids are being taught.

Pete_D 10y ago

There's a great segment in one of Richard Feynman's books where he talks about picking up tricks like this, and ends up beating an abacus salesman in a cube-root-finding race. I think it involved memorizing log tables.

pmoriarty 10y ago
The story of "Feynman vs The Abacus" can be read here: [1] While it's a great story and is very entertainingly written, as is typical of Feynman, I don't think he was being very fair to the merits of the two approaches to mental math -- his own vs that of the soroban. Feynman clearly hadn't done any research on the merits of the soroban method, he just had this single run in with a random soroban salesman. From this one experience, and without very much thought on the subject, he just kind of dismisses the method almost out of hand. Such offhand dismissals are unfortunately also pretty typical of Feynman regarding subjects he is mostly ignorant of and has little experience with (such as his dismissals of philosophy in other talks). In the story, the random soroban salesman, who clearly hasn't even very much proficiency with the soroban (as evidenced by his need for a physical soroban, vs soroban-trained practitioners who no longer need the physical device but can apply the soroban methods in their head) is matched against one of the greatest physicists in history, a Physics Nobel Prize winner who has advanced math training. That's really not a fair match. It would have been great to see how Feynman would have fared against someone like this guy: [2] or against one of the national soroban competition winners, or against a top mathematician or physicist who was soroban-trained. That would be a fairer match and the results more indicative of the merits of the two approaches. [1] - http://www.ee.ryerson.ca/~elf/abacus/feynman.html http://www.ee.ryerson.ca/~elf/abacus/feynman.html [2] - https://www.youtube.com/watch?v=rPTKZ4PLkMc https://www.youtube.com/watch?v=rPTKZ4PLkMc
- gpawl 10y ago
  You miss the point of the story. The story is about "calculating" vs "thinking", and how refusing to think holds one back.

dahart 10y ago

> Add and Subtract From Left to Right This one's my favorite, and I like to combine it with doing all borrows/carries in a separate pass, which means intermediate "digits" are sometimes no longer 0..9. For example: 78 + 89 = 7+8|8+9 = 15|17 = 1|6|7 = 167

mathiium 10y ago
78+89 = 80+90-3=167
Existenceblinks 10y ago
90 + 78 - 1 is faster (to me). Actually, I believe it just depends on ways which either fits your brain or you gets used to it.
- dahart 10y ago
  For this example, totally. What I'm talking about is more useful with larger numbers, when working on paper, and simply as an alternative way to think. I'm not making any claims about what's "faster", that's up to individual experience and preference. That said, rounding tricks get hard to use on 4+ digit numbers.

george_ciobanu 10y ago

I'm a bit surprised - I do almost all these things in my head anyway, never thought them as tricks just that I'm weird and cannot model school methods like others do. (Not trying to brag)

LordKano 10y ago
I thought pretty much the same thing as I was reading this. I don't think of these as "tricks", just "how I do it in my head".
irrational 10y ago
But, how did you learn them? I didn't do many of these things naturally, but I've read about them over the years and have learned to do them from reading so now I do almost all these things in my head too.
- george_ciobanu 10y ago
  Laziness? I didn't like remembering to carry when multiplying 45x16 so I did 40x16 + 5x16 - easy to add two numbers in your head, hard to remember two shifted partial results (the way it's done on paper). Different media require different methods. 45x 16 on paper: 270+ 45 It's easy to see here but in your head you have to remember that 45 is shifted left when added to 270. Visually it makes sense but otherwise hard. Breaking things into multiplications by 10, five or other easy substeps is recursive and easy to trace back up. Truth is I don't know how I came up with it, I think I was just lazy.

dahart 10y ago

While these are simple tricks almost everyone here knows, there are great books on advanced techniques. I heard about this one from a math blog and got it. Goes into gory detail about to manually calculate logs and roots, reciprocals and transcendentals. It was a fascinating read, I expected it to help me sleep at night and it did the opposite. "Dead Reckoning: Calculating Without Instruments" https://www.amazon.com/gp/product/0884150879/ https://www.amazon.com/gp/product/0884150879/

pmoriarty 10y ago
How would you respond to the negative Amazon reviews that say the book is very difficult, not well explained, and has no exercises?
- dahart 10y ago
  All those things are true. ;) And the author admits them in the introduction, which you should be able to read from the amazon page. It is still an amazing book. A lot of the reviews clearly expected this to be a way to learn quick tricks to impress their friends. There are lots of books like that, but this isn't one of them, this one goes deep into the theory and long lost methods for doing difficult problems without a computer. For this audience, the difficulty level will be mixed and not necessarily all hard. Anyone who's done some programming will find plenty of material that is both accessible and super interesting.

dancek 10y ago

There was a great MITx course called Street-Fighting Math. Most of the techniques and ideas taught apply well to mental math. The course materials are freely readable after logging into edX and enrolling. https://www.edx.org/course/street-fighting-math-mitx-6-sfmx https://www.edx.org/course/street-fighting-math-mitx-6-sfmx EDIT: Apparently the similarly-titled MIT Press book is freely available as PDF. Now that the course isn't currently running, I'd probably prefer the book. https://mitpress.mit.edu/books/street-fighting-mathematics https://mitpress.mit.edu/books/street-fighting-mathematics

imranq 10y ago
this is great - thanks!

pmoriarty 10y ago

Lately I've been picking up the soroban[1], the Japanese abacus, and it's been tons of fun. It feels a little like solving a Rubik's cube with arithmetic, or maybe like working with a finite state machine. There are different algorithms to apply depending on the state of the soroban, and applying the right sequence of these algorithms will get you to the right result. I find it to be a little addictive, and sometimes find it a bit hard to stop. I always feel like wanting to improve my skills a little more, become a little faster at it, and increase the number of digits and terms I can handle without making a mistake. The soroban is a great tool for developing concentration, a memory for numbers, a facility for performing a relatively complex series of steps in a certain sequence, and eventually for lightning fast mental arithmetic. In Japan, soroban use is taught to young kids[2], who after a while develop enough proficiency not to need the physical device any longer and can perform the calculations on an imaginary soroban, and eventually can achieve some really amazing feats of mental arithmetic, such as this example from their national competitions: [3] Forward to the 1'45" mark in the following video for another really impressive display of soroban-trained mental math prowess: [4] Finally, there's a great subforum for mental math in the Art of Memory Forum: [5] [1] - https://en.wikipedia.org/wiki/Soroban https://en.wikipedia.org/wiki/Soroban [2] - https://www.youtube.com/watch?v=Px_hvzYS3_Y https://www.youtube.com/watch?v=Px_hvzYS3_Y [3] - https://www.youtube.com/watch?v=7ktpme4xcoQ https://www.youtube.com/watch?v=7ktpme4xcoQ [4] - https://www.youtube.com/watch?v=rPTKZ4PLkMc https://www.youtube.com/watch?v=rPTKZ4PLkMc [5] - http://mt.artofmemory.com/forums/mathematics-and-mental-calculation http://mt.artofmemory.com/forums/mathematics-and-mental-calc...

rabboRubble 10y ago
Agree. I taught myself in order to teach a young relative, and it really improved my ability to handle mental calculations at the simple end of the mathematical operational spectrum. What was really weird was doing the abacus work, moving beads, all in chunks of 5 or 10, and then looking at the final results without any preconceived notion of what the results were. Totally fresh eyes as I counted the final set of beads. The manual movement was one thing, but the motion didn't cause me to visualize "numbers" as I understand them. Completely separate pathway to the same answer.

Tycho 10y ago

I don't care too much about mental arithmetic, but I care a lot about my "mathematical literacy." I find that reading anything "mathy" can quickly become hard to follow. Sometimes it seems the authors are being very obtuse by introducing variables and not specifying what they represent (also doesn't help that they are all greek letters which I forget the names of). But regardless, I'd love to be able to read pages of mathematics as easily as I can read pages of source code. Anyone got tips for that? Is there some sort of bootcamp for learning the language of maths? I feel the language part can be disentangled from the problem solving part. The same way one doesn't need to be able to come up with brand new algorithms to read and understand the algorithms used in computer science. I don't want math superpowers, just math fluency...

cshenton 10y ago
I've found a good basis in abstract math makes a really wide range of mathy stuff accessible. The first two chunks I would tackle from a high school math + college calculus foundation would be set theory and group theory, and though it's an oldie, I'd honestly recommend Russel and Whitehead's "Principles of Mathematics" for the former as a start.
westoncb 10y ago
The 'Princeton Companion to Mathematics' has the best introduction I've seen to understanding what mathematics is, and what the most important, orientation-giving concepts are. I think your problem is missing implicit context; once you begin understanding what mathematicians value and how various branches interrelate, and how parts of it developed historically—it becomes much easier to follow all kinds of mathematical writing. The intro is about 150 pages, and somewhat dense, but with work definitely accessible to someone with little knowledge of mathematics. I wish they would sell it separately from the rest of the book because it's that good, but unfortunately it comes attached to another thousand pages of material you probably won't have a use for unless you get into mathematics research.
nandemo 10y ago
Try "What is Mathematics?" by Courant et al. However, you might be mistaken about language and problem solving being independent.

penglish1 10y ago

Worth a read: "Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks" by Arthur Benjamin, one of my wife's college professors.

vitohuang 10y ago

'Mathemagics' https://www.amazon.co.uk/Mathemagics-Genius-without-Really-Trying/dp/0737300086 https://www.amazon.co.uk/Mathemagics-Genius-without-Really-T... This is awesome book, it teach you all sort of tricks from 3 digits addition to 3 digits multiplication. I know its written for school teens, but still pretty helpful life skills.