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Improving Your Mental Math Skills?
Infrared-Archer asks: "I want to learn how to do most math calculations in my head. That way I won't have to reach for the calculator for problems I should be able to do mentally. Of course there are various websites (beat the calculator) that show many tricks, but I am looking for a comprehensive solution (books, websites) that shows how to solve of wide range of math problems mentally. Any suggestions?"
A good way to practice is every time you get a bus or train (or get any sort of ticket with numbers on) add up the digits on it in your head. After doing this for a while you'll get quicker and more accurate.
For added challenge translate every letter on there into a number using its place in the alphabet (or even its ascii number) and add them on.
You can then make up your own versions using other arithmetical operators and fractions.
After your arithmetic is up to scratch other areas of maths will be easier to do in your head (although beyond anything simple it is still best to write it down)
http://www-gap.dcs.st-and.ac.uk/~history/HistTopic s/Mental_arithmetic.html
1000s Warcraft Gold while you sleep
I picked this book up a few years ago second-hand.
It's a really great book.
I went from functionally innumerate to someone who can perform tricks with multiplication/division in my head,
It seems to use some of the vedic tricks mentioned in previous comments, but it's far more simpler to learn and put into practice.
I'm actually looking for one myself. A few weeks ago, I met some young (2nd graders to middle school) students in a mall who were demonstrating their math skills from an abacus class. The thing is, they weren't using abaci in their demo. They were able to do the basic math operations (up to division by three digit numbers) in their heads instantly using abacus principles.
These days, I have a new baby to worry about (Jaime, a girl, Mar 4, 5 lbs 13 oz) so I haven't had a chance to play with one yet. After meeting those kids, though, I do want to take a look and see if it could help me.
-- John Truong
4 hours?!! I find that boggling. But then, I'm a math tutor so my skills are fairly sharp. If I had some idea of what you were doing during those four hours, I'd probably be a better tutor.
My deviantArt site
Like everyone else, I say practice makes perfect. I do a lot of UI layout at work, and to conform to interface guidelines, I do a lot of "that control's left plus that control's width plus 14". Little things like that can make all the difference in the world.
Now, so that I don't get modded as redundant ;) Try this:
Take 1000 and add 40 to it. Now add another 1000. Now add 30. Add another 1000. Now add 20. Now add another 1000.Now add 10. What is the total?
Did you get 5000? The correct answer is actually 4100.
I also find this site very fun: perplexus.info, with one of my favorite problems being the ant on the rubber band.
Land of serious mental mathematicians, not just Ramanujan the theoretician.
I remember reading, Guiness Book perhaps, of someone in India extracting high roots of many digit numbers.
Sometimes, even in Europe, mental mathematicians lead interesting and unpredictable lives.
"Provided by the management for your protection."
"Cube roots! He wants to do cube roots by arithmetic! It's hard to find a more difficult fundamental problem in arithmetic. It must have been his topnotch exercize in abacus-land.
"He writes a number on some paper--any old number--and I still remember it: 1729.03. He starts working on it, mumbling and grumbling: "Mmmmmmmmagmmmmbrrr"--he's working like a demon! He's poring away, doing this cube root.
Meanwhile I'm just sitting there.
One of the waiters says, "What are you doing?"
I point to my head. "Thinking!" I say. I write down 12 on the paper. After a little while I've got 12.002.
The man with the abacus wipes the sweat off his forehead: "Twelve!" he says.
"Oh, no!" I say. "More digits! More digits!" I know that in taking a cube root by arithmetic, each new digit is even more work than before. It's a hard job."
Feynman goes on to explain the approximate method he used to get the result, and then gives his analysis:
"I realized something: he doesn't know numbers. With the abacus, you don't have to memorize a lot of arithmetic combinations; all you have to do is learn how to push the little beads up and down. You don't have to memorize 9 + 7 = 16; you just know that when you add 9 you push a ten's bead up and pull a one's bead down. So we're slower at basic arithmetic, but we know numbers.
Furthermore, the whole idea of an approximate method was beyond him, even though a cube root often cannot be computed exactly by any method. So I never could teach him how I did cube roots or explain ho lucky I was that he happened to choose 1729.03."
The rest of that chapter (entitled "Lucky Numbers") talks about his experiences in trying to improve his mental math skills. Definitely worth a read.
Run a google-search on "trachtenberg math".
You're looking for sites like Trachtenberg Speed System or Trachtenberg Math (Multiplication).
Professor Jakow Trachtenberg was a brilliant mathematician. Imprisoned by the nazis during WWII, he kept his mind busy to survive by applying advanced mathematical techniques to numeric computation. Eventually developing a number of techniques that provide for rapid mental computation without massive rote memorization.
For example:
Total agreement. I'd been helping some kids with schoolwork and was amazed that they needed the calc for times tables. I was amazed, but noticed that my own skills were a bit rusty (too much excel and the HP12 was a crutch, so I started doing any and all four function stuff in my head prior to reaching for the calculator. I recalled enough tricks to be close in estimating higher level stuff to ensure that I punched it in correctly. In about six months I've brought my arithmetic back up to a refined level.
Degaussing scares the bad magnetism out of the monitor and fills it with good karma.
I use this method a bunch, in various forms...distributing wierd values over things.
What's 19*19?
Well, it's 20*20, minus 20 (20*19) minus 19 (19*19).
Which is (20-1)*(20-1), which is (20*20)-19-19+1, or (20*20)-20-19.
Where did I learn this? I'm not really sure, since it was never actually taught to me, but I think I might have picked it up from Schoolhouse Rock. Go figure.
This space for rent. Call 1-800-STEAK4U
When I was a kid I found this already old book called (?) "The Art of Ciphering". That's a guess since I haven't seen the book in probably 35 years. But I remember some of the techniques in it. I was a farm kid at the time so while doing field work I'd have long blocks of time (as much as 10-12 hours a day) without much to occupy my mind. So I filled the time doing math in my head. I got pretty good at multiplying 4-digit x 4-digit, 5-digit x 5-digit, etc. in my head. Also extracting square roots, doing Roman fractions, and other stuff.
As I did these arithmetic problems, I found that my mind developed a kind of blackboard. I could visualize the problem and effectively "write" the answer without worrying about keeping track of everything as separate digits.
My advice: Find a good algorithm, practice a lot (yep, hours and hours), draw a picture in your mind.
The bonus of doing this is that later when I started studying math, the visualization I'd developed helped lots in advanced courses. I could "see" solutions almost instantly that would take others awhile to derive and even then they wouldn't really understand the relationships which led to the solution.
I can't speak for the parent post, but I can tell you I'm really bad at math. And I had some TERRIBLE experiences with tutors in my day, so I had to struggle with a lot of loathing/hostility before writing this post. Under other circumstances, I would have just passed it by, but you must have caught me in a helpful mood.
If you're really good at math, I'll guess that you're very bad at drawing/sketching. (That's always been my experience, but I'm sure there are plenty of exceptions to that rule, so forgive me for presuming.) If you're actually good at drawing things, then mentally substitute something else you're very bad at - playing a musical instrument, perhaps, or repairing a car.
If you really want to know what your students are going through, get a pencil and a stack of plain white paper, and sit down at your desk. Give yourself a two-hour time limit, and try to draw something. Don't use any reference material or models - math work is all theoretical, and drawing from a model is comparing apples to oranges. Draw something you're not particularly familiar with - if you own a dog, draw a cat, and vice versa.
It's hard, isn't it? You know what you're working towards, but you don't necessarily know how to get there. You may feel as if you're fumbling in the dark, knowing that millions of people could just sketch a dog in a few quick lines and be done with it. Your brain will probably feel as if it's being pushed in a direction it was never meant to go, and the venture may feel as if it's taking a thousand times more effort than it should.
Now picture a professional artist sitting across from you. You have hired the artist to help you get better at sketching dogs. What would you have the artist say to you? What sort of assistance would be helpful?
You will see, on the artist's face, an acknowledgement of how very hard you are trying, as well as how poorly you are doing. You can tell that the artist knows you're doing your best, but it's obvious to both of you that your sketched dog is TERRIBLE. This knowledge will not help your morale. You may lose patience with the venture and get snappish. The artist may lose patience with you, and snap back. Or maybe they shake their head sadly and say "You're not very good at this, are you?" in a tone of faux-sympathy. Perhaps they start barking out directions - "Make the paws rounder! ROUNDER!"
Um. Not that I'm bitter, you understand.
HTH.
For example, if you knew what you were looking for, such as calories or joules or centimetres, that's one part of it. If you know the formula relevant to the situation, that's another. Then you get to basic arithmetic skills- it doesn't do you any good to know the formula if you can't add or multiply the numbers.
My favourite way to tutor math- and how i learned it as an adult (i never took the SATs and was fortunate to have a tutor who could teach me high school math even though i'm 27) - is to use basic math issues that everyone sees, every day. Like the label on food. If this equals x% of your USRDA, how much is the USRDA? Putting the problems in everyday life situations may make you more comfortable with the math,a nd it will definitely leave you with an idea of the numbers involved.
'An idea of the numbers...' by which i mean a feel for the numbers, and what they stand for. A lot of people have trouble connecting the numbers to reality- and if you can understand in a concrete way the relationship between the distance around a pipe and the distance across it, the math may stick better for real world use later on.
The other trick? Estimate where you can, and use the information that's easily accessible to you..
For example: What's 5% of the time in a week?
well, you know that there's 24 hours per day. Add the big numbers first- 20 times seven, that's 140, right? plus four times seven- 28. Right off the bat, you're up to 168 hours in a week. Ten percent of a number is easy, ten percent of this number is 16.8. Half of that will give you the five percent that you're looking for- 8.4. You've just figured out that 8.4 hours is 5% of a week. Convert that .4 into minutes- forty percent of an hour is a little less than half. (sixty minutes, times ten percent, is six minutes. That's ten percent. Four times six is twenty four minutes. That's forty percent.) The answer? Eight hours, 24 minutes.
I use this with others because it teaches people how to think about numbers, that they are reachable things, not just the provenance of mathemagicians. The biggest barrier to doing math is the belief that math is too difficult. (i also play for people Tom Lehrer's wonderful song, New Math, and assure them that we're going to ignore base 8.)
Good luck with it, and try to use it in the real world where you can get a feel for what the numbers attach to. Figure out what you know and what you need to know, and just practice. There will always be more math to attempt; there will always be stuff that's intimidating. The only way to learn it is to do it, a piece at a time from the information that you can grasp easiest.
Oh, and in high school, in that science class? i got a C. Worked hard for it, i've never been prouder of a grade then or since. And i've never forgotten the real stuff i learned there- that being able to describe what you're reaching for is as important as the math skills to get you that answer.
"I'd say 'Have a good time,' but arson is still illegal.
Isaac Asimov wrote a story called 'A Feeling of Power' (also reprinted as 'A Long Forgotten Technique') that takes place in an advanced society in which all calculations are performed by machines. One day, a bored technician figures out how to add without a calculator. He theorizes that long ago, man must have had to perform calculations without machines, so he goes about trying to re-invent other machine-free calculating techniques. The ability to compute without relying on machines gives him a great feeling of power.
20*20 - 20 = 20*19
20*19 - 19 = 19*19
I am a programmer and my metal arithmatic skills have lowered to the level that I will not sum more than two simple two digit numbers or multiply more than two simple single digit numbers in my head. Why? The reason is I don't need to. My ability to do these things has reduced over time because I do not need to use these skills. I know people emphasise mathematics to be a good programmer, but I just don't need mental arithmatic for it. Sometimes I need algebra, but this rarely involves large numbers and is more about manipulate.
My point is, if your maths skills have reduced over time, it's because you don't exercise them which means you haven't needed them for your daily use. So, what's the point of practising something you don't need?
The guy who wrote it, Tirthaji, was a fraud. Every word and every claim in the book reg. the history is fabrication. The math is also pure junk and utterly useless.
Seriously. I did a term paper on it last year. You don't have to take my word, of course: read this article by Prof. S. G. Dani, School of Mathematics, Tata Inst. of Fundamental research (the premier research inst. in India.) There's also a much more detailed version.
Unfortuntely, the book fits the political ideology of the current Hindu-fascist government in power in India, and so they've been promoting it big time.
Sorry, your example makes an extra step:
95 * 23 = (100 * 23) - (5 * 23) = 2300 - 115 = 2185
I recognize that this is mildy picky, but the point is to show people how easy it is. What you're really trying to do here is to use numbers that you can do easy math on (5, 10, 50, 100, etc) and then account for the differences. This example works because 95 * 23 is the same as (100 - 5) * 23 is the same as (100 * 23) - (5 * 23), which is an easy mental calculation.
You don't have to think about this algebraically. I usually approximate and refine. When I look at 95 * 23, I immediately figure it's about 2300 because I know 100 * 23 without thinking. I think figure well, it's less than that because 95 is less than 100. It's less than 2300 by 5 23's (100 - 95). 5 23's is 115, so it's 1185.
You can take this one step further (which is where the original post went) and apply it to all the numbers. You can start with 100 * 25 = 2500. This is useful in the event that neither of your numbers are close to something simple like 100. You then have to account for your increase or decrease of each number. 2500 is too high by 2 100's (25 - 23). Now you're at 2300, and can pick up from above. Remember to reduce one number at a time - you can't subtract out 2 * 100 and 5 * 25, that won't be correct. You can either do 2 * 100 and 5 * 23 or 2 * 95 and 5 * 25, which ever is simpler in your head.
No, seriously ... but it is ancient Indian (think outsourcing and not White Bear ;)) knowledge, a lot of which is popular as Vedas.
The article seems to be picking issues with the author of the book, rather than Vedic Maths itself. OK, Vedic Maths is a misnomer (not much math is from the Vedas)
Also, the sutras are useful to a large extent. (Though most of the sutras have exceptions, and blah blah)
To end, yeah, Vedic Mathematics (sic) is a very useful tool in mental maths.