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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?"
All the tricks are fine, but there is no way around it, you have to practice and keep your skills up. Start adding things up when shopping, calculate tips and sales taxes, etc. When ever you rach for the calculator, see if you can't do it in your head first, at least for a quick estimate.
The best way is to simply limit your calculator usage. I like to show off with the folks I tutor by doing their calculations in my head before they can type them into calculators. A strong basis in algebra can help you beak apart calculations into managable chunks, the trick is remembering how to put those chuncks back together. For instance (contrived example so not great but...): 95*23=100*25-100*2-5*23=2500-200-115=2185
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)
You could also try a google search I found some interesting websites
http://www.vedicmaths.comu /~rgupta/vedic.html
http://www1.ics.uci.ed
http://vedmaths.tripod.com
Hope this helps.
Slashdot: Tabloid for the nerds. Stuff that doesn't matter.
Awhile ago I realized that since highschool my own math skills had deteriorated beyond belief. The breaking point was when I was going to buy a 21" monitor and I wanted to figure out what the height and width of the screen would be so I could actually get a feel for what it was I was paying $400 for. It took me about 4 hours of racking my brain trying to remember old algerbra rules to transform the pythagorean theorem to use the diagonal (20" viewable) and a generic aspect ratio 1.333 to derive the height / width.
My point is that if you want to get quicker with your mental math skills or keep your current pace, you have to keep using it or else it will atrophy like everything else. Translation: college math courses or at home math excercises, but either way don't expect to be able to ever be "done" with it.
Good luck with that by the way, you're a better man than I.
Research proves there is no trick or secret. People who rely on calculators are poor at mental math because of lack of practice. While some people do have innate skills in maths, everyone has the ability to train the brain to to basic math. Take a look at this study
Memory, mental arithmetic and mathematics
1000s Warcraft Gold while you sleep
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
Some links (click the 1's). Some are for dylexics but still relevent for all since pretty much all of us are capable of visual thought...:
1 1 1 1 1 1 1 1 & similar 1 1 1A blog I run for the wealth
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.
I like estimating tricks.
:) Anyone else have logarithm tricks to share?
The rule of 72 helps to figure out how long it takes for something to double or halve. Divide 72 by the percentage rate of growth or decrease and you'll get the number of time periods in which something will double or halve. For example, let's assume Moore's law says double CPU speeds every 18 months. 72/18=4. So CPU speeds increase by 4% every month. Or another example: your phat mutual fund gets 12% per year, so 72/12=6. So your money will double in 6 years.
This trick is so simple that even the finance guys always know it.
http://tinyurl.com/4ny52
"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:
An excellent way to do truly astounding mathematics is to train your subconscious to work for you. Your subconscious records lots of things and basically remembers them forever. Your conscious mind often has trouble recalling certain memories or details though, but that doesn't mean it's not still there.
The trick then is to let your subconscious do the math for you, and then find a way to "pull out" the answer (like recalling a distant memory, almost). You can train your subconscious to do math a variety of ways, but one of the most effective is to electrically stimulate nerves (in your hand or arm or thigh, whatever) to count out numbers. So for instance, if you wanted to do 22+34, you'd count out 56 quick electic pulses. Practicing this for a few months, your subconscious will eventually get the idea that when you hear numbers, you want them added. The electric shocks will no longer be necessary, but your subconscious will still internall 'tick' out the answer. It works for multiplication, too, and through various mathematical tricks, you can use it to subtract and divide.
The only remaining difficulty is training your conscious mind to retrieve the result. This is accomplished via a hypnosis-like state. You can get good at it so that it only takes you a half-second to pull out the resulting number. No eyes rolling back or chanting or anything like that.
Heh, ok, not really.
First, there is no substitute for exposure to a great math teacher. I had the fortune to have had a couple great math teachers through elementary and high school that led me to major in math in college.
Second, knowing a few tricks isn't enough. Understanding the tricks and why they work is the key to improving your math skills. Beyond access to a teacher to help you with this, you may want to try some resources available on the web like MIT's OpenCourseWare. They have a lot of information available on their courses, including lecture notes and text books. However, quite a few of their courses online deal with mathematical theory and may not fit with what you are looking for, try some of their "applied" courses.
Third, as one previous poster mentioned, understanding algebra will help with breaking larger calculations into smaller, and easier, parts to calculate quickly in your head. A good source for learning materials would be a local college book store. Focus on algebra textbooks that cover the basics and how to teach them (If a local college offers Education majors, they should have at least one course that will fit your needs, find out which course and the accompanying books they recommend).
Finally, go to your local high school and find out what text they use in their first year algebra classes. If you mainly what to be able to calculate angles or lengths of object sides quickly, texts for high school geometry and trigonometry classes will offer more information. Understanding these texts will help you to improve.
I hope this helps.
"Don't worry about people stealing an idea. If it's original, you will have to ram it down their throats." --Howard Aike
Long ago in high school, I competed in what was then called "Number Sense" - doing math problems mentally, no aid of scratch paper. (Calculators were an expensive novelty - 4 functions, Nixie tube displays, plugged into the wall, had 4 functions.) The system we all worked from is now called the "Trachtenberg Speed System of Basic Mathematics", and it had lots of tricks for converting decimals to fractions and vice versa, multiplication of pairs of 4 digit numbers, etc. There are a lot of drills on visualization that helps in holding intermdiate results in the head. See http://www.speed-math.com or find the book on Amazon.
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.
It works better on craps, as the odds are tighter, in roulette, the green squares are both losers giving you about about a 47% chance of winning. The pass (or nopass, but you get dirty looks from the shooter) line is north of 49% (a bit better if you take odds). The issue is that in strings of random numbers long sequences of the same result are more common than conventional wisdom would have us believe and you only get about 9 losses before most table limits kick in (do the math on the doublings between minimums and maximums) You could increase your initial bet (which is what you win back each sequence), if you had a big enough starting stake.
Degaussing scares the bad magnetism out of the monitor and fills it with good karma.
http://www-gap.dcs.st-and.ac.uk/~history/HistTopic s/Mental_arithmetic.html
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 have been a math tutor for 3 years. I also have a BS in Math (for whatever that is worth).
But there is one thing that I *always* tell my students. That is this: There are many, many, MANY ways of going about doing a math problem. Sometimes the way the book describes it, or the way the prof tells you to do it doesn't make as much sense to you. For instance, some people understand fractions better than decimals, or vice versa. As a statistician (or future statistician at the time) I would always convert fractions to decimal before I worked with them because it made more sense to me. (I just had to remember to convert them back when i was done)
Point being...there are many correct ways to come to a correct answer. When we learned to multiply and do long division in elementry school we were taught an algorithm for doing so. However, as some people have already posted their 'tricks', there are other algorithms out there. You just have to make sure it actually yields a correct answer before you utilize it. (If you don't want to formally prove it, like me, then you can try it on at least 3 different sets of varied number sets. Don't pick simple numbers, they can often lead you to a wrong conclusion)
Find what works best for you. (as long as its correct!) I'm a big fan of rounding numbers, calculating them and then adjusting them from there. e.g. 17 x 4 is almost 20 x 4 = 80, but we left out 3 of the 4's so the answer is 80-12 = 68. (IMHO the algorithm we learned in elementary school for multiplying is the worst way of trying to calculate something in one's head!!!)
A good trick I use when calculating discounts in stores (i.e. 70% off, 25% off etc.) is to figure out how much 10% of the price is. This is easy, just shift the decimal point. Then if its 70% off, I'll take the 10% off price and multiply by 3. Unless it is easier to calculate it the other way around. If it is 25% off, I'll divide the price by 4 and then subtract that.
Anyhow, I haven't really given any specifics or good examples, but explore thinking about the problems in slightly different manners and then making small adjustments to the final answer. Do what makes sense to you.
DATA comments; PROC SORT DATA = comments BY score; PROC DELETE comments >> 1; RUN; DATA entertainment SET commen
One activity you can indulge in can simultanesouly improve your memory, make you feel good and allow you to show off in front of your friends so they will think that you are a really intelligent person (which I am not saying you aren't, but people who aren't really into this kind of brainy and "geeky" activity will surely be very impressed) is to memorize 1000 digits of pi. It's funner than you may think, as it's a real challenge and over time will increase your capacity to use the full potential of your memory properly.
"Really, I'm not out to destroy Microsoft. That will just be a completely unintentional side effect" -- Linus Torval
"Consider a Spherical Cow" and there's a 2nd book "Consider a Cylindrical Cow" :-) - which is about how to do "back-of-the-envelope" estimates. How many pairs of shoes can be made from a single cow? Consider a spherical cow. :-)
And a Dover reprint: "How to Calculate Quickly" which has many of the tricks and rules of thumb people used to all know before calculators.
From my "antiquarian" collection I have a number of "arithmetic" textbooks (all pre-1930) that have lots of little rules of thumb for checking sums and products - many are familiar to accountants. Also great chapters like "Arithmetic of Thrift", "Arithmetic of Agriculture", etc. with problems like "...girls in a class in millinery need 20 yd. of ribbon..."
If I attempt to perform a calculation in my head, I can often see the tricks to make it doable, but can't hold on to more than a couple of intermediate values. Particularly if I'm trying to keep track of mantissas and exponents at the same time. I usually need some random access storage (pen and paper) to hold the temporary variables.