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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.
Actually, devices exist that work completely transparently, unlike the old fashioned calculator. Even tux uses one! Here's a pic of him modeling one of the later models.
He who laughs last is stuck in a time dilation bubble.
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
Practice,
Practice,
rinse,
repeat.
You want to improve your mental arithmetic skills not your mental mathematics skills. The distinction is that arithmetic involves applying simple algorithms, memorization, and other techniques to carry out computations. Mathematics involves dealing with purely abstract concepts, moving between different levels of abstraction, working with formalism, and related concepts. At any rate practice helps with both.
Stay in school.
No Comment.
Am I the only one who thought
:P
This is slashdot, we don't think. We slashdot.
But seriously, I find that question quite exciting
That said, "Rapid Math Tricks and Tips: Thirty Days to Number Power" by Edward H. Julius is great. It is a little cheesy, but very practical. It allows you to do much of the same calculations that a 'child prodgy' can even if you're old.
It does not help with number theory, though it can help give you a much better feel for numbers.
A firewall can not protect you from yourself. Turn off what you do not need. Do not use the firewall to do your work.
"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
Yes there was a book called Fingermath back in the 70's. Of course there's the social stigma of being seen counting on one's fingers.
It comes down to practice, and the only way you will practice is when you have to do it. So get a job where you have to do this.
When I worked carpendry I got really got a multipling by 1.42 (guess why[1]) because that is something we had to do often, and calculators didn't last more than a week on the job so we rarely had one. (The foreman would buy one if he knew a lot of calculations were coming up, but he often had to do math by hand) In that job there there is plany of surface to work with so we wrote everything out. Normally on the stud right next to us with a carpenders pencil, not optimal but it worked.
When I worked at McDonalds I soon learned that an extra value meal was $3.18 with tax in my state. At the time I knew all the common combonations, but I never added them up, I just memorized if off the register. On the slow days I would take drive through orders, and the customer would see me running to the door while giving his total and "please pull ahead". Different job, different set of skills that I got good at.
[1]I don't remember my trig terms, but if you combine thoughs of trig with 45 degree angle you should be able to figgure out what 1.42 is.
Anyway, what he would do is get various slides with various simple multiplication problems of the form xx * x, and show them for all of 1.5 seconds each before skipping the next one. (The x * x form was something learned early on.) The object was to be able to know the answer to said problem immediately on sight. EG, you see 12 * 5, and 60 theoretically registers immediately.
Expand from there. Who knows, it might work.
This sig no verb.
Similar story - once, after a trip to a casino, I got it into my head that red/black roulette betting could be won all the time using the simple strategy of "Always bet the same color, and when you lose, double your bet." Mathematically it works. I even heard Artie Lang (sp?) quote it on the Howard Stern show talking about the pass line in craps.
The problem is that this is a known, flawed, strategy. First, it implies that you might need a huge stake of cash up front if you hit a losing streak (kinda like the old "doubling a grain of rice on the chessboard" story). Second, you only ever win 1. Think about it -- you bet 1, you lose, you bet 2, you lose, you bet 4, you win 4 - but you've already lost 3, so you're up 1. So technically it does work, but it is not realistic to build up any real winnings.
www.HearMySoulSpeak.com
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.
was Calculator's Cunning by Karl Menninger.
I believe it's out of print now, but was an excellent text, covering all of the tricks.
If you search bestwebbuys you can see that it is for sale used.
A simple "trick" that might help regular folks.
When adding two numbers, say 27 + 36
Round both numbers to next whole multiple of 10
so you get 30 + 40 which is obviously 70
then add the two differences 3 + 4 which is 7
subtract this from the total 70 - 7
which is obviously 63 and there you have the answer in a so much easier fashion than adding those two ugly numbers.
And yes, this is how my mind works.
I only look human.
My mother is a halfling and my dad is an ogre, so that makes me an Ogreling
no clue about author or publisher, but there's a little black book with colorful numbers (at least the edition I had) on it called "the art of calculating quickly".
It cost me less than $5 back in '98, and gave me the superpower of being the idiot who knows the answer to arithmetic problems.
If that's what you seek, that's the book.
http://www-gap.dcs.st-and.ac.uk/~history/HistTopic s/Mental_arithmetic.html
in a school far far away I used to mentally multiply two 5-digit numbers during certain boring lessons. (Religion, anyone ?)
No tricks involved. I did it just like you would do it on paper.
A benefit seems to be that I'm able to remember phone/pin/account numbers and random passwords easily. And I avoided being brainwashed...
Flourescent (adj): smelling like ground wheat.
Right now, 4 + 3 = 7,
(4-1)+(3-1)=3+2=5
(5-1) = 4
(All units karma)
You cunning karma whore...you got a full point more for your 'negligence'!
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.
EXERCISE!! An activity like jogging promotes the release of neurotrophins and dendritic branching. Healthy neurons equal healthy mental operation.
never underestimate the power of boredom and innocense in sweet harmony
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.
A pleasant exercise is the Doomsday algorithm (invented I think by John Conway) and described on rudy.ca/doomsday.html whereby you can calculate in your head the day of the week of any given calendar date in the last century. (It takes a minute or so, faster if you have been practicing).
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?
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
Listen to this CD every day and your skills will VASTLY improve:
Buy the Numbers CD
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
Just go to any bookstore that has a math section. Every single one I've been to always has a book or two about doing mental math that is very comprehensive. And, these books are almost always really cheap, because no one else wants to buy them. :)
Here's 2 tricks when you need to know if a divisor evenly divides into the numerator. (e.g. is N mod D ?= 0)
/, sqrts, etc, see the Human Calculator, by Scott. Flansberg
To tell if a number is divisible by 3:
- sum up the digits
- if the sum divides by 3 with no remainder, the orginal number is divisible by 3 with no remainder.
The proof is pretty trivial to work out. It only takes a few lines to prove it.
Another trick, that isn't well know, and that I can't take credit for is
A number is divisible by 7 if
- take the last digit off a number and double it
- subtract the doubled number from the remaining digits.
If the remainder (which may be negative or zero!) evenly divides by 7, the orginal number is divisible by 7.
i.e.
4158
415 - (16) = 399
Repeat the process, since adding a multiple of the denominator does't change the mod result.
39 - 18 = 21
Therefore 4158 is divisble by 7.
I know a mathematician by proxy of a good friend of mine, who noticed the 7 trick. He also suggests that there is a rule for any prime number, but I haven't seen any proofs.
For *learning* some really cool tricks for +, -, *,
Peace
"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..."
Here's 2 tricks that I accidently discovered, being forced to constantly convert between the two.
... real answer is 55.92
To convert a number, K from kmph to M mph
M = (K / 2) + (K / 10)
i.e.
90 kmp = 45 + 9 = 54
The relative error is 3.4%, which isn't too bad.
I usually drop the fractions, so the formula becomes
M = int(K / 2) + int(K / 10)
Even though the relative error will be a tad higher at low speeds, and oscillate around 3 to 6% for the most part, the absolute error is at most off by 3 mph for speeds less then 100 kmph. Not too bad at all.
Conversly, to convert anumber, M from mph to K kmph
K = (M * 16 ) / 10
And since 16 = 2^4,
K = ((((M * 2) * 2) * 2) *2) * 10
Cheers
or... you could do what i do! augment your brain with new hardware, rather than learning a few new tricks. i use a pda, an extension of my brain that is almost always with me in my pocket. i have many good software packages for doing maths. a lot better than carrying around a ti-xx with you all the time for various reasons.
god bless the public school calculator generation!
Working toward a usable PDA environment in the spirit of Newton OS: Dynapad
Math Magic by Scott Flansburg. I've seen the guy demo his skills on TV and he's amazing.
and don't forget the Doomsday Algorithm which is actually useful on an almost daily basis.
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.
Try becoming a game programmer. You'll have math out the wazoo.
Part of what I had to learn in primary was my times tables; we'd have to memorize everything from 1*1 up to 12*12 (and all the numbers in between). It was very boring and I hated it at the time, but I'm glad of it now, as I can multiply pretty well in my head.
Mental math depends heavily on the availability and reliability of short-term memory. You need to be able to hold two or three numbers in memory while you manipulate two or three more. The most reliable way to acquire this mental RAM is to practice, so here is the mental equivalent of push-ups:
... they had sticks and rocks, and they were happy to have 'em!
To square any two-digit number X, decide what number N it will take to raise or lower X to P, the nearest multiple of 10. Add the opposite of N to X to get Q, multiply P times Q, and add N^2 to the result. For instance:
67^2 =
(67 + 3) * (67 - 3) + 3^2 =
70 * 64 + 9 =
4489
The hard part is 70 * 64, but if you teach yourself to ignore the zero at the end of the 70 and multiply from left to right, it sounds like this:
"Seven sixes make forty-two, times ten makes four hundred and twenty. Seven fours make twenty-eight, plus four hundred and twenty makes four hundred and forty-eight, times ten makes forty-four hundred and eighty, plus three squared--that's nine--makes forty-four hundred and eighty-nine."
Presto, you've figured out the answer in less time than it takes to say it. Note: don't be discouraged if you forget what number you needed to add at the end, or what number you were originally squaring; they're going to drop out of your short-term memory storage until you practice enough.
If you get this down you will win bar bets, impress your co-workers, and shut your hardass father's mouth next time he starts droning on about how they didn't have calculators when he was in school
I can do md5sum by head. Now, beat me!
There you are, staring at me again.
During highschool, my junior and senior math teachers made me tutor (I was horrible about doing day to day work, but killed the curve on tests. So it was either tutor or flunk). I found that the best way to teach these folks math was a combination of doing and showing. I always started the sessions with a two steps back, one step forward progression. Basically you build their confidence by going through easy exercises (simple X+Y=Z problems) then you add to the complexity ({XY}+Z=A), etc. I raised all my students average by two points (mostly D grade students started getting B's) It's not that they were bad at math, just that many teachers, and other tutors didn't Keep It Simple. If you can get the concept for theory across using simple numbers (1,2,3,5,9,10) then why confuse them with larger numbers?
Many of the other posters have it right you have to use it or you lose it. I find I have to apply the same formula I used as a tutor to myself. When I want to calculate something in my head, sometimes I have to back up to the previous basics that built towards the theorems I used in other mathematics. In your exercise, if the art tutor starts with paint by numbers, or shadowing some ink drawings, then has you sketch and watches where your eye goes when drawing in relation to what's being drawn, then they can help teach you to draw better. But if all they are doing is critiquing what's being done, then all they are doing is criticizing. And let's face it, most critics only help themselves feel better about what they do. Two steps back and then one forward.
TANSTAAFL
I'm quite good at math and bad at sketching. I like your example.
Let me say this about being good at math. Not surprisingly, it's (almost) all about practice. I'm not denying the possibility of natural talent playing a role, but I think probably anybody could reach the calculus level without much need for natural talent, as long as there's no time pressure and you do a lot of work.
You have to do lots and lots of example problems. As you do, you start to develop a "feel" for them. If you do it enough to develop that feel before moving on to the next level, you'll be fine. If you're forced to go to the next level before you develop the feel, you won't "see" the new things properly, and you can't deal with what you can't "see".
You'll then be forced to figure out ways to get by that aren't real learning -- some combination of cheats and dodges and settling for less, etc. Then, without the proper foundation, if you're forced yet again to move on, things will go into a death spiral of lost confidence, bad grades, poor skills, tricks, playing the system, etc.
You need to go back as far as you need to go back, get a source of LOTS of sample problems, and do them again and again and again until you get the feel for it. Then move on in steps that are as slow and gradual as they need to be and keep doing LOTS of problems.
The natural talent issue may make you slower at this than some other people, but it probably can't stop you as long as you don't quit.
I solve math problems now for the most part by seeing through them, meaning that I recognize them and have a feel for them that comes from familiarity gained over years of practice. You may not have the interest to put that much time into it, which is fine, but then don't accuse yourself of being "bad at math" if you willingly decide not to take the time to become good at math. You probably could be, but you may be underestimating both yourself and the amount of work needed for anybody to become good at math.
Maybe a good tutor could help, maybe some good tutorial books could help, but NOTHING helps like practice, practice, practice.
"Those who have never entered upon scientific pursuits know not a tithe of the poetry by which they are surrounded."
Most people's primary learning style is either visual, auditory or kinsthetic. To figure out which one yours is, consider how you think of things and how you like to organize information. If you learn best from looking at charts and graphs, you're likely either visual or kinsthetic (a lot of kinsthetic learners pick up very quickly how to 'translate' information from other states, else they get labled as 'learning disabled'), if you learn best from reading words, you're either auditory or visual (or a very adaptive kinsthetic), if you learn best from lecture (without considering visuals from an overhead projector/blackboard/etc) you're almost certainly auditory. If you're one of those people who has a heck of a time learning something from reading or lecture, but picks it up quickly from doing it, you're very likely kinsthetic. There's a lot of little quizzes around to help you figure this out, I'd suggest taking a few (try googling on 'learning styles') and seeing which style keeps coming up. Most people can learn using all three styles, but the vast majority do prefer one style over the others.
;)
If you're auditory, sorry, I have no really good advice. Someone else probably will, though. I recommend googling for 'auditory learners' - and don't ignore the pages meant for those who are learning disabled or their parents, as they often have really good ideas whether or not you are learning disabled.
I use a technique that translates well for both visual and kinsthetic learners (I'm kinsthetic) -- in my head I use 'blocks' to do math, and then translate the answers from looking at how many blocks I have. If you're a visual learner you can practice this outside of your head using graph paper and a pen. If you're a kinsthetic learner, the graph paper will work, but legos or similar physical blocks will probably work better. For a very basic problem, say 52+34, one would 'block out' (either on the graph paper or with the blocks) 52 blocks and 34 blocks, and then mentally (or physically, if you're practicing outside your head) putting the blocks together, then seeing how many blocks there are. With practice this is much faster than a calculator or even doing the math the standard way in one's head. For more complex problems, using algebraic simplification, the various 'rules' for multiplication and division, etc can be combined with this method for very quick calculations, even of relatively complex (for humans. Let's not get into complex-for-computers issues) problems. I
It helps immensely if from the very beginning you group your blocks in groups of five or ten, and group those in groups of 50-100 (total blocks) and group those in groups of 500 or 1000 (again, total blocks) and so on to simplify the final count up. (five is a good number because most people can 'eyeball' five without having to explicitely count each piece -- if you can't actually eyeball five, you can almost certainly eyeball two and three, and know that that makes five. Then ten is just eyeballing two groups of five)
Incidently, I've gotten people who were terrified of math and believed that they couldn't do math to do calculations quickly and easily using this method. This probably has more to do with the lousy way math is taught in our schools than the inherent wonderfulness of my brain, though
(on a somewhat related note, fractions are best done using circle charts and putting them together in your head, the same way as the blocks. One circle becomes the same as one block. I imagine you could also use squares the same way, but it's easier to make 1/3rd a circle than 1/3rd a square, and you don't end up playing games trying to make your funny shapes fit together in the square -- a wedge is a wedge is a wedge in a circle)
The best time to develop your FPU is as an adolescent. All you can hope for as an adult, with your inferior mathetmatically, perma-wrinkled, old dog, old tricks brain is to memorize a lot of both formulas and shortcuts, and keep practicing them indefinitely unless you want them to deteriorate. That isn't "learning", btw. And don't expect to change the field, it's so diversely and extremely specialized now that the only ones in the future who will do that are prodigies. You will be a layman.
If you are an adolescent, read and absorb Organon, Principia Mathematica and Grundgesetze der Arithmetik and then wise up, stop teaching yourself and enroll in university classes after you ace the entrance exams.
**** *. *. *****, professor emeritus *.*.*.
What if you need to program the computer to square numbers that are larger than the word size can contain? (i.e. arbitrary precision arithmetic)
If you know the math tricks, you can do this easily. If you don't, you have to struggle to do it analytically, which is a pain.
Quick, how can you figure out the lowest set bit of a number?
X&-X