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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
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
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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.
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
synthesthesia for your case? not quite. it's strongly more complex than that. it involves total sensory abnormal function. seeing colors excites smell. sounds produce other sensations. normally, the brain can only interpret receptors input from a hair cell as hearing, or from the recepters in the retina as light (law of specific nerve energies). that is why rubbing your eyes produces spots of light, even though your are only applying pressure to them. people with synesthesia do not follow this law. i don't think performing math calculations by imagining them as colors (albeit very interesting) or thinking about letters has anything to do with what you think synesthia is.
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.
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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.
In actual fact, if the percentage rate of growth is i, and the number of periods p, then the relation (1 + i/100) ^ p = 2 must hold. Simple rearrangement gives p= ln(2)/ln(1+i/100). Approximating ln(2) by 0.72 (actual value closer to 0.69), and ln(1+i/100) by i/100 using a Taylor expansion truncated to first order, gives p = 0.72 / (i/100) = 72 / i.
This trick is so simple that even the finance guys always know it. :) Anyone else have logarithm tricks to share?
This trick is a simplification of such a simple thing that only finance guys could come up with it.
- a finance guy, formerly a mathematician