5 ms·

If you like this kind of stuff, there are a couple of fun mental math books I can recommend: http://www.amazon.com/Speed-Mathematics-Secret-Skills-Calculation/

• colanderman 15y ago
  Along these lines, here's how to do square roots in your head. Let s be the number whose square root you wish to find, and p be the nearest perfect square. You can then approximate √s ≈ √p+(s-p)/(2√p). Example: √33 ≈ √36+(33-36)/(2√36) = 6 - 3/12 = 6-¼ = 5¾ = 5.75. Actual √33 = 5.74456… I am a math tutor and taught this to one student who didn't have a square root button on their calculator (albeit I taught it as a calculator trick rather than mental math). He picked up on it pretty quickly.
  • kragen 15y ago
    You can look at this as linear extrapolation using d[x²]/dx = 2x, which is Newton's Method. Taking √p as your initial guess, your new guess is (√p + s/√p)/2 = √p + (s/√p - √p)/2 = √p + (s-p)/(2√p). You can go for another iteration if you want a really accurate square root, which roughly doubles your number of accurate digits each time: 5.75 + (33 - 5.75²)/(2·5.75) ≈ 5.744565, while the correct answer is closer to 5.74456264653802865980. At that point, though, I think it's probably easier to calculate it as (5.75 + 33/5.75)/2. You can even use this approach for taking square roots of fairly large numbers, just using the squares of single digits — factor the initial number into a power of 100 and a number between 1 and 100. For example, 80802363 is 80.802363 × 100³, so its square root is close to 9 × 10³. Edit: fixed stupid braino in first sentence.
  • happy4crazy 15y ago
    I also really like the simple-but-awesome tricks for multiplications: 1) (a + b)(a + c) = a(a + b + c) + bc. Example: 13 14 = 10 * (13 + 4) + 34. You learn to "see" 13 14 --> 170 + 12 = 182. 2) (a + b)(xa + c) = a(xa + c + xb) + bc Example: 22 * 62 = 20(62 + 32) + 22 = 1364. Here you would see that 60 = 3 20, so you add 3 * 2 to 62 to get 68, multiply by 20 to get 1360, and then add 2*2. Edit: Er, my stars are turning into italics. You get the idea.
    • jerf 15y ago
      Try copying and pasting × or ⋅.
      • kragen 15y ago
        Or use a compose key: http://canonical.org/~kragen/setting-up-keyboard.html http://canonical.org/~kragen/setting-up-keyboard.html.
• tokenadult 15y ago
  It's really baffling that people don't teach this stuff in school. It's not taught at school because schoolteachers are already struggling with teaching much simpler mathematics. http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf http://www.nctq.org/resources/math/ http://www.nctq.org/resources/math/ http://www.ams.org/notices/200502/fea-kenschaft.pdf http://www.ams.org/notices/200502/fea-kenschaft.pdf http://www.aft.org/pdfs/americaneducator/fall2009/wu.pdf http://www.aft.org/pdfs/americaneducator/fall2009/wu.pdf I do teach casting out nines as part of the curriculum I have developed for the supplementary mathematics courses I teach as my main current occupation. I have found some very old books in which casting out nines was a routine subject (a check on pencil-and-paper arithmetic back when almost no one had access to an adding machine) as well as theoretical treatments of number theory that explain why casting out nines (or casting out elevens) works--most of the time--to catch computation errors. Many of these techniques, and other techniques mentioned in the interesting replies in this thread, are standard parts of the mathematics curriculum in some other countries to this day. Even today when everyone in developed countries have access to inexpensive hand calculators, it's still considered good mathematics pedagogy to expose learners to topics of this kind in Singapore, in Taiwan, in the better schools in China, and in much of eastern Europe.
  • happy4crazy 15y ago
    Thank you for posting those. I'll take a more careful look tomorrow morning, but my quick perusal of the Kenschaft article has been sobering :(
• deleted 15y ago
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