The Magic Cafe Forum Index » » Magical equations » » Short-cut mathematics (0 Likes) Top Hat Inner circle We peed on you! 1077 Posts  Posted: Feb 27, 2008 02:37 am 0 I have a short-cut for mentally calculating the square of any two-digit number. For numbers where the second digit is 5 or less, proceed as in this example: 73 x 73 Think: (70 + 3) x (70 + 3) Square the larger number (70 x 70): 4900 Multiply the two numbers in a bracket (70 x 3 = 210) and double it: 420 Add this on to get: 5320 Square the smaller number (3 x 3): 9 Add this on to get final result: 5329 For numbers where the second digit is 6 or more, proceed as in this example: 46 x 46 Think: (50 - 4) x (50 - 4) Square the larger number (50 x 50): 2500 Multiply the two numbers in a bracket (50 x 4 = 200) and double it: 400 Take this off to get: 2100 Square the smaller number (4 x 4): 16 Add this on to get final result: 2116 There's nothing particularly special about the above, except that it is a way of looking at the task that makes it easier to do in your head. Do you have any other short-cuts for other mental math tasks? TH     Scott Cram Inner circle 2677 Posts  Posted: Feb 27, 2008 10:53 am 0 I do two-digit squares a bit differently, so that there's only two numbers to add, instead of three. First, I learn the squares of the numbers 1-25 so that I can just look at them and see their square. Now, for numbers from 26-50, I'm going to use their distance from 50 as a reference: 37 * 37? 37 is halfway between 50 and . . . 24 (50 is 37 + 13, so we do 37-13 to get 24). Take 24 * 50, which is 1,200 (This is why I use 50, it's so easy to multiply by! Just take half the other number and add 2 zeroes!) We've moved 13 each way, so what is 13 squared? 169 (Remember, you need to know the squares from 1-25 by heart). Take the 1,200, add 169, and you've got 1,369! How about 42 squared? That's 34 * 50 (8 each way), which is 1,700. 8 squared is 64. 1,700 + 64 = 1,764! For numbers from 50 to 75, you use the same process. 56 squared? 56 is halfway between 62 and 50, so 62 * 50=3,100 We went 6 each way, and 6 squared is 36. 3,100 + 36 = 3,1,36! From 75-100, We use the same process again, but with 100 as the reference number (100 is veeeeery easy to multiply by!) 84 squared? 84 is halfway between 68 and 100. 68 * 100 = 6,800! 16 each way, means 16 squared, which is 256, 6,800 + 256 = 7,056! Finally, you can use this method with numbers from 100-125, as well! How about 119 squared? 119 is halfway between 100 and 138, and 138 * 100 = 13,800. 19 each way means 19 squared, which is 361. 13,800 + 361 = 14,161! Do I have any other mental math shortcuts? Mental math is a favorite hobby of mine, so here are a few goodies for you: Figuring Cube, Fifth and Square Roots Doug Canning's "Mental Shopper" Multiplying two 3-digit numbers together Arthur Benjamin's "Mathemagics" Course BEATCalc (Over 600 mathematical shortcuts!) If you explore my site, linked in the signature below, you'll find plenty of mathematical wonders! Grey Matters:Blog|Videos|Mental Gym|Presentation|Store Bill Hallahan Inner circle New Hampshire 3223 Posts  Posted: Apr 19, 2008 05:45 pm 0 (a + b)(a + b) = a*a + 2*a*b + b*b In the example in the first post, a = 70, and b = 3. Humans make life so interesting. Do you know that in a universe so full of wonders, they have managed to create boredom. Quite astonishing. - The character of ‘Death’ in the movie "Hogswatch" TomasB Inner circle Sweden 1144 Posts  Posted: Apr 20, 2008 11:27 am 0 I think it's fun to try to reshape that expression to get further shortcuts when the number to be squared fullfills certain criterias. For example it's good to write the number A to be squared as 10a+b so that a and b are the digits of the number A. A^2 = (10a + b)^2 = 10a(A + b) + b^2 which gives us a remarcably easy expression to do in the head when a is small, like 1 or 2. Let's assume that a=1 and that expression simply means: Add the first digit, b, to the whole expression, then write the square of b to the right of it. Just remember that if b^2 is larger than 9 you have to carry the left digit to the number you already wrote. 12^2 means that you write 12 + 2 first then write 2^2 to the right of it. 144 in other words. 17^2 means that you write 17+7=24 first, then write 49 to the right of it. The 4 in 49 however needs to be carried to the first number giving us 289. This is also a good one for when a=2 but I would brobably not use it for a=3 even though you could. For a=2 the expression says: Add the last digit to the whole number and _double_ it. Write the square of the last digit to the right of it (and remember to carry if it goes above 9). So 26^2 means that we add 6 to 26 to get 32 which we double to get 64. 36 (square of 6) is written to the right of it but the 3 is carried to get 676. The original expression can be written differently if we only go for number ending in 5. A^2 = (10a + 5)^2 = 100a(a + 1) + 25 which simply means: Take the left digit times a number one higher than itself and write this down. Write 25 to the right of it. Never any special cases since 25 takes up the two rightmost digits and the a(a+1) is multipled by 100 to never interfere. Example: 35^2 means that we take 3*4 to get 12. Write 25 to the right of it for the result 1225. 65^2 means 6*7 = 42 and you write 25 to the right of it to get 4225. Yet another way to reshape the expression is to work around the number 50: A^2 = (50 + d)^2 = 100(25 + d) + d^2 where d doesn't have to mean a digit, but is the distance from 50, which may also be negative. The expression means: see what distance from 50 the number is (might be negative) and add it to 25. Write the square of the distance to the right of it to fill the two rightmost positions. You see that there are no special cases as long as d^2 < 100 so this strategy is very easy for numbers from 41 to 59. Example: 56^2 means that we are 6 from 50. Add 6 to 25 to get 31 and write 6^2=36 to the right of it and the result is 3136. For 48^2 we have the distance -2 from 50 which we add to 25 to get 23. Put 2^2 to the right of it (need to fill two positions) to get 2304. For numbers around 100 you can look at the first expression again, which turns into A^2 = (100 + d)^2 = 100(A + d) + d^2 which means that you add the distance (might be negative) from 100 to the original number then write the square of the distance to the right of it to fill two digits. Example: 108^2 means that we are 8 from 100 so we add 8 to the original to get 116. Write 8^2=64 to the right of it to get 11664. 91^2 has distance -9 from 100. So we subtract 9 from 91 to get 82 and write 9^2=81 to the right of it to get 8281. Other ideas for squaring numbers that fullfill certain criterias? I know this is going the opposite direction from the nice and general approach Scott uses. /Tomas Harry Lorayne V.I.P. New York City 8511 Posts  Posted: May 10, 2008 04:42 pm 0 You might want to check my short-cut methods for squaring two-digit numbers, particularly those ending with a 5, in MATHEMATICAL WIZARDRY - or not. Best - HARRY LORAYNE. [email]harrylorayne@earthlink.net[/email] http://www.harrylorayne.com http://www.harryloraynemagic.com TomasB Inner circle Sweden 1144 Posts  Posted: May 10, 2008 06:48 pm 0 Does your method for squaring numbers ending in 5 differ from what I wrote above? I'm very interested in reading any more shortcuts if you have any for those and other numbers. /Tomas stanalger Special user St. Louis, MO 996 Posts  Posted: May 10, 2008 08:16 pm 0 Tomas, I've read MATHEMATICAL WIZARDRY and can answer the question you asked Harry Lorayne. The answer is "NO." I don't think Mr. Lorayne read your post. If he did, his follow-up post doesn't make any sense. Nir Dahan Inner circle Munich, Germany 1390 Posts  Posted: May 11, 2008 04:11 am 0 There is not a single post where he doesn't plug his own books/videos... Adventures in ASIC digital design Harry Lorayne V.I.P. New York City 8511 Posts  Posted: May 11, 2008 09:31 am 0 Hi: Yes, the method for the squaring of any 2-digit number that ends in 5 is the same. Obviously, I didn't realize it because I scanned it. I scanned before I got to that part because I DON'T KNOW ALGEBRA (I have only one year of high school), and when I saw all those =s and +s and >s and s and
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