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| montecristo0612 Quote | Reply | | Shortcuts in Multiplying posted on: 8/26/2002 3:21:47 AM Anyone knows some shortcuts in multiplication?  Math rules!!! c  |
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tranquilium
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Re: Shortcuts in Multiplying
replied on: 8/26/2002 3:24:24 AM Hi montecristo0612, thanks for posting! I'm not sure what you mean though when you say shortcuts in multiplication. Could you please give me some more detail? Sorry for my ignorance 
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montecristo0612
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Re: Shortcuts in Multiplying
replied on: 8/26/2002 3:35:51 AM Example: Shortcut in multiplying numbers from 11-19. -or- How to square a number easily? Math rules!!!
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Soroban
Quote | Reply | This message was updated on 8/26/2002 11:32:47 PM by Soroban |
Re: Re: Shortcuts in Multiplying
replied on: 8/26/2002 6:26:59 PM Example: 75^2 Multiply the tens-digit (7) by the next integer (8): 56 Append a "25" on the end. Answer: 5625 Example: 125^2 Multiply the "tens-digit" (12) by the next integer (13): 156 Append "25". Answer: 15,625 Instead, say "Fifteen thousand six hundred twenty-five." It's far more impressive! Proof: Let N = 10T + 5 Then N^2 = (10T + 5)^2 = 100T^2 + 100T + 25 = 100T(T+1) + 25 We can now see that N^2 ends in 25 and T(T+1) appears in the hundreds place. |
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Soroban
Quote | Reply | This message was updated on 8/26/2002 11:29:15 PM by Soroban |
Re: Re: Shortcuts in Multiplying
replied on: 8/26/2002 6:33:29 PM We now know that 75^2 = 5625. To find the next square, add on twice 75 plus 1. 5625 + 151 = 5776 = 76^2 Fact: Consecutive squares differ by consecutive odd numbers. We just saw that 75^2 and 76^2 differ by 151. Hence, 76^2 and 77^2 differ by 153, 77^2 and 78^2 differ by 155, and so on. We can make a list of consecutive squares by simple addition. That is, 75^2 = 5625 5625 + 151 (twice 75 + 1) = 5776 = 76^2 5776 + 153 = 5929 = 77^2 5929 + 155 = 6084 = 78^2, etc. Proof: Consider two consecutive squares, N^2 and (N+1)^2 We can see that (N^2) and (N^2 + 2N + 1) differ by 2N + 1. |
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Soroban
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Re: Re: Shortcuts in Multiplying
replied on: 8/26/2002 11:36:32 PM Multiplying numbers that differ by 2 Example: 16 x 18 Square the "middle number" (17) and subtract 1. 16 x 18 = 17^2 - 1 = 289 - 1 = 288 Example: 34 x 36 35^2 - 1 = 1225 - 1 = 1224 |
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Soroban
Quote | Reply | This message was updated on 8/27/2002 1:33:31 AM by Soroban |
Re: Re: Shortcuts in Multiplying
replied on: 8/26/2002 11:37:21 PM Example: 16 x 18 Square the "middle number" (17) and subtract 1. 16 x 18 = 17^2 - 1 = 289 - 1 = 288 Example: 34 x 36 34 x 36 = 35^2 - 1 = 1225 - 1 = 1224 Proof: The two numbers are of the form a - 1 and a + 1. And, of course, their product is a^2 - 1. This can be extended to "Numbers differing by 4", etc. Since (a - 2)(a + 2) = a^2 - 4, then 23 x 27 = 25^2 - 4 = 621 |
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Soroban
Quote | Reply | This message was updated on 8/27/2002 7:44:00 PM by Soroban |
Shortcuts: Cube Roots
replied on: 8/27/2002 1:46:09 AM Have someone cube a two-digit number and give you the answer. You can immediately name the cube root. This requires a small investment of effort, but I think it's worth it. First, we must memorize the cubes of the digits 1 through 9. 1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64 5^3 = 125 6^3 = 216 7^3 = 343 8^3 = 512 9^3 = 729 We also note the last digits of the cubes (in bold face). These are easily memorized. Some of them are "self-enders": 1,4,5,6,9 (The 1 and 9 on the ends, and the 4,5,6 in the middle.) The rest are paired with a sum of 10: 2-cubed ends in 8, 8-cubed ends in 2, 3-cubed ends in 7, 7-cubed ends in 3. We are now ready to perform some amazing mental magic! Example: 97,336 Disregard the last three digits and consider the rest (97). Find the largest cube in 97. (4^3 = 64) The tens-digit is 4. Examine the last digit (6). It's a "self-ender"; the units-digit is 6. The cube root of 97,336 is 46. Example: 300,763 Disregard the last three digits and consider the 300. The largest cube is 300 is 6^3 = 216. The tens-digit is 6. The last digit is 3. It's a "sum-of-10". The units-digit is 7. The cube root of 300,763 is 67. |
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Soroban
Quote | Reply | This message was updated on 8/27/2002 10:14:45 PM by Soroban |
Re: Re: Shortcuts in Multiplying
replied on: 8/27/2002 1:49:52 PM I tried to delete this post. Somehow it got posted while I was previewing it. Anyone know how to delete a post? |
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Soroban
Quote | Reply | This message was updated on 8/28/2002 9:17:28 AM by Soroban |
Shortcuts: Square Roots
replied on: 8/27/2002 1:58:55 PM Have someone square a two-digit number and give you the result. You instantly name its square root. This is similar to cube root extraction with an extra step. First, memorize the squares of the digits 1 to 9 and note their final digits. 1^2 = 1 2^2 = 4 3^2 = 9 4^2 = 16 5^2 = 25 6^2 = 36 7^2 = 49 8^2 = 63 9^2 = 81 The only "self-ender" is 5. The others appear in pairs (with a sum of 10): the squares of 1 and 9 end in 1, the squares of 2 and 8 end in 4, the squares of 3 and 7 end in 9, the squares of 4 and 6 end in 6. We are ready... Example: Find the square root of 2116 Disregard the last two digits. Find the largest square in what remains, 21. Remember this "remainder". 4^2 = 16. The tens-digit is 4. The last digit is 6. It could have come from 4^2 or 6^2. Multiply the tens-digit (4) by the next larger digit (5): 20 Compare the "remainder" to 20. The remainder 21 is greater than 20. Hence, we use the greater units-digit, 6. The square root of 2116 is 46. Example: 5329. Dropping the last two digits, the "remainder" is 53. The largest square in 53 is 7^2 = 49. The tens-digit is 7. The last digit is 9. It could have come from 3^2 or 7^2. Multiply the tens-digit (7) by the next number (8): 56 The remainder 53 is less than 56. Hence, we use the lesser units-digit, 3. The square root of 5329 is 73. |
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montecristo0612
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Re: Shortcuts in Multiplying
replied on: 9/11/2002 12:23:18 AM Wow! Lots of shortcuts. A big thanks for you! I really needed some of your shortcuts there to help me whenever I have encountered problems similar to that one. Hope you could post more!!! Math rules!!!
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BobG
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Shortcuts in Multiplying
replied on: 3/28/2004 7:48:35 PM A quick shortcut for 2 digit squares is to convert them to Roman numerals and plug into (x + y + z)^2 x is the tens y is either 5 or zero z ranges from -2 to +2 (okay, we modified the Roman numeral system to make III read as IIV) This turns into a 5 step process, of which, only the 4th step takes any thought. x^2 is easy. Square the significant digit and double the zeroes. 2yx is also easy. I changed the order because of the commutative property and because it is easier to think of this way. It's either 10 times your x value or zero. Add to the result of step 1. y^2 is either 25 or zero. Either one is as easy as the other. Add to the result from step 2. The fourth step is the only one that takes any thought. 2xz and 2yz are combined via the distributive property to make (x+y)*2*z. z is 0, 1, or 2 (plus or minus - just tack the sign on after getting the magnitude. This is the easiest order to think about this. For most people, it's mentally easier to multiply by 2 twice than by 4 once. Add to the result from step 3. (The addition or subtraction part of this step is the hardest part of this process). z^2 Either 0, 1, or 4. Add to the result from step 4. With a little practice (especially if you know some addition/subtraction shortcuts), this is quicker than reaching for your calculator. Quick enough that you can estimate the square roots of 4 digit numbers and refine them to AT LEAST the nearest whole number. With a little feel for x^2 curve, you can interpolate the square roots of 4 digit numbers to the nearest hundredth. |
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PythagorasisHOT
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Shortcuts in Multiplying
replied on: 11/21/2004 6:22:27 PM hey tranquilim, i like your avatar... is it pythy??? |
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