My mom taught me how to do it as a kid, since she took math classes in an earlier generation when they assumed you'd have to do more things by pencil and paper I've kept it in my back pocket ever since as a party trick
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It's beyond me to explain it in words concisely but it's a long-division-like process based on the algebra for squaring a binomial - (a+b)^2 = a^2 + 2ab + b^2
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So it's like, the square root of 7 First I come up with the "biggest number I can square that fits into 7 without a remainder", just like long division So that's 2
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2 is now the "a" in a binomial expressed as (10a + b)^2, which expands to 100a^2 + 20ab + b^2 Now to continue to the next decimal place of the solution I need to know what b is
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I expand 7 to 7.00 and subtract 4 (this is me subtracting 100a^2 from the expansion) I am left with 3.00, which is 20ab + b^2, or b(20a + b) The correct value for b is one that leaves the smallest remainder from the number I actually have without going over, like long division
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So using the formula (2a + b)*b, I double the digit I just wrote down, 2 I get 4, I make the 4 into 4_ and try to guess a next digit (forty-what?) that I can multiply it by (41*1? 42*2?) so that it "fits into" 300
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The highest digit I can plug into 4_ * _ that "fits into" 300 is 6 -- 46*6 is 276, 47*7 is 329
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So the next digit of my expansion is 6 The first two digits of the square root of 7 are 2.6 Quick sanity check, 2.6^2 is 6.76, 2.7 is 7.29, so I'm on the right track Subtract 276 from 300 and I get 24 Bring down the next two decimal places of the operand, 2400
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Now I'm working on the number 2400 in order to get the third digit of the square root This all takes much more time to talk about than to actually do but you can see that like long division it's tedious but simple and mechanical
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It's the kind of thing you learn how to do if you want to impress people, like memorizing trivia
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You can see that knowing how to do this is kind of cool and gives you a certain degree of understanding of how numbers work, like the explanation (it's applying the formula we all learned in algebra for expanding a binomial squared to decimal places!) ks legitimately elegant
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At the same time, you can seriously just type "What is the square root of 7" into the search box of Google
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