I think this article - "A Mathematician's Lament" by Paul Lockhart - is something everyone should read Not exactly an answer to your questions and not exactly aimed at math beginners, unfortunately, but it's a really engaging rant about this topic https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf …https://twitter.com/graciegcunning/status/1298804338727489536 …
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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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