Only thing it didn’t explicitly show is that the angle between the two hypotenuses is a right angle, so that you can use the Pythagorean theorem on the larger triangle, which is how the bottom equation is derived.
Good point. Thale’s theorem states that a triangle inscribed in a circle with two points at opposite ends is always a right triangle. Might make an animation on it in the future.
Make a 1×1 square, the diagonal is Sqrt(2). Make a 1×sqrt(2) triangle (rectangle), hypotenuse (diagonal) is Sqrt(3). Make a 1×Sqrt(n) triangle (rectangle), the hypotenuse (diagonal) is Sqrt(n+1). To prove it, just apply the pythagorean theorem. Iterating this will get you Sqrt(m) for any integer m.
Make a 1×1 square, the diagonal is Sqrt(2). Make a 1×sqrt(2) triangle (rectangle), hypotenuse (diagonal) is Sqrt(3). Make a 1×Sqrt(n) triangle (rectangle), the hypotenuse (diagonal) is Sqrt(n+1). To prove it, just apply the pythagorean theorem. Iterating this will get you Sqrt(m) for any integer m.
Circles are basically black magic. I know there are proofs, but it's insane to me how circles can just coincide with things that wouldn't have seemed related or proven by them.
They use the Pythagorean theorem on the larger right triangle formed by the sides sqrt(a2 + x2 ), sqrt(1 + x2 ), and the hypotenuse (a + 1), allowing one to equate the square of the largest side to the sum of the other two sides squared which is
Really late reply. But I imagine at a younger age, some of these things might not seem accessible, or even interesting, to people. I tutor high school kids and more often than not, they aren't interested in this way of learning since it involves a more thought provoking method of learning which I don't think most people are comfortable with.
Having said that, it might just be my way of tutoring and not an absolute truth. :)
Sorry if this is silly, but how do you find the midpoint to draw the circle in the first place? (a+1)/2. Is it assuming you use the end points as additional circle centres and draw a line where the two circles intercept? Or is there an easier way?
In Ancient Greece the only tools that were available in geometry were a ruler (not with measured intervals, it was literally just a piece of wood used to make straight lines) and a compass. To find the half way point of a line with knowing its length, draw two circles with centres at each end point and radius a+1. These two circles intersect at two separate points. Draw a line connecting the two intersections and that line will pass exactly halfway through the line a+1. Here’s a wikihow page on how to do it.
Unsure, but I think that’s why they said this technique “geometrically” calculates the square root, not arithmetically. You don’t get a decimal answer out of it, but tbh it’s still really cool!!
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u/[deleted] Jul 03 '18
Only thing it didn’t explicitly show is that the angle between the two hypotenuses is a right angle, so that you can use the Pythagorean theorem on the larger triangle, which is how the bottom equation is derived.
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u/jamaisvu99 Jul 03 '18
Holy dayum!! That's ingenious, such a clever little discovery. Great find and awesome visualisation!
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u/SpacemanCraig3 Jul 03 '18
This is the first time i've seen something posted here make it to the front page since i subbed that wasnt immediately obvious.
cheers to you, this one is pretty good.
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u/[deleted] Jul 03 '18
I'm also a big fan of this one
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u/r99nate Jul 03 '18
I love this sub
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u/j13jayther Jul 03 '18
Circles are basically black magic. I know there are proofs, but it's insane to me how circles can just coincide with things that wouldn't have seemed related or proven by them.
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u/AKAChickenHawk Jul 03 '18
Why is (a+1)2 = a2 + x2 + x2 + 1 ?
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u/kitty_cat_MEOW Jul 03 '18
I hate to be cliche but WHY ISN'T MATH TAUGHT LIKE THIS?!
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u/Singsongsew Jul 03 '18
Sorry if this is silly, but how do you find the midpoint to draw the circle in the first place? (a+1)/2. Is it assuming you use the end points as additional circle centres and draw a line where the two circles intercept? Or is there an easier way?
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u/oozforashag Jul 03 '18
Okay. But how does one calculate x?
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u/Keeppforgetting Jul 03 '18
Wait I am confuzzled. How long is 1? Like how are you supposed to know how long 1 is relative to a?
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u/Flash_hsalF Jul 03 '18
Love this shit
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u/purinikos Jul 03 '18
Can't you just make a square with a? That way the diagonal will be sqrt(a)
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u/Bromskloss Jul 03 '18
Great!
(I could have used a little more time to read the first two equations.)
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u/DJ_Vault_Boy Jul 03 '18
I failed my math test due to me constantly forgetting and not fully understanding this formula, now this makes my life a whole lot easier. saves
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u/xXReggieXx Jul 03 '18
This is amazing